Unit Quadratic Functions and Modeling 4
Unit Cluster (F.F.4, F.F.5, F.F.6) Unit Cluster (F.F.7, F.F.9) Interpret functions that arise in applications in terms of a contet Analyzing functions using different representations Cluster :.. Interpret key features; intercepts, intervals where increasing and decreasing, intervals where positive and negative, relative maimums and minimums, symmetry, end behavior, domain and range, periodicity.. Relate the domain of a function to its graph or a contet..3 Average rate of change over an interval: calculate, interpret, and estimate from a graph. Cluster :.. Graph functions from equations by hand and with technology showing key features (square roots, cube roots, piecewise-defined functions including step and functions, and absolute value)... Graph linear and quadratic functions and show intercepts, maima, and minima..3 Compare properties (key features) of functions each represented differently (table, graph, equation or description) VOCABULARY The domain is the set of all first coordinates when given a table or a set of ordered pairs. It is the set of all -coordinates of the points on the graph and is the set of all numbers for which an equation is defined. The domain it is written from the least value to the greatest value. The range is the set of all second coordinates when given a table or a set of ordered pairs. It is the set of all y-coordinates of the points on the graph. When modeling real world situations, the range is the set of all numbers that make sense in the problem. The range is written from the least value to the greatest value. Eample: Find the domain and range of f = + 3. Domain. Find any values for which the function is undefined. The square root function has real number solutions if the epression under the radicand is positive or zero. This means that + 0 therefore.. Write the domain in interval notation. The domain is [, ). 5
Range. Find all values for which the output eists. The square root function uses the principal square root which is a positive number or zero ( y 0 ). However, the function has been shifted down three units so the range is also shifted down three units y 3.. Write the range in interval notation. The range is [ 3, ) Eample: Find the domain and range of the function graphed to the right. Domain. List all the -values of the function graphed. If you were to flatten the function against the -ais you would see something like this: The function is defined for all the -values along the -ais.. Write the domain in interval notation. The domain is (, ). Range. List all the y-values of the function graphed. If you were to flatten the function against the y-ais you would see something like this: The function is defined for all the y-values greater than or equal to -.. Write the range in interval notation. The range is [, ). 6
Eample: The path of a ball thrown straight up can be modeled by the equation h t = 6t + 0t + 4 where t is the time in seconds that the ball is in the air and h is the height of the ball. What is the real world domain and range for the situation? Domain. Find all the values that would make sense for the situation. The domain represents the amount of time that the ball is in the air. At t = 0 the ball is thrown and enters the air shortly afterwards so the domain must be greater than zero. The ball will hit the ground at.45 seconds. Once it is on the ground it is no longer in the air so the domain must be less than.45 seconds. The ball is in the air for 0 < t <.45 seconds.. Write the domain in interval notation. The domain is (0,.45). Range. Find all the values that would make sense for the situation. The ball will not go lower than the ground so the height must be greater than zero. The ball will go no higher than its maimum height so the height must be less than or equal to 0.5 feet. The range will be 0 < h 0.5.. Write the range in interval notation. The range is (0,0.5]. Practice Eercises A: Find the domain and range.. f = 3 + 3. f = 3. 4. 5. Your cell phone plan charges a flat fee of $0 for up to000 tets and $0.0 per tet over 000. 6. The parking lot for a movie theater in the city has no charge for the first hour, but charges $.50 for each additional hour or part of an hour with a maimum charge of $7.50 for the night. 7
VOCABULARY The -intercept is where a graph crosses or touches the -ais. It is the ordered pair ( a,0). Where a is a real number. The y -intercept is where a graph crosses or touches the -ais. It is the ordered pair ( 0,b ). Where b is a real number. A relative maimum occurs when the y-value is greater than all of the y-values near it. A function may have more than one relative maimum value. A relative minimum occurs when the y-value is less than all of the y-values near it. A function may have more than one relative minimum value. Eample: =. Find the intercepts of the function f -intercept. Substitute y in for f(). y =. Substitute 0 in for y. 0 = 3. Solve for. = = 4. Write the intercept as an ordered pair.,0 y-intercept. Substitute 0 in for. y = (0). Solve for y. y = 0 y = 3. Write the intercept as an ordered pair. ( 0, ) 8
Eample: f = 3 5. Find the intercepts of the function -intercept. Use your graphing calculator to graph the function.. Use the Calculate Menu ( nd, Trace, Zero) to find the -intercepts. (Zero is another name for the -intercept) y-intercept The -intercepts are,0 3,0. and. To find the y-intercept, replace each with 0. y = 3 0 5 0. Solve the equation for y. y = 0 0 y = 3. Write the intercept as an ordered pair. ( 0, ) Eample: f = + 4 + 4. Find the maimum of To find the maimum use your graphing calculator to graph the function. Then use the Calculate Menu ( nd, Trace, Maimum). Enter a number that is to the left of the maimum, for eample 0, then push enter. Then enter a number that is to the right of the maimum, for eample 4, then push enter. You can guess the value of the maimum or just push enter again and the maimum will be calculated. The maimum is (, 8). Eample: = + 3. Find the minimum of f ( ) To find the maimum use your graphing calculator to graph the function. Then use the Calculate Menu ( nd, Trace, Minimum). Enter a number that is to the left of the minimum, for eample -3, then push enter. Then enter a number that is to the right of the minimum, for eample -, then push enter. You can guess the value of the minimum or just push enter again and the minimum will be calculated. The minimum is (-, -3). 9
Practice Eercises B Find the and y-intercepts for each function.. + 5y = 0 f = +. 4 7 f = 30 3. f = 4 3 4. f = + 3 8 5. f = 3 + 4 6. Find the relative maimums or minimums of each function. 7. f ( ) = + 3 8. f ( ) = 3 + 7 f = 4 6 8 9. 0. f ( ) = 5 4 f = 3 8 + 3. f = 8 + 4. VOCABULARY An interval is a set of numbers between two -values. An open interval is a set of numbers between two -values that doesn t include the two end values. Open intervals are written in the form (, ) or < <. A closed interval is a set of numbers between two -values that does include the two end values. Closed intervals are written in the form [, ]. or A function f is increasing when it is rising (or going up) from left to right and it is decreasing when it is falling (or going down) from left to right. A constant function is neither increasing nor decreasing; it has the same y-value for its entire domain. A function is positive when f ( ) > 0 or the y-coordinates are always positive. A function is negative when f ( ) < 0 or the y-coordinates are always negative. Eample: Find the intervals where the function a. increasing b. decreasing c. constant d. positive e. negative f = + 3 is: 30
Increasing/Decreasing/Constant. Find the maimums or minimums. The minimum is (-, -4).. Determine if the function is rising, falling, or constant between the maimums and minimums. 3. Write the intervals where the function is increasing, decreasing, or constant using interval notation. To the left of the minimum the function is falling or decreasing. To the right of the minimum the function is rising or increasing. The function is increasing on the interval (, ). The function is decreasing on the interval (, ). The function is never constant. Positive/Negative. Find all the -intercepts of the function. The -intercepts are at (-3, 0) and (, 0).. Determine if the function has positive or negative y-values on the intervals between each -intercept by testing a point on the interval. 3. Write the intervals where the function is positive or negative using interval notation. < 3 3 < < > = 4 f ( 4) = 5 Positive = 0 f (0) = 3 Negative = f () = 5 Positive The function is positive on the intervals (, 3) and (, ). The function is negative on the interval ( 3,). Eample: Find the intervals where the function 3 a. increasing b. decreasing c. constant d. positive e. negative f = + is: Increasing/Decreasing/Constant. Find the maimums or minimums. There are no maimums or minimums.. Determine if the function is rising, falling, or constant on its entire domain. 3. Write the intervals where the function is increasing, decreasing, or constant using interval notation. The function is rising from left to right so it is increasing on its entire domain. The function is increasing on the interval,. The function is never decreasing nor is it constant. 3
