= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background

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1 Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic functions In this unit ou will Graph quadratic functions using transformations Graph quadratic functions using standard form You can use the skills in this unit to Graph parabolas using translations and identif translations of quadratic functions from a graph and/or table. Show the maimum and minimum (verte) on the graph of a quadratic function. Show intercepts and the ais of smmetr on the graphs of quadratic functions. Graph parabolas with and without technolog. Vocabular Ais of smmetr A line that passes through a figure in such a wa that the part of the figure on one side of the line is a mirror reflection of the part on the other side of the line. Dilation A transformation in which a figure is enlarged (stretched) or reduced (shrunk). Domain The input values of the function. For a quadratic function, the domain is all real numbers. Maima The largest value (highest point) of the function. Minima The smallest value (lowest point) of the function. Parabola The U-shaped graph of a quadratic function. Parent Quadratic function The simplest quadratic function, f( ) =. Quadratic function A function where the highest eponent of the variable is a square. Range The output values of a function. Reflection A transformation in which ever point of a figure is mapped to a corresponding image across a line of smmetr. Transformation A change in the position, size, or shape of a figure or graph. Translation A figure is moved from one location to another on the coordinate plane without changing its size, shape or orientation. Verte The point of intersection of a parabola and its line of smmetr. X-intercept Where the graph of a function intersects the -ais. Y-intercept Where the graph of a function intersects the -ais. Alg II Notes Unit Graphing Quadratic Functions Page 1 of 17 8/7/014

2 Algebra II Notes Quadratic Functions Unit Essential Questions How does multipling b and/or adding a constant to a function change the graph? What are the ke features of the graphs of quadratic functions? Overall Big Ideas The location and value of a constant within a function affects its graph. Quadratic equations can be written in verte form = a ( h) + k, where the verte of the graph of the equation is at ( hk, ) and the ais of smmetr is the line = h. The value of k determines the graph s vertical translation. The value of h determines the graphs horizontal translation. The value of a determines the direction of opening of the graph and the shape of the parabola. A quadratic function is represented b a U-shaped curve, called a parabola, intercepts one or both aes and has one maimum or minimum value. Skill To use transformations to graph quadratic functions. To use standard form to graph quadratic functions. Related Standards F.IF.C.7a Graph linear and quadratic functions and show intercepts, maima, and minima. *(Modeling Standard) F.BF.B.3-1 Identif the effect on the graph of replacing f( ) b f( ) + k, kf( ), f( k), f( + k) for special values of k (both positive and negative) and find the value of k given the graphs. Eperiment with cases and illustrate an eplanation of the effects on the graph using technolog. Include linear, eponential, quadratic, and absolute value functions. Alg II Notes Unit Graphing Quadratic Functions Page of 17 8/7/014

3 Algebra II Notes Quadratic Functions Unit Parent Function: The most basic quadratic function is Notes, Eamples, and Eam Questions f( ) =. This function is often called the parent quadratic function. Once ou understand the parent function ou can shift, reflect, and stretch the parent graph to get graphs of other quadratic functions. The graph of the function f( ), f(), is in the domain of f. We match domain values along the -ais with their range values along the -ais to get the ordered pairs that ield the graph of = f( ). = is the set of all points ( ) Graphing a Parabola Using a Table E 1 Graph the parent quadratic function =. Step One: Make a table of values (t-chart) Step Two: Plot the points on a coordinate grid and connect to draw the parabola Note: The verte is a minimum at ( 0,0 ), and the ais of smmetr (the vertical line that passes through the verte) is = 0. What is the domain (the input values) of { 0} f( ) =? {all real numbers} What is the range (the output values)? For what value of is f( ) = increasing? For what values of is it decreasing? f( ) = is increasing for > 0 and decreasing for < 0. Wh is (0,0) the verte of the graph of f( ) =? It is the lowest point on the parabola. Alg II Notes Unit Graphing Quadratic Functions Page 3 of 17 8/7/014

4 Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions Using a Table E 8 Graph f = + b using a table. ( ) Make a table. Plot enough ordered pairs to see both sides of the curve. f( ) = 6+ 8 (, f( )) 1 f (1) = 1 6(1) + 8 = 3 (1, 3) f () = 6() + 8 = 0 (,0) f (3) = 3 6(3) + 8 = 1 (3, 1) 4 f (4) = 4 6(4) + 8 = 4 (4,0) f () = 6() + 8 = 3 (,3) Verte: (3, 1) Ais of Smmetr: = 3 Domain: All real numbers Range: 1 E 3 Graph the parabola = Step One: Make a table of values (t-chart) Step Two: Plot the points on a coordinate grid and connect with a smooth curve to draw the parabola. Note: The verte is a maimum at ( 0,0 ), and the ais of smmetr is = Describe the transformation of the parent graph It is a reflection across the ais. f( ) = What is the domain of f( ) =? {all real numbers} What is the range? { 0} For what value of is f( ) = increasing? For what values of is it decreasing? f( ) = is increasing for < 0 and decreasing for > 0. Wh is (0,0) the verte of the graph of f( ) =? It is the highest point on the parabola. Alg II Notes Unit Graphing Quadratic Functions Page 4 of 17 8/7/014