Positive/Negative. Find all the -intercepts of the function. The -intercept is (-, 0).. Determine if the function has positive or negative y-values on the intervals between each -intercept by testing a point on the interval. 3. Write the intervals where the function is positive or negative using interval notation. < > = = 0 f ( ) = f (0) 0.6 Negative Positive The function is positive on the interval (, ). The function is negative on the interval (, ). Eample: Find the intervals where the function f a. increasing b. decreasing c. constant d. positive e. negative + + 3, = is: 3, > Increasing/Decreasing/Constant. Find the maimums or minimums and any breaks in the domain.. Determine if the function is rising, falling, or constant between each maimum or minimum and each break in the graph. 3. Write the intervals where the function is increasing, decreasing, or constant using interval notation. There is a maimum at (-, 0) and a break in the domain at = -. The function is rising (increasing) to the left of the maimum. It is falling (decreasing) to the right of the maimum. It is constant to the right of = -. The function is increasing on the interval,. It is decreasing on the interval (, ). It is constant on the interval (, ). 3
Positive/Negative. Find all the -intercepts of the function and any places where there is a break in the domain.. Determine if the function has positive or negative y-values on the intervals between each -intercept by testing a point on the interval. 3. Write the intervals where the function is positive or negative using interval notation. The -intercept is (-5, 0). There is a break in the domain at = -. < 5 5 < < > = 6 f ( 6) = Negative = 3 f ( 3) = Positive = 0 f (0) = 3 Positive The function is positive on the intervals ( 5, ) and (, ). The function is negative on the interval (, 5). Practice Eercises C Find the intervals where the function is: a. increasing b. decreasing c. constant d. positive e. negative f = + 3. f = + 3. 3. f = 3 f = 3 4. 3 5. f = 4 + f 6., < 0 =, > 0 33
VOCABULARY GRAPHICALLY ALGEBRAICALLY A function is symmetric with respect to the y-ais if,, y on the for every point graph, the point (, y) is also on the graph. In other words, if you substitute in for every you end up with the original function. When looking at the graph, you could fold the graph along the y-ais and both sides are the same. f = + 5 f ( ) = + 5 f = f ( ) = + 5 A function is symmetric with respect to the origin, y on if, for every point the graph, the point (, y) is also on the graph. In other words, if you substitute in for every you end up with the opposite of the original function. When looking at the graph, there is a mirror image in Quadrants & 3 or Quadrants & 4. f = 8 3 f ( ) = 8 f ( ) = f = 8 3 3 An equation with no symmetry. If you substitute in for every you end up with something that is neither the original function nor its opposite. When looking at the graph, you could not fold the graph along the y-ais and have both sides are the same or it doesn t reflect a mirror image. Eample: f = + f ( ) = ( ) + ( ) f = f f 34
Determine what kind of symmetry, if any, f = has. Test for y-ais Symmetry Replace with and see if the result is the same as the original equation. f = f ( ) = This is not the same as the original equation. Test for Origin Symmetry Replace with and see if the result is the opposite of the original equation. f = f ( ) = This is not the opposite of the original equation. The function f = has no symmetry. Graph Eample: f = Determine what kind of symmetry, if any, Test for y-ais Symmetry Replace with and see if the result is the same as the original equation. f = ( ) f ( ) = This is equal to the original equation. The function Test for Origin Symmetry Replace with and see if the result is the opposite of the original equation. f = ( ) f ( ) = This is not the opposite of the original equation. f = has y-ais symmetry. Graph Eample: Determine what kind of symmetry, if any, the function graphed at the right has. Pick a point (, y) Test for y-ais Symmetry, y on the graph and see if is also on the graph. The point (-, ) is on the graph but the point (, ) is not. The function does not have y-ais symmetry. The function graphed has origin symmetry. Pick a point (, y) Test for origin symmetry, y on the graph and see if is also on the graph. The point (-, ) is on the graph and the point (, -) is also on the graph. The function has origin symmetry. 35
VOCABULARY End behavior describes what is happening to the y-values of a graph when goes to the far right. ( + ) or goes the far left End behavior is written in the following format: Right End Behavior: Left End Behavior: lim f = c lim f = c Eample: Find the end behavior of f 4 3 = +. As gets larger the function is getting more and more negative. Therefore, the right end behavior is lim f =. As gets smaller the function is getting more and more positive. Therefore the left end behavior is lim f =. Eample: f = 3. Find the end behavior of As gets larger the function is getting more and more negative. Therefore, the right end behavior is lim f =. As gets smaller the function is getting more and more negative. Therefore the left end behavior is lim f =. Eample: Find the end behavior of f =. As gets larger the function is getting more and more positive. Therefore, the right end behavior is lim f =. The domain is restricted to numbers greater than or equal to, therefore this graph has no left end behavior. 36
Practice Eercises D Graph each function below and find the: a. Domain and Range b. Intercepts, if any c. Determine whether the function has any symmetry. d. List the intervals where the function is increasing, decreasing, or constant. e. List the intervals where the function is positive or negative. f. Find all the relative maimums and minimums. g. Find end behavior. f 5 = 4. f = 3 3. f = 3 + 5. f = + 5 3. f = + 4 6. 3, f = +, < 3 VOCABULARY Periodicity refers to a function with a repeating pattern. The period of this function is 6 horizontal units. Meaning the pattern will repeat itself every 6 horizontal units. 37
You Decide Mr. Astro s physics class created rockets for an end of the year competition. There were three groups who constructed rockets. On launch day the following information was presented for review to determine a winner. Group A estimated that their rocket was easily modeled by the equation: y = 6 + 76 + 3. Group B presented the following graph of the height of their rocket, in feet, over time. height (feet) time (seconds) Group C recorded their height in the table below. Time (seconds) 0 4 6 8 0 Height (feet) 3 56 38 377 46 0 Who should be the winner of the competition? Use mathematical reasons to support your conclusion. 38
Unit Cluster (F.IF.7b) Graphing Square Root, Cube Root, and Piecewise-Defined Functions, Including Step Functions and Absolute Value Functions Cluster : Analyzing functions using different representations..b Graph functions from equations by hand and with technology showing key features (square roots, cube roots, piecewise-defined functions including step functions, and absolute value) VOCABULARY There are several types of functions (linear, eponential, quadratic, absolute value, etc.). Each of these could be considered a family with unique characteristics that are shared among the members. The parent function is the basic function that is used to create more complicated functions. Square Root Function Parent Function / f = = Domain: [ 0, ) Range: [ 0, ) Intercepts: -intercept Key Features 0,0, y-intercept ( 0,0 ) Intervals of Increasing/Decreasing: increasing ( 0, ) Intervals where Positive/Negative: ( 0, ) Relative maimums/minimums: minimum at( 0,0 ) Symmetries: none End Behavior: right end behavior lim = ; left end behavior lim = 0 + 0 39