5 Algebra II Notes Quadratic Functions Unit Reflection, Shrinking and Stretching, Vertical and Horizontal Translations Comparing = and = : The verte is( 0,0 ), and the ais of smmetr is = 0 for both graphs. When a is positive, the parabola opens up and its verte is a minimum; when a is negative, the parabola opens down and its verte is a maimum. DISCOVERY: Calculator Eploration: Transformations with = and =. Use the graphing calculator to investigate the graphs of quadratic functions. Describe the effect on the graphs of = and =. (Note: In the calculator graphs shown, = or = is graphed as a dotted line.) 1. = Compare to = Verte: Same ( 0,0 ) Opens narrower than =. 1 3 = Compare to = Verte: Same ( 0,0 ) Opens wider than = 3. = + 3 Compare to = Verte: Up 3 ( 0,3 ) Opens the same as = 4. = Compare to 4 = Verte: Down 4 ( 0, 4) Opens the same as =. = ( + 3) Compare to = Verte: left 3( 3, 0) Opens the same as = 6. = ( 3) Compare to = Verte: right 3( 3, 0 ) Opens the same as = Alg II Notes Unit Graphing Quadratic Functions Page of 17 8/7/014

6 Algebra II Notes Quadratic Functions Unit Transformations of Quadratic Functions = a( h) + k a indicates a reflection across the -ais and/or a vertical stretch or compression. h indicates a horizontal translation k indicates a vertical translation Vertical Stretch: a > 1 Vertical Shrink: 0< a < 1 Reflection over -ais: a Reflection over -ais: Horizontal Translation: h < 0 moves left h > 0 moves right Vertical Translation: k < 0 moves down k > 0 moves up Other: Ais of Smmetr = h Verte is ( hk, ) If a is positive the parabola opens up. If a is negative the parabola opens down. E 4 a. Describe the shift, reflect and stretch of the parent function. f( ) 1 f( ) = ( + ) 4 = Vertical Shrink b a factor of 1 4 (wider) Horizontal Shift units left. b. c. f = ( ) ( ) 1 f Horizontal Shift units right Vertical Shift 1 units down = + ( ) ( 3) 4 Reflect across the ais Vertical Stretch b a factor of Horizontal Shift 3 units right Vertical Shift 4 units up Alg II Notes Unit Graphing Quadratic Functions Page 6 of 17 8/7/014

7 Algebra II Notes Quadratic Functions Unit E Graphing a Quadratic Function in Verte Form Graph the quadratic function = 1 ( + 3) + 4. State the verte and ais of smmetr. 1 Step One: Determine if the graph opens up or opens down. Because a is the graph opens down. Step Two: Identif the verte and ais of smmetr. Note: Another wa of writing the function is = 1 ( ( 3 )) + 4. So the verte is ( 3, 4) and the ais of smmetr is = 3. Step Three: Plot the verte and sketch the graph. Notice the parent graph is reflected over the ais, is wider (vertical shrink), shifted to the left 3 (opposite of what ou think), and is translated up four units Optional: Make a table of values. When choosing -values for the T-table, use the verte, a few values to the left of the verte, and a few values to the right of the verte. (Note: Because of the fraction, ou ma want to choose values that will guarantee whole numbers for the -coordinates.) Five to seven points will give a nice graph of the parabola E 6 Writing the Equation of a Quadratic Function in Verte Form Write an equation for the parabola in verte form. The verte is at ( 1, 7). So the verte form of the equation is = a( 1) 7. To solve for a, we will choose a point on the parabola and substitute it into the equation for (, ). = a ( 4 1) 7 10 Choose ( 4, ). = 9a 7 1 = a = 1 7. So the verte form of the equation is ( ) E 7 Automotive Application -10 The minimum braking distance d in feet for a vehicle on dr concrete is approimated b the function dv ( ) =.03v, where v is the vehicle s speed in miles per hour. If the vehicle s tires are in poor condition, the braking-distance function is d ( ).06 p v = v. What kind of transformation describes this change and what does the transformation mean? Alg II Notes Unit Graphing Quadratic Functions Page 7 of 17 8/7/014

8 Algebra II Notes Quadratic Functions Unit Eamine both functions in verte form. dv ( ) =.03( v 0) + 0 d v v p ( ) =.06( 0) + 0 The value a has increased from.03 to.06. The increase indicates a vertical stretch. Find the stretch factor b comparing the new a-value to the old a value: dp () v dv ( ) = d p d The function d p represents a vertical stretch of d b a factor of approimatel 1.9. Because the value of each function approimates braking distance, a vehicle with tires in poor condition takes about 1.9 times as man feet to stop as a vehicle with good tires does. 7 6 Graph: Graph both functions on a graphing calculator. The graph of d p appears to be 4 3 verticall stretched compared with the graph of d Eplore: The height h in feet of a baseball on Earth after t seconds can be modeled b the function ht ( ) = 16( t 1.) + 36 where 16 is a constant in ft / s due to Earth s gravit. a. What if? The gravit on Mars is onl 0.38 times that on Earth. If the same baseball were thrown on Mars, it would reach a maimum height 9 feet higher and. seconds later than on Earth. Describe the transformations that must be applied to make the function model the height of the baseball on Mars. b. Write a height function for the baseball thrown on Mars. QOD: Describe the shift, reflect, and stretch of a quadratic equation in verte form. Write About It: Describe the graph of f = + + without graphing it. ( ) 999,999( ) The graph would b a ver narrow parabola opening upward with its verte at (-, ) Closure: What tpe of graph would a function of the form would it be? a h k = ( ) + have if a = 0? What tpe of function Alg II Notes Unit Graphing Quadratic Functions Page 8 of 17 8/7/014