Cube Root Function Parent Function f = 3 = /3 Key Features Domain: (, ) Range: (, ) Intercepts: -intercept ( 0,0 ), y-intercept ( 0,0 ) Intervals of Increasing/Decreasing: increasing (, ) Intervals where Positive/Negative: positive( 0, ), negative (,0) Relative maimums/minimums: none Symmetries: origin End Behavior: right end behavior lim behavior lim 3 = 3 = ; left end Absolute Value Function Parent Function f = Key Features Domain: (, ) Range: [ 0, ) Intercepts: -intercept ( 0,0 ), y-intercept ( 0,0 ) Intervals of Increasing/Decreasing: increasing ( 0, ), decreasing (,0) Intervals where Positive/Negative: positive,0 0, Relative maimums/minimums: minimum at ( 0,0 ) Symmetries: y-ais symmetry End Behavior: right end behavior lim = ; left end behavior lim = Piecewise-Defined Functions A piecewise-defined function is a function that consists of pieces of two or more functions. For +, < eample f =, 0 is a piecewise-defined function. It has a piece of the + 5, > 0 40
function f = + but only the piece where <. It also contains the function f ( ) =, but only where 0 > 0.. Finally, it contains the function f 5 = + but only where Piecewise-Defined Function +, < f =, 0 + 5, > 0 Key Features Domain: (, ) Range: (,5] Intercepts: -intercept (.5, 0 ), y-intercept ( 0, ) Intervals of Increasing/Decreasing: increasing (, ), decreasing ( 5, ) Intervals where Positive/Negative: positive,0 and 0,.5, and.5, ( ), negative Relative maimums/minimums: none Symmetries: none lim + 5 = ; left End Behavior: right end behavior end behavior lim ( ) + = Step Functions are piecewise-defined functions made up of constant functions. It is called a step function because the graph resembles a staircase. Step Function f = int Key Features Domain: (, ) Range: { y y is an integer} Intercepts: -intercept = [ 0, ) and y = 0, y-intercept ( 0,0 ) Intervals of Increasing/Decreasing: neither increasing nor decreasing Intervals where Positive/Negative: positive(, ), negative (,0) Relative maimums/minimums: none Symmetries: none End Behavior: right end behavior lim int behavior lim int = = ; left end 4
Unit Cluster (F.IF.6) and Cluster 5(F.LE.3) Quadratic Functions and Modeling Cluster : Interpret Functions that Arise in Applications in Terms of a Contet..3 Average rate of change over an interval: calculate, interpret, and estimate from a graph. Cluster 5: Constructing and comparing linear, quadratic, and eponential models; solve problems.5. Eponential functions will eventually outgrow all other functions VOCABULARY The average rate of change of a function over an interval is the ratio of the difference (change) in y over the difference (change) in. y y y average rate of change = = Eample: Find the average rate of change for f = 3 + on the interval [0, ]. First, find the value of the function at each end point of the interval. f 0 = (0) 3(0) + f = () 3() + f ( 0) = 0 0 + f ( 0) = f = 4 6 + f = 8 6 + f = 3 ( 0,) (,3) Net, find the slope between the two points (0, ) and (, 3). 3 m = = = 0 The average rate of change of f = 3 + on the interval [0, ] is. 4
Eample: The per capita consumption of ready-to-eat and ready-to-cook breakfast cereal is shown below. Find the average rate of change from 99 to 995 and interpret its meaning. Years since 990 0 3 4 5 6 7 8 9 Cereal Consumption 5.4 6. 6.6 7.3 7.4 7. 6.6 6.3 5.6 5.5 (pounds) The year 99 is two years since 990 and 995 is 5 years since 990, therefore the interval is [, 5]. Find the slope between the two points (, 6.6) and (5, 7.). 7. 6.6 0.5 m = = = 0.6 5 3 The average rate of change from 99 to 995 is 0.6 pounds per year. This means that each household increased their cereal consumption an average of 0.6 pounds each year from 99 to 995. Eample: Joe is visiting the Eiffel Tower in Paris. He accidentally drops his camera. The camera s height is graphed. Use the graph to estimate the average rate of change of the camera from 4 to 7 seconds and interpret its meaning. At 4 seconds the height of the camera is approimately 650 feet. At 7 seconds the height of the camera is approimately 00 feet. Find the slope between the points (4, 650) and (7, 00). Height in feet 00 650 550 m = = = 83.3 7 4 3 Time in seconds The negative indicates that the camera is falling. The camera is picking up speed as it is falling. This means that for each second the camera is falling from 4 to 7 seconds, it increases in speed an average of 83.3 feet per second from 4 to 7 seconds. 43
Practice Eercises A Find the average rate of change for each function on the specified interval. f = 3 + 5 on [-, 3]. f = + 4 on [-4, -] 3. f = 4 + + 9 on [-3, 0]. f = 6 on [-, 0] 4. Find the average rate of change on the specified interval and interpret its meaning. 5. Many of the elderly are placed in nursing care facilities. The cost of these has risen significantly since 960. Use the table below find the average rate of change from 000 to 00. Years since 960 Nursing Care Cost (billions of $) 0 0 4 0 8 30 53 40 96 50 57 7. The net sales of a company are shown in the graph below. Estimate the average rate of change for 007 to 009. 6. The height of an object thrown straight up is shown in the table below. Find the average rate of change from to seconds. Time (seconds) Height (feet) 0 40 6 5 3 0 4 36 8. The graph below shows fuel consumption in billions of gallons for vans, pickups and SUVs. Estimate the average rate of change for 005 to 0. Net Sales in millions of $ Fuel Consumption Years since 999 Years since 980 44
Practice Eercises B Complete the tables. f = g = h( ) = - - - - - - 0 0 0 3 3 3 4 4 4 5 5 5 Practice Eercises C Find the average rate of change for functions f, g,and h for the specified intervals. Determine which of the three functions is increasing the fastest.. [0, ]. [3, 5] 3. [-, 5] 4. [0, 3] 5. [-, 0] 6. [0, 5] 45
Practice Eercises D. Graph the following functions on the same coordinate plane. a. 3 k = b. p = 3 5 c. = 3 7 r. Find the average rate of change for functions k, p, and r for the specified intervals. Determine which of the three functions is increasing the fastest. a. [-4, ] b. [3, 5] c. [0, 0] You Decide Use eercises C and D to help you answer the following questions.. For each eercise, determine which function has the greatest average rate of change on the interval [ 0, )?. In general, what type of function will increase faster? Eplain your reasoning. 46
FACTORING (To be used before F.IF.8) VOCABULARY Factoring is the reverse of multiplication. It means to write an equivalent epression that is a product. Each of the items that are multiplied together in a product is a factor. An epression is said to be factored completely when all of the factors are prime polynomials, that is they cannot be factored any further. The greatest common factor is the largest epression that all the terms have in common. FACTOR OUT A COMMON TERM Eample: + 6 + 8 What is the largest factor that evenly divides,6, and 8? : 6 : 3 8 : The common numbers are and. Multiply them and the product is the greatest common factor. 6 8 + + ( + + ) 3 4 Divide each term by the greatest common factor. Rewrite with the common term on the outside of the parenthesis and the simplified terms inside the parenthesis. Eample: 4 3 8w 3w + 5w What is the largest factor that evenly divides 4 3 8 w,3 w, and 5w? 4 8 w : w w w w 3 3 w : w w w 3 5 w : w w 5 8w 3w 5w + w w w 4 3 ( 8 3 + 5) w w w The common numbers are w and w. Multiply them and the product is the greatest common factor. Divide each term by the greatest common factor. Rewrite with the common term on the outside of the parenthesis and the simplified terms inside the parenthesis. 47
Eample: 3 9 3 + 5 z z z z z z z 3 9 : 3 3 3 : 3 z z z 5 z : 3 5 z 3 9 3 5 z z + z 3z 3z 3z ( + ) 3z 3z z 5 What is the largest factor that evenly divides 3 9 z, z, and 5z? The common numbers are 3 and z. Multiply them and the product is the greatest common factor. Divide each term by the greatest common factor. Rewrite with the common term on the outside of the parenthesis and the simplified terms inside the parenthesis. Practice Eercises A Factor out the greatest common factor.. 4 6. 5 + 0 + 5 5 4 3 3. 8 3 + 6 4 3 4. 3 3 5. 7 + 36 8 6. 4 + 49 FACTOR A TRINOMIAL WITH A LEADING COEFFICIENT OF When factoring a trinomial of the form a + b + c,where a =, and b and c are integers, find factors of c that add to equal b. If p and q are the factors, the factored form looks like ( + p)( + q). Eample: + 5 + 6 Factors of 6 Sum (adds to be) 6 7 3 5 - -6-7 - -3-5 ( )( 3) 48 Find factors of 6 that add to be 5. The factors are and 3. + + This is the factored form.
Another way to look at factoring is with an area model like the one pictured below. The rectangular area represents the trinomial + 5 + 6. The width across the top is + + + or + 3 and the length down the side is + + or +. To obtain the area of the rectangle, you would multiply the + + 3. length times the width or Notice that this result is the same as when we found factors of the constant term that added to the coefficient of the term. When factoring a trinomial of the form a b + c,where a =, and b and c are integers, find factors of c that add to equal b. If p and q are the factors, the factored form looks like ( p)( q). Eample: 5 + 6 Factors of 6 Sum (adds to be) 6 7 3 5 - -6-7 - -3-5 ( )( 3) Find factors of 6 that add to be -5. The factors are - and -3. This is the factored form. This eample can also be modeled with an area model as the picture below demonstrates. The rectangular area represents the trinomial 5 + 6. The width across the top is or 3 and the length down the side is or. To obtain the area of the rectangle, you would multiply the 3. length times the width or Notice that this result is the same as when we found factors of the constant term that added to the coefficient of the term. 49
When factoring a trinomial of the form a + b c,where a =, and b and c are integers, find factors of c that add to equal b. If p and q are the factors, the factored form looks like ( + p)( q). Eample: + 5 6 Factors of 6 Sum (adds to be) -6-5 -3 - - 6 5-3 ( )( 6) Find factors of -6 that add to be 5. The factors are - and 6. + This is the factored form. This eample can also be modeled with an area model as the picture below demonstrates. The rectangular area represents the trinomial + 5 6. The width across the top is + + + + + + or + 6 and the length down the side is. To obtain the area of + 6. Notice the rectangle, you would multiply the length times the width or that this result is the same as when we found factors of the constant term that added to the coefficient of the term. When factoring a trinomial of the form a b c,where a =, and b and c are integers, find factors of c that add to equal b. If p and q are the factors, the factored form looks like ( + p)( q). 50
Eample: 5 6 Factors of 6 Sum (adds to be) -6-5 -3 - - 6 5-3 ( )( 6) Find factors of -6 that add to be -5. The factors are and -6. + This is the factored form. This eample can also be modeled with an area model as the picture below demonstrates. The rectangular area represents the trinomial 5 6. The width across the top is or 6 and the length down the side is +. To obtain the area of + 6. Notice the rectangle, you would multiply the length times the width or that this result is the same as when we found factors of the constant term that added to the coefficient of the term. Practice Eercises B Factor each epression.. 4. + 4 3. 4. 9 + 8 5. 7 6. 5 36 7. + 5y + 4y 8. + 3y + y 9. 7y + 7y 5
VOCABULARY A perfect square is a number that can be epressed as the product of two equal integers. For eample: 00 is a perfect square because 0 0 = 00 and is a perfect square because =. FACTOR USING THE DIFFERENCE OF TWO SQUARES When something is in the form a b, where a and b are perfect square epressions, the a b a + b. factored form looks like Eample: 49 = and 49 are both perfect squares and you are finding the difference between them, so you can use the difference of two squares to factor. 49 = 7 7 Therefore: a = and b = 7 7 + 7 This is the factored form. Eample: 5 36y 5 = 5 5 36 = 6 6 y y y ( 5 6y)( 5 6y) 5 and 36y are both perfect squares and you are finding the difference between them, so you can use the difference of two squares to factor. Therefore: a = 5 and b = 6y + This is the factored form. 5
Practice Eercises C Factor each epression.. 49 5 3. 9 4 5. 36. 64y 4. 6 8y 6. 00 64y FACTOR BY GROUPING When factoring a trinomial of the form a + b + c,where a, b, and c are integers, you will need to use the technique of factoring by grouping. Eample: 6 5 (6)(5) = 90 Factors of 90 90 45 3 30 5 8 6 5 9 0 Multiply the leading coefficient and the constant. Choose the combination that will either give the sum or difference needed to result in the coefficient of the term. In this case the difference should be -, so 9 and -0 will give you the desired result. Or in other words, when you combine 9 and 0 you will end up with. 6 + 9 0 5 ( 6 + 9 ) + ( 0 5 ) ( + ) ( + ) 3 3 5 3 ( + 3) ( + 3)( 3 5) Rewrite the equation using the combination in place of the middle term. Group the first two terms and the last two terms together in order to factor. Factor the greatest common factor out of each group. Write down what is in the parenthesis (they should be identical). This is one of the factors. Add the left-overs to obtain the second factor. 53
Eample: + 7 0 ()(0) = 0 Factors of 90 0 60 3 40 4 30 5 4 6 0 8 5 0 8 + 5 0 ( 8 ) + ( 5 0 ) ( ) + ( ) 4 3 5 3 ( 3 ) ( 3 )( 4 + 5) Multiply the leading coefficient and the constant. Choose the combination that will either give the sum or difference needed to result in the coefficient of the term. In this case the difference should be 7, so -8 and 5 will give you the desired result. Or in other words, when you combine 8 and 5 you will end up with 7. Rewrite the equation using the combination in place of the middle term. Group the first two terms and the last two terms together in order to factor. Factor the greatest common factor out of each group. Write down what is in the parenthesis (they should be identical). This is one of the factors. Add the left-overs to obtain the second factor. Eample: 4 5 (4)(5) = 00 Factors of 00 00 50 4 5 5 0 0 0 Multiply the leading coefficient and the constant. Choose the combination that will either give the sum or difference needed to result in the coefficient of the term. In this case the difference should be 0, so -0 and 0 will give you the desired result. Or in other words, when you combine 0 and 0 you will end up with 0. 54
4 0 + 0 5 ( 4 0 ) + ( 0 5 ) ( ) + ( ) 5 5 5 ( 5) ( 5)( + 5) Rewrite the equation using the combination in place of the middle term. Group the first two terms and the last two terms together in order to factor. Factor the greatest common factor out of each group. Write down what is in the parenthesis (they should be identical). This is one of the factors. Add the left-overs to obtain the second factor. Practice Eercises D Factor the epression.. + 3 + 6. 4 + 3 3. 3 + 8 4. 6 5. + 4 + 6. 3 6 + 3 7. 0 6 8. 6 7 0 9. + 7 + 6 FACTORING GUIDELINES #: Always look for a greatest common factor. Then factor it out if there is one. #: Count the number of terms. If there are two terms, determine if you can use the difference of two squares. If you can, factor. If not, proceed to #3. #3: If there are three terms, check the leading coefficient. If it is, then find factors of the constant term that add to the coefficient of the -term. If not, proceed to #4. #4: If the leading coefficient is not, factor by grouping. Mied practices E Factor the epression.. 50. + 6y 3y 3. 3 5 4. 5 + 0 + 5 5. 5 64y 6. 3 7 7. y + 4 4 y 8. + 3 + 4 9. 4 + + 9 0. 6. 9 + 4. 8 + 3 55
Unit Cluster (F.IF.8), Unit 3 Cluster (A.SSE.a) and Unit 3 Cluster (A.SSE.3a,b) Forms of Quadratic Functions Cluster : Analyzing functions using different representations.. Writing functions in different but equivalent forms (quadratics: standard, verte, factored) using the processes of factoring or completing the square to reveal and eplain different properties of functions. Interpret these in terms of a contet. Cluster : Interpret the structure of epressions 3..a Interpret parts of an epression, such as terms, factors, and coefficients Cluster : Writing epressions in equivalent forms and solving 3.. Choose an appropriate from of an equation to solve problems (factor to find zeros, complete the square to find maimums and minimums VOCABULARY Forms of Quadratic Functions Standard Form: f = a + b + c, where a 0. Eample: f = + 4 6 3 Verte Form: f = a( h) + k, where a 0. Eample: f = ( + 3) + 5 Factored Form: f = a( p)( q ), where a 0. Eample: f = ( 4)( + 7) A zero of a function is a value of the input that makes the output f ( ) equal zero. The zeros of a function are also known as roots, -intercepts, and solutions of a b c + + = 0. The Zero Product Property states that if the product of two quantities equals zero, at least one of the quantities equals zero. If ab = 0 then a = 0 or b = 0. Finding Zeros (Intercepts) of a Quadratic Function When a function is in factored form, the Zero Product Property can be used to find the zeros of the function. If f = a ( p) then a( p) = 0 can be used to find the zeros of If 0 = a( p) then either a = 0 or ( p) = 0. Therefore, either = 0 or = p. f. 56
Eample: Find the zeros of f = ( + 7) f = ( + 7) ( + 7) = 0 Substitute zero in for f(). = 0 or + 7 = 0 Use the zero product property to set each factor equal to zero. = 0 or = 7 Solve each equation. The zeros are ( 0,0 ) and ( 7,0) Write them as ordered pairs. If f = ( p)( q) then ( p)( q) 0 f ( ). = can be used to find the zeros of If ( p)( q) = 0 then either ( p) = 0 or ( q) 0 Therefore, either = p or = q. Eample: Find the zeros of f = ( 5)( + 9) f = ( 5)( + 9) ( 5)( 9) 0 =. + = Substitute zero in for f(). 5 = 0 or + 9 = 0 = 5 or = 9 The zeros are ( 5,0 ) and ( 9,0) Use the zero product property to set each factor equal to zero. Solve each equation. Write them as ordered pairs. NOTE: If a quadratic function is given in standard form, factor first then apply the Zero Product Property. Eample: Find the zeros of f = + 4 + 4 = 0 ( )( ) 8 3 = 0 8 = 0 or 3 = 0 f = + 4 Substitute zero in for f(). Factor the trinomial. (See factoring lesson in Unit for etra help.) Use the zero product property to set each factor equal to zero. = 8 or = 3 Solve each equation. The zeros are ( 8,0 ) and ( 3,0 ) Write them as ordered pairs. 57
f = 4 4 5 Eample: Find the zeros of f = 4 4 5 4 4 5 0 = Substitute zero in for f(). ( )( ) 5 + 3 = 0 Factor the trinomial. 5 = 0 or + 3 = 0 = 5 = 3 5 or 3 = = 5 The zeros are,0 and 3,0 Use the zero product property to set each factor equal to zero. Solve each equation. Write them as ordered pairs. Practice Eercises A Find the zeros of each function. f = + 7. 4. f = ( )( 3) f = + 8 + 7. f = 9 5 0.. f = ( 6) f = 7 + 6 5. f = + 0 4 8. f = 5 4. 3. f = ( + 3)( 4) f = 6. f = 4 9. f = 3 + 7 + 0. COMPLETING THE SQUARE To complete the square of b ± b, add. In other words, divide the coefficient by two and square the result. + b + b + b + b b + + b + + 6 + 6 + 6 + + 6 + 3 ( + 3)( + 3) ( + 3) + 6 + 9 58
An area model can be used to represent the process of completing the square for the epression + 6 +. The goal is to arrange the pieces into a square. The pieces are divided evenly between the + 3 long. two sides so that each side is However, there is a large piece of the square that is missing. In order to complete the square you need to add 9 ones pieces. To complete the square of b + b, add. In other words, divide the coefficient by two and square the result. To complete the square of a + b, factor out the leading coefficient, a, giving b you a +. Now add a b, which is the square a of the coefficient of divided by two. + b + b + b + b b + + a a b + + b + b + + a b b a + + a a b b a + + a a b a + a + 5 + 5 + 5 + 5 + 5 + 4 5 5 + + 5 + 3 6 3 3 6 + 3 ( + ( ) ) 3( )( ) 3 3 + ( ) 6 6 + 3 3 59
Practice Eercises B For each epression complete the square.. + 0 +. 7 + 3. + 4. 4 6 + 5. + + 6. 5 + 0 + Finding Maimum/Minimum (the verte) Points of a Quadratic Function VOCABULARY Remember when a quadratic function is in verte form = ( ) + h, k is the verte of the parabola. The value of a determines whether the parabola opens up or down. f a h k the point The verte of a parabola that opens up, when a > 0, is the minimum point of a quadratic function. The verte of a parabola that opens up, when a < 0, is the maimum point of a quadratic function. Eample: Find the verte of f ( ) or minimum point. = + 3, then determine whether it is a maimum ( ) f = + 3 ( ) f = + 3 Rewrite the equation so it is in the general verte form f = a( h) + k. Verte: (-, -3) h = and k = 3 The verte is a minimum. The leading coefficient is, which makes a > 0 60
Eample: Find the verte of f ( ) or minimum point. ( ) f = 5 8 + 4 ( ) f = 5 8 + 4 = 5 8 + 4, then determine whether it is a maimum Verte: (8, 4) h = 8 and k = 4 The verte is a maimum. This equation is already in the general verte form f = a( h) + k. The leading coefficient is -5, which makes a < 0. Practice Eercises C Find the verte and determine whether it is a maimum or minimum point.. f ( ) = 4 5 3 4. f = ( + 6). f ( ) = + 3 + 7 5. f ( ) = 5 + 3 3. f = 6 + 5 6. f ( ) = 7 NOTE: If a quadratic function is given in standard form, complete the square to rewrite the equation in verte form. Eample: Find the verte of f = + + 7, then determine whether it is a maimum or minimum point. f = + + 7 f = + + + 7 f = + + + 7 ( + + ( 6) ) ( 6) f = + 7 Collect variable terms together inside parenthesis with constant term outside the parenthesis. b Complete the square by adding inside the parenthesis. Now subtract b outside the parenthesis to maintain equality. In other words you are really adding zero to the equation. 6
f = + + 36 + 7 36 Simplify ( ) f = + 6 9 Factor and combine like terms. Verte: (-6, -9) h = 6 and k = 9 The verte is a minimum. The leading coefficient is, which makes a > 0 Eample: Find the verte of f = 3 + 8, then determine whether it is a maimum or minimum point. f = 3 + 8 f = 3 + 8 + f = 3 + 6 + 6 6 f = 3 + 6 + 3 ( + ( 3) ) 3 ( 3) f = 3 + 6 f = 3 + 6 + 9 7 ( ) Collect variable terms together inside parenthesis with constant term outside the parenthesis. Factor out the leading coefficient. In this case 3. b Complete the square by adding inside the parenthesis. Notice that everything in the parenthesis is multiplied by 3 so we need to subtract b 3 outside the parenthesis to maintain equality. In other words you are really adding zero to the equation. Simplify f = 3 + 3 9 Factor and combine like terms. Verte: (-3, -9) h = 3 and k = 9 The verte is a minimum. The leading coefficient is 3, which makes a > 0 6
Eample: Find the verte of f = 4 8 + 3, then determine whether it is a maimum or minimum point. f = 4 8 + 3 f = 4 8 + + 3 f = 4 + + + 3 f 4 3 4 = + + + ( ) ( + ) 3 ( 4) f = 4 + + f = 4 + + + 3+ 4 Simplify ( ) Collect variable terms together inside parenthesis with constant term outside the parenthesis. Factor out the leading coefficient. In this case -4. b Complete the square by adding inside the parenthesis. Notice that everything in the parenthesis is multiplied by -4 so we need to subtract b 4 outside the parenthesis to maintain equality. In other words you are really adding zero to the equation. f = 4 + + 7 Factor and combine like terms. Verte: (-, 7) h = and k = 7 The verte is a maimum. The leading coefficient is -4, which makes a < 0 Practice Eercises D Find the verte of each equation by completing the square. Determine if the verte is a maimum or minimum.. f = + 0 0 3. f = 5 0 9 5. f = + 8 + 0. f = 4 + 4. f = + 6 + 6 6. f = 9 63
The ais of symmetry is the vertical line that divides a parabola in half. The zeros will always be the same distance from the ais of symmetry. The verte always lies on the ais of symmetry. When completing the square we end up with Eample: Ais of symmetry b f = a + + k a b f = a + k a f = a h + k k f = 3 + h = = = ( 3) 6 = f () = 3() + () = b Notice the -coordinate of the verte is. a The y-coordinate can be found by evaluating b the function at. a The point (, ) is the verte. Since 3 < 0, (, ) is the maimum point of the function. Therefore, another method for finding the verte (h, k) from a standard form equation is to use h = b a and b k = f a. Practice Eercises E Identify the verte of each function. Then tell if it is a maimum or minimum point.. f = 4 + 8 7 3. f = + 3. f = + 30 f = + 4 4. 64
YOU DECIDE A model rocket is launched from ground level. The function h( t) = 6t + 60t models the height h (measured in feet) of the rocket after time t (measured in seconds). Find the zeros and the verte of the function. Eplain what each means in contet of the problem. Practice Eercises F Solve. The height h(t), in feet, of a weeping willow firework display, t seconds after having been launched from an 80-ft high rooftop, is given by h( t) = 6t + 64t + 80. When will it reach its maimum height? What is its maimum height?. The value of some stock can be represented by V = 8 + 0, where is the number of months after January 0. What is the lowest value V() will reach, and when did that occur? 3. Suppose that a flare is launched upward with an initial velocity of 80 ft/sec from a height of h t = 6t + 80t + 4. How long 4 ft. Its height in feet, h(t), after t seconds is given by will it take the flare to reach the ground? 4. A company s profit can be modeled by the equation p = + 980 3000 where is the number of units sold. Find the maimum profit of the company. 5. The Rainbow Bridge Arch at Lake Powell is the world s highest natural arch. The height of an object that has been dropped from the top of the arch can be modeled by the equation h( t) = 6t + 56, where t is the time in seconds and h is the height in feet. How long does it take for the object to reach the ground? 6. The amount spent by U.S. companies for online advertising can be approimated by a( t) = t t + 8, where a(t) is in billions of dollars and t is the number of years after 00. In what year after 00 did U.S. companies spend the least amount of money? 65
Unit 6 Cluster 3 (G.GPE.): Parabolas as Conics Cluster 3: Translating between descriptions and equations for a conic section 6.3. Find the equation of a parabola given the focus and directri parallel to a coordinate ais. VOCABULARY A parabola is the set of all points,, P y, in a plane that are an equal distance from both a fied point, the focus, and a fied line, the directri. 66
Equation Direction Standard Form for the Equation of a Parabola Verte at (0, 0) Verte at (h, k) y = y 4 p k = 4 h p Opens upward if p > 0 Opens upward if p > 0 Opens downward if p < 0 Opens downward if p < 0 Focus (0, p) (h, k + p) Directri y = p y = k p Graph Eample : Use the Distance Formula to find the equation of a parabola with focus ( 0,3 ) and directri y = 3. PF = PD ( ) + ( y y ) = ( ) + ( y y ) 0 ( y 3) ( ) ( y 3) A point P(, y) on the graph of a parabola is the same distance from the focus F ( 0,3) and a point on the directri D (, 3). + = + + Substitute in known values. y y + ( 3) = + 3 Simplify. ( + ( y 3) ) = ( y + 3) + ( y 3) = ( y + 3) Square both sides of the equation and use the properties of eponents to simplify. 67
= ( y + 3) ( y 3) ( 6 9) ( 6 9) y y y y = + + + = y y = Solve for y. Eample: Use the Distance Formula to find the equation of a parabola with focus (-5, 3) and directri y = 9. PF = PD ( ) + ( y y ) = ( ) + ( y y ) ( ) 5 ( y 3) ( ) ( y 9) A point P(, y) on the graph of a parabola is the same distance from the focus F ( 5,3) and a point on the directri D(,9). + = + Substitute in known values. ( 5 ) ( y 3) ( y 9) + + = Simplify. ( 5 ) ( y 3) ( y 9 + + = ) + 5 + ( y 3) = ( y 9) ( + 5) = ( y 8y + 8) ( y 6y + 9) ( ) + 5 = ( y 9) ( y 3) + 5 = y + 7 ( + 5) = ( y 6) ( + 5) = y 6 Square both sides of the equation and use the properties of eponents to simplify. Combine the terms on one side of the equation and the y terms on the other side of the equation. Practice Eercises A Use the distance formula to find the equation of parabola with the given information.. focus ( 0, 5) directri y = 5 4. focus (,6 ) directri y = 8. focus ( 0,7 ) directri y = 7 5. focus ( 3, 4 ) directri y = 3. focus ( 0, 3) directri y = 6 6. focus ( 3,3 ) directri y = 7 68
Equation Direction Standard Form for the Equation of a Parabola Verte at (0, 0) Verte at (h, k) y h = ( y k ) 4 p 4 p Opens to the right if p > 0 Opens to the right if p > 0 Opens to the left if p < 0 Opens to the left if p < 0 Focus ( p,0) ( h + p, k ) Directri = p = h p y y D(-p, y) P(, y) D(h - p, y) P(, y) Graph (h, k) F(p, 0) (h, k) F(h + p, k) = - p = h - p Eample: Use the Distance Formula to find the equation of a parabola with focus (,0 ) and directri =. PF = PD ( ) + ( y y ) = ( ) + ( y y ) ( y 0) ( ) ( y y) A point (, y) on the graph of a parabola is the same distance from the focus (,0 ) and a point on the directri (, y). + = + + Substitute in known values. ( ) y ( ) + = + Simplify. ( ) y ( + = + ) + y = ( + ) Square both sides of the equation and use the properties of eponents to simplify. 69
8 y = + ( 4 4) ( 4 4) y y y = + + + = 8 = Solve for. Eample: Use the Distance Formula to find the equation of a parabola with focus ( 4,3 ) and directri = 6. PF = PD ( ) + ( y y ) = ( ) + ( y y ) ( 4) ( y 3) ( 6) ( y y) A point (, y) on the graph of a parabola is the same distance from the focus ( 4,3 ) and a point on the directri ( 6, y ). + = + Substitute in known values. ( 4) ( y 3) ( 6) + = Simplify. ( 4) ( y 3 ) ( 6 + = ) ( 4) + ( y 3) = ( 6) ( y 3) = ( 6) ( 4) ( y 3) = + 36 ( 8 + 6) ( y ) 3 = 4 + 0 ( y 3) = 4( 5) ( y 3) = 5 4 Square both sides of the equation and use the properties of eponents to simplify. Combine the terms on one side of the equation and the y terms on the other side of the equation. Practice Eercises B Use the distance formula to find the equation of parabola with the given information.. focus ( 4,0) directri = 4 4. focus (, 3) directri = 5. focus ( 5,0 ) directri = 5 5. focus (, 4) directri = 6 3. focus ( 3,0 ) directri = 3 6. focus (,) directri = 5 70
Practice Eercises C Determine the verte, focus, directri and the direction for each of the following parabolas.. ( y + ) = ( 3) 4. 6( y 3) = ( + ). ( + 4) = 6( y ) 5. ( y + 3) = ( ) 3. ( y ) = 4( + 5) 6. ( y 6) = 6( 4) You Decide A parabola has focus (-,) and directri y = -3. Determine whether or not the point (,) is part of the parabola. Justify your response. 7
Unit 6 Cluster 3 Honors (G.GPE.3) Deriving Equations of Ellipses and Hyperbolas Cluster 3: Translate between the geometric description and the equation for a conic section H.5. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. VOCABULARY An ellipse is the set of all points in a plane the sum of whose distances from two fied points (called foci), F and F, is constant. The midpoint of the segments connecting the foci is the center of the ellipse. P(, y) d F(-c, 0) d F(c, 0) An ellipse can be elongated horizontally or vertically. The line through the foci intersects the ellipse at its vertices. The segment whose endpoints are the vertices is called the major ais. The minor ais is a segment that is perpendicular to the major ais and its endpoints intersect the ellipse. 7
Deriving the Standard Equation of an Ellipse PF + PF = a + c + y 0 + c + y 0 = a ( + c) + ( y 0) = a ( c) + ( y 0) ( ( + c) + ( y 0) ) = ( a ( c) + ( y 0) ) ( + c) + ( y 0) = 4a 4a ( c) + ( y 0) + ( c) + ( y 0) + c + c + y = 4a 4a c + y 0 + c + c + y 4c = 4a 4a c + y 0 4c 4a = 4a c + y 0 4c 4a 4 ( 0) = a c + y ( 0) c a a c y + = + ( 0) a c = a c + y ( a c) = a ( c) + ( y 0) 4 a a c + c = a c + y 0 a a c + c = a c + c + y 4 a a c + c = a a c + a c + a y 4 a + c = a + a c + a y 4 a a c = a c + a y 4 a a c = a c + a y a b = b + a y Let The sum of the distance from a point P, y on the ellipse to each foci, c,0 and c,0, is equal to a. Use the distance formula and substitute in known values. Isolate one of the radicals. Square each side then simplify. Isolate the radical again. Square each side then simplify. Combine all the terms containing and y on one side. b = a c. 73
a b b + a y = a b a b y = + a b Divide by a b. Equation Major Ais Minor Ais Standard Form for the Equation of an Ellipse Centered at (0, 0) Horizontal Ellipse y + =, a > b a b Along the -ais Length: a Along the y-ais Length: b Vertical Ellipse y + =, a > b b a Along the y-ais Length: a Along the -ais Length: b Foci ( c,0 ) and ( c,0) ( 0, c) and ( 0, c) Vertices ( a,0 ) and ( a,0) ( 0, a) and ( 0, a) Pythagorean Relation a = b + c a = b + c Basic Graph Eample: Locate the vertices and foci for the ellipse 5 + 4y = 00. Graph the ellipse. 5 + 4y = 00 5 4y 00 + = 00 00 00 y + = 4 5 The standard equation of an ellipse is equal to. Divide each side of the equation by 00 and simplify. 74
y + = 4 5 a = 5 = 5 b = 4 = 5 = 4 + c = c = c Vertices: ( 0, 5 ) and ( 0,5) Foci: ( 0, ) and ( 0, ) Identify a and b. Remember a > b. Note: a and b are lengths therefore the positive square root will ALWAYS be used. Use a and b to find c. Remember a = b + c. The vertices are ( 0, a) and ( 0, a) foci are ( 0, c) and ( 0, c) ellipse is vertical. and the because the Begin graphing the ellipse by plotting the center which is at (0, 0). Then plot the 0, 5 0,5. vertices which are at ( ) and Use the length of b to plot the endpoints of the minor ais. b = so the endpoints are units to the left and right of the,0 and center (0, 0). They are at (,0 ). Connect your points with a curve. 75
Eample: Write an equation in standard form for an ellipse with foci located at (,0 ) and (,0) and vertices located at ( 6,0 ) and ( 6,0). a y b + = 6 = b + The ellipse is horizontal because the foci and vertices are along the -ais. Use the standard equation for a horizontal ellipse. a = 6 36 = b + 4 c = 3 = b Find b using a = b + c. y + = Substitute in known values. 36 3 Practice Eercises A Locate the vertices and foci of the ellipse, then graph.. y + =. 6 7 y + = 3. 5 y + = 7 36 4. 3 4y + = 5. 9 4y 36 + = 6. = 4y Write an equation in standard form for the ellipse that satisfies the given conditions. 7. Foci: ( 5,0 ) and ( 5,0) Vertices: ( 8,0 ) and ( 8,0) 8. Foci: ( 0, 4 ) and ( 0, 4) Vertices: ( 0, 7 ) and ( 0, 7) 9. Foci: ( 0, 3 ) and ( 0,3) Vertices: ( 0, 4 ) and ( 0, 4). Major ais endpoints: ( 0, ± 6) Minor ais length 8 0. Foci: ( 6,0 ) and ( 6,0) Vertices: ( 0,0 ) and ( 0,0). Endpoints of aes are ( ± 5,0) and ( 0, ± 4) 76
Ellipses Centered at (h, k) Standard Form for the Equation of an Ellipse Centered at (h, k) Equation ( h) ( y k ) ( h) ( y k ) a + = a > b b, b + = a > b a, Center ( h, k ) ( h, k ) Major Ais Minor Ais Parallel to the -ais Length: a Parallel to the y-ais Length: b Parallel to the y-ais Length: a Parallel to the -ais Length: b Foci ( h c, k ) and ( h + c, k ) ( h, k c) and ( h, k + c) Vertices ( h a, k ) and ( h + a, k ) ( h, k a) and ( h, k + a) Pythagorean Relation Eample: a = b + c a = b + c Locate the center, the vertices and the foci of the ellipse ( ) ( y ) the ellipse. ( ) ( y ) ( + ) ( y ) + 3 + 4 = 6 3 4 6 + = 6 6 6 ( + 3) ( y ) + = 6 4 ( + 3) ( y ) + = 6 4 a = 6 = 4 b = 4 = 6 = 4 + c = c = c 3 = c Center: ( 3,) ( 3 4, ) and ( 3 + 4, ) Vertices: ( 7, ) and (, ) Foci: ( 3 3, ) and ( 3 + 3, ) + 3 + 4 = 6. Graph The standard equation of an ellipse is equal to. Divide each side of the equation by 6 and simplify. Identify a and b. Remember Use a and b to find c. Remember a = b + c. a > b. h = 3 and k = The ellipse is horizontal, therefore the h a, k and h + a, k and vertices are the foci are ( h c, k ) and ( h c, k ) +. 77
Begin graphing the ellipse by plotting the 3,. Then plot the center of the ellipse vertices ( 7, ) and (, ). Use the length of b to plot the endpoints of the minor ais. b = so the endpoints are units above and below the center 3,. 3,0 3,4. ( ) They are at ( ) and Connect your points with a curve. Eample: Write an equation in standard form for an ellipse with foci at (, ) and (,5) vertices at (, ) and (, 7). ( h) ( y k ) b + = 4 = b + b 6 = + 4 = b a + + 7 = Center:, (,3) ( + ) ( y 3) and The ellipse is vertical because the foci and vertices are parallel to the y-ais. Use the standard equation for a horizontal ellipse. a = 7 ( ) c = 5 a = 8 a = 4 Find b using c = 4 c = a = b + c. The center is the midpoint of the vertices. + = Substitute in known values. 6 78
Practice Eercises B Locate the center, vertices and foci of the ellipse, then graph.. ( ) ( y ) + =. 9 4 ( 4) ( y + ) + = 3. 9 5 ( 3) ( y + ) + = 9 6 4. ( 3) + 9( y + ) = 8 5. 9( ) + 4( y + 3) = 36 6. ( ) ( y ) + 4 + 4 + 3 = 4 Write an equation in standard form for the ellipse that satisfies the given conditions. 7. Foci: (, 4 ) and ( 5, 4) Vertices: ( 0, 4 ) and ( 6, 4) 9. Foci: ( 4, ) and ( 6, ) Vertices: (, ) and ( 8, ). Vertices: ( 5, ) and ( 3, ) Minor ais length is 6. 8. Foci: ( 3, 6 ) and ( 3, ) Vertices: ( 3, 7 ) and ( 3,3) 0. Foci: (, 0 ) and (, 4) Vertices: (, ) and (, 5). Vertices: 0, and 6, Minor ais length is. 79
VOCABULARY A hyperbola is the set of all points in a plane whose distances from two fied points in the plane have a constant difference. The fied points are the foci of the hyperbola. The line through the foci intersects the hyperbola at its vertices. The segment connecting the vertices is called the transverse ais. The center of the hyperbola is the midpoint of the transverse ais. Hyperbolas have two oblique asymptotes that intersect at the center. Deriving the Standard Equation of a Hyperbola PF PF = ± a The difference of the distance from a point P, y on the hyperbola to each foci, c,0 and c,0, is equal to ± a. 80
c + y 0 + c + y 0 = ± a c + y + c + y = ± a c + y = ± a + + c + y ( ) c + y = ± a + + c + y 4 4 c + y = a ± a + c + y + + c + y 4 4 c + y + c y = a ± a + c + y c + c + c + c = 4a ± 4a + c + y 4c 4a = ± 4a + c + y 4c 4a 4 4c = 4a ± 4a + c + y = ± a + c + y c a = ± a + c + y c a = ± a + c + y 4 c + a c + a = a + c + y c + a c + a = a + c + c + y 4 c + a c + a = a + a c + a c + a y 4 c + a = a + a c + a y 4 c a a y = a c a 4 ( ) = ( ) c a a y a c a b a y = a b Let b a y a b = a b a b y = a b Use the distance formula and substitute in known values. Isolate one of the radicals. Square each side then simplify. Isolate the radical again. Square each side then simplify. Combine all the terms containing and y on one side. b = c a. Divide by a b. 8
Equation Transverse Ais Conjugate Ais Standard Form for the Equation of a Hyperbola Centered at (0, 0) Opens Left and Right y = a b -ais Length: a y-ais Length:b Opens Up and Down y = a b y-ais Length: a -ais Length:b Foci ( c,0 ) and ( c,0) ( 0, c) and ( 0, c) Vertices ( a,0 ) and ( a,0) ( 0, a) and ( 0, a) Pythagorean Relation Asymptotes c = a + b b y = ± a c = a + b a y = ± b Basic Graph Eample: Find the vertices, foci and asymptotes of the hyperbola hyperbola. 4 9y 36 =. Then graph the 4 9y = 36 4 9y 36 = 36 36 36 y = 9 4 The standard equation of an ellipse is equal to. Divide each side of the equation by 36 and simplify. 8
y = 9 4 a = 9 = 3 b = 4 = c c c = = 9 + 4 = 3 3 Vertices: ( 3,0 ) and ( 3,0) Foci: ( 3, 0 ) and ( 3, 0) Asymptotes: y = and 3 y = 3 Identify a and b. Use a and b to find c. Remember c = a + b. This hyperbola opens left and right so the a,0 a,0 and the foci vertices are ( ) and are ( c,0 ) and ( c,0). b The asymptotes are y = ±. a Begin graphing the hyperbola by plotting the center at (0, 0). Then plot the vertices 3,0 and 3,0. at Use the length of b to plot the endpoints of the conjugate ais. b = so the endpoints are units above and below the center 0, 0,. ( 0,0 ). They are at ( ) and Construct a rectangle using the points. 83
Draw the asymptotes by drawing a line that connects the diagonal corners of the rectangle and the center. Use the asymptotes to help you draw the hyperbola. The hyperbola will open left and right and pass through each verte. Eample: Write an equation in standard form for the hyperbola with foci ( 0, 3) and conjugate ais has length 4. 0,3 whose y a b b = 4 = b = 3 = a + = a + 9 4 5 = a The foci are along the y-ais so the hyperbola s branches open up and down. The conjugate ais is length 4. Use it to solve for b. Use b = and c = 3 to solve for Remember c = a + b. a. y = Substitute in known values. 5 4 84
Practice Eercises C Locate the center, vertices, foci and asymptotes of the hyperbola, then graph.. y =. 4 6 y = 3. 5 36 y = 9 4. 0y 5 = 00 5. 4y 6 = 64 6. 4y = 6 Write an equation in standard form for the hyperbola that satisfies the given conditions. 7. Foci: ( 0, ) and ( 0, ) Vertices: ( 0, ) and ( 0,) 9. Foci: ( 0, 7 ) and ( 0, 7) Vertices: ( 0, 5 ) and ( 0,5). Vertices: ( 4,0 ) and ( 4,0) Conjugate ais length is 0. 8. Foci: ( 5,0 ) and ( 5,0) Vertices: ( 3,0 ) and ( 3,0) 0. Foci: ( 0,0 ) and ( 0,0) Vertices: ( 6,0 ) and ( 6,0). Vertices: ( 0, 3 ) and ( 0,3) Conjugate ais length is 6. Equation Transverse Ais Conjugate Ais Standard Form for the Equation of a Hyperbola Centered at (h, k) Opens Left and Right ( h) ( y k ) = a b Parallel to -ais Length: a y-ais Length:b Opens Up and Down ( y k ) ( h) = a b Parallel to y-ais Length: a -ais Length:b Foci ( h c, k ) and ( h + c, k ) ( h, k c) and ( h, k + c) Vertices ( h a, k ) and ( h + a, k ) ( h, k a) and ( h, k + a) Pythagorean Relation c = a + b b a c = a + b Asymptotes y k = ± ( h) y k = ± ( h) a b 85
Eample: Find the center, vertices, foci and asymptotes of the hyperbola ( + ) ( y 5 ) Then graph the hyperbola. =. 9 49 ( + ) ( y 5) = 9 49 ( + ) ( y 5) = 9 49 a = 9 = 3 b = 49 = 7 c c c = = 9 + 49 = 58 58 Center: (,5) ( 3, 5 ) and ( + 3, 5 ) Vertices: ( 5,5 ) and (,5 ) Foci: ( 58,5 ) and ( + 58, 5) 7 = 3 + and 7 y 5 = ( + ) 3 Asymptotes: y 5 ( ) Identify a and b. Use a and b to find c = a + b. c. Remember that The hyperbola s branches open left and h a, k and right so the vertices are ( h + a, k ). The foci are ( h c, k ) ( h + c, k ). and Begin graphing the hyperbola by plotting the center at (-, 5). Then plot the 5,5 and,5. vertices at 86
Use the length of b to plot the endpoints of the conjugate ais. b = 7 so the endpoints are 7 units above and below the,5., and center ( ) They are at (,). Construct a rectangle using the points. Draw the asymptotes by drawing a line that connects the diagonal corners of the rectangle and the center. Use the asymptotes to help you draw the hyperbola. The hyperbola will open left and right and pass through each verte. 87
Eample: Write an equation in standard form for the hyperbola whose vertices are (, ) ( 8, ) and whose conjugate ais has length 8. and ( y k ) ( h) = a b + 8 +, = 3, a = 8 ( ) Center: a = 0 a = 5 b = 8 b = 4 ( 3) ( y + ) = 5 6 The foci are parallel to the -ais so the hyperbola s branches open left and right. The midpoint of the vertices is the center of the hyperbola. The vertices are at (, ) and ( 8, ). Use the distance between them to find a. The conjugate ais is length 8. Use it to solve for b. Substitute in known values. Practice Eercises D Locate the center, vertices, foci and asymptotes of the hyperbola, then graph.. ( y 5) ( 6) =. 5 6 + 5 y = 3. 4 36 ( + ) ( y 3) = 49 6 4. 4( y + ) ( + 6) = 6 5. ( y 6) 5( 4) = 00 6. ( ) ( y ) 7 + 4 4 + = 8 Write an equation in standard form for the hyperbola that satisfies the given conditions. 7. Foci: (,9 ) and (, ) Vertices: (,7 ) and (,3 ) 9. Foci: ( 8, 4 ) and ( 4, 4) Vertices: ( 7, 4 ) and ( 3, 4). Vertices: ( 3,6 ) and ( 3, ) Minor ais length is 8. 8. Foci: (, 5 ) and ( 8, 5) Vertices: ( 0, 5 ) and ( 6, 5) 0. Foci: ( 3,5 ) and ( 3, ) Vertices: ( 3, ) and ( 3, 7). Vertices: 7, and 3, Minor ais length is 6. 88
Unit Cluster 3 (F.BF.) Building Functions That Model Relationships Between Two Quantities Cluster 3: Building functions that model relationships between two quantities.3. Focus on quadratics and eponentials to write a function that describes a relationship between quantities ( nd difference for quadratics).3. Determine an eplicit epression or steps for calculation from contet..3. Combine functions using arithmetic operations. Vocabulary A function is a relation for which each input has eactly one output. In an ordered pair the first number is considered the input and the second number is considered the output. If any input has more than one output, then the relation is not a function. For eample the set of ordered pairs {(,), (3,5),(8,)} is a function because each input value has an output value. The set {(, ) (, 3), (6, 7)} does not represent a function because the input has two different outputs and 3. Linear Function- a function that can be written in the form y = m + b,where m and b are constants. The graph of a linear function is a line. A linear function can be epressed in two different ways: Linear notation: y = m + b Function notation: f = m + b f = + Linear functions can model arithmetic sequences, where the domain is the set of positive integers, because there is a common difference between each successive term. The common difference can also be called the first difference. Linear functions can model any pattern where the first difference is the same number., 3, 5, 7,... + + + st difference 89
Eponential Function- a function of the form f = ab where a and b are constants and a 0, b > 0, and b. Eponential functions are most easily recognized by the variable in the eponent. The values of f() are either increasing (eponential growth) if a > 0 and b > or decreasing (eponential decay) if a > 0 and 0 < b <. Eponential functions can model geometric sequences, where the domain is the set of positive integers, because each successive term is multiplied by the same number called the common ratio. Eponential functions can model any pattern where the net term is obtained by multiplying each successive term by the same number. f ( ) =, 3, 9, 7,... 3 3 3 common ratio Quadratic Function- a function that can be written in the form f = a + b + c where a 0. Quadratic functions are most easily recognized by the term. The graph is a parabola. A quadratic function can be formed by multiplying two linear functions. The quadratic function to the right can also be written as ( 3)( ) f = +. To determine if a pattern or a sequence can be modeled by a quadratic function, you have to look at the first and second difference. The second difference is the difference between the numbers in the first difference. If the first difference is not the same number but the second difference is, then the pattern or sequence can be modeled by a quadratic function. f = 5 3, 4, 9, 6,... +3 + +5 + +7 st difference nd difference 90
Eample: Determine if the pattern, 3, 9, 9, would be modeled by a linear function, an eponential function, or a quadratic function. Answer: Check the first difference to see if it is the same number each time. For this pattern, it is not the same, so it will not be modeled by a linear function. Check to see if each term is being multiplied by the same factor. For this pattern, it is not the same, so it will not be modeled by an eponential function. Check the second difference to see if it is the same number each time. For this pattern, it is the same, so the pattern can be modeled by a quadratic function., 3, 9, 9,... + +6 +0, 3, 9, 9,... 3 3., 3, 9, 9,... + +6 +0 Conclusion: The pattern can be modeled by a quadratic function. +4 +4 Practice Eercises A Determine if the pattern would be modeled by a linear function, an eponential function, or a quadratic function... 3 4 3. 4. 0, 8, 8, 3 3 4 5. 8, 7, 9, 6. 8, 6, 4, 9
Eample: Using a graphing calculator determine the quadratic function modeled by the given data 3 4 5 6 f() 9 3 43 69 0 Input the data into a TI-83 or TI-84 calculator list Enter the information into your lists by pushing STAT followed by Edit. If you have values in your lists already, you can clear the information by highlighting the name of the list then pushing CLEAR and ENTER. Do not push DEL or it will delete the entire list. Enter the values into L and the f() values into L. Push nd MODE to get back to the home screen. Make a scatter plot Push nd Y= to bring up the STAT PLOT menu. Select Plot by pushing ENTER or. Turn Plot on by pushing ENTER when ON is highlighted. Make sure that the scatter plot option is highlighted. If it isn t, select it by pushing ENTER when the scatter plot graphic is highlighted. The Xlist should say L and the Ylist should say L. If it doesn t, L can be entered by pushing nd and L by nd. To view the graph you can push GRAPH. If you want a nice viewing window, first push ZOOM arrow down to option 9 ZOOMSTAT and either push ENTER or push 9. Creating a quadratic regression equation You do not have to graph a function to create a regression, but it is recommended that you compare your regression to the data points to determine visually if it is a good model or not. From the home screen push STAT, arrow right to CALC and either push 5 for QuadReg or arrow down to 5 and push ENTER. (To do an eponential regression, push 0 for EpReg or arrow down to 0 and push ENTER.) Type nd, (the comma is located above 7) nd, VARS arrow right to Y-VARS select FUNCTION and Y then push ENTER. f = 3. It has been The quadratic regression is pasted into Y so that you can push GRAPH again and compare your regression to the data. 9
Practice Eercises B Find the regression equation. Round to three decimals when necessary.. Given the table of values use a graphing calculator to find the quadratic function. 0 3 4 5 f() -6 - -40-57 -66-6. Use a graphing calculator to find a quadratic model for the data. 3 4 5 6 f() 3 3 7 3 3. From 97 to 998 the U.S. Fish and Wildlife Service has kept a list of endangered species in the United States. The table below shows the number of endangered species. Find an appropriate eponential equation to model the data. Year 97 975 978 98 984 987 990 993 996 Number of 9.6 57.5 07.3 73 359.4 473.3 63. 80.5 080.3 species 4. The cell phone subscribers of the small town of Herriman are shown below. Find an eponential equation to model the data. Year 990 995 000 005 00 Subscribers 85 80,59 6,360 7,904 Eample: When doctors prescribe medicine, they must consider how much the effectiveness of the drug will decrease as time passes. The table below provides information on how much of the drug remains in a person s system after t hours. Find a model for the data. t (hours) 0 4 6 8 0 Amount (mg) 50 5.6 03.6 83.8 65.9 49.7 Answer: Sometimes it is helpful to look at the graph of the points. For this particular eample, it is difficult to determine if this should be modeled by an eponential or a quadratic function 93
from the graph. Therefore, consider the contet of the eample. The amount of the drug will continue to decrease unless more is given to the patient. If the patient does not receive more medication, at some point there will only be trace amounts of the drug left in the patient s system. This would suggest a function that continues to decrease until it reaches a leveling off point. An eponential model would be better suited for this situation. Use the regression capabilities of your graphing calculator to find an eponential model for the data. Follow the instructions for the previous eample but make sure that you select option 0: EpReg. The function that models the data is: f ( ) = 49.977( 0.950). Practice Eercises C Determine if the data is best modeled by an eponential or quadratic function. Then find the appropriate regression equation.. The pesticide DDT was widely used in the United States until its ban in 97. DDT is toic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. The half-life of DDT can be 5 or more years. Half-life is the amount of time it takes for half of the amount of a substance to decay. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal. Write an equation to model the data below. Year 97 98 99 0 Amount of DDT (in grams) 50 9.8.9 0.4. Use a graphing calculator to find a model for the data. 3 4 5 6 f() 0-7 -4 80 85 94