9 Algebra II Notes Quadratic Functions Unit Graphing a Quadratic Function in Standard Form Standard Form of a Quadratic Function = a + b + c, when a 0 ; a, b, and c are real numbers b b Verte: the verte is the point, f a a Ais of Smmetr: b = -intercept: c a Minimum Value: When a parabola opens upward, the -value of the verte is the minimum value. Maimum Value: When a parabola opens downward the -value of the verte is the maimum value. Ais of Smmetr: the vertical line that passes through the verte of a quadratic function Verte Maimum Ais of Smmetr Verte Minimum Ais of Smmetr D:{ } R:{ k} The domain is all real numbers The range is all values greater than or equal to the minimum. D:{ } R:{ k} The domain is all real numbers The range is all values less than or equal to the maimum. E 8 Graph the quadratic function = 6 1. State the verte and ais of smmetr. Step One: Determine whether the graph opens upward or downward. If a is positive the graph opens upward. If a is negative the graph opens downward. Since a =1 the parabola opens upward. Step Two: Find the ais of smmetr. (the -coordinate of the verte) = b a a = 1, b= 6 6 = = 3 The ais of smmetr is the line = 3 1 ( ) Alg II Notes Unit Graphing Quadratic Functions Page 9 of 17 8/7/014

10 Algebra II Notes Quadratic Functions Unit Step Three: Find the verte. The verte lies on the ais of smmetr, so the -coordinate is. The -coordinate is the value of the function at this - value, or f (3). f (3) = (3) 6(3) 1 = 10 The verte is (3, 10) Step Four: Find the -intercept. Because c = 1, the -intercept is 1. Step Five: Graph b sketching the ais of smmetr and then plotting the verte and the intercept point (0, 1). Use the ais of smmetr to find another point on the parabola. Notice that (0, 1) is 3 units left of the ais of smmetr. The point on the parabola smmetrical to (0, 1) is 3 units to the right of the ais at (6,0). Connect points with a smooth curve to draw the parabola. Verte: ( 3, 10) 10 Ais of Smmetr: = Optional: To check, make a table of values. When choosing -values for our T-table, use the verte, a few values to the left of the verte, and a few values to the right of the verte Note: When calculating the -coordinate of points to the right and left of the verte, notice the smmetr. E 9 Find the minimum or maimum value of f( ) = +. State the domain and range. Step One: Determine whether the function has a minimum or maimum value. Because a is positive, the graph opens upward and has a minimum value. Alg II Notes Unit Graphing Quadratic Functions Page 10 of 17 8/7/014

11 Algebra II Notes Quadratic Functions Unit Step Two: Find the -value of the verte. b ( ) 1 = = = = a () 4 b Step Three: Then find the -value of the verte, f a. f = + = = 4 Minimum Value: 1 4 or 4.. Domain: all real numbers,. Range: all real numbers greater than or equal to 1 4 { 4.} or [4., ) Check with Graphing Calculator. Graph f( ) = +. The graph and table support the answer. Using a Quadratic Model E 10 A basketball s path can be modeled b = , where represents time (in seconds) and represents the height of the basketball (in feet). What is the maimum height that the basketball reaches? Press the Y= ke and graph the function in Y 1 and find the maimum (in the CALC menu). The maimum is the verte. The maimum height of the basketball is the -coordinate of the verte, which is approimatel 9.ft. E 11 A baseball is thrown with a vertical velocit of 0 ft/sec from an initial height of 6 ft. The height h in feet of the baseball can be modeled b ht ( ) = 16t + 0t+ 6, where t is the time in seconds since the ball was thrown. Approimatel how man seconds does it take the ball to reach it maimum height? About 1.6 seconds What is the maimum height that the ball reaches? About 4 ft. Alg II Notes Unit Graphing Quadratic Functions Page 11 of 17 8/7/014

12 Algebra II Notes Quadratic Functions Unit Closure: Show equivalent quadratic functions in both standard form and verte form. Review how to use each form to determine the -intercept, ais of smmetr, verte, and maimum/minimum value. Use a graph to check. f ( ) = f ( ) = ( 1) + 4 -intercept: 6 Ais of Smmetr: = 1 Verte: (1, 4) Minimum Value: 4 SAMPLE EXAM QUESTIONS 1. What is the equation of the parabola shown? (A) (C) = 4 (B) 1 = (D) = = 1 4 B. Find the verte of and state if it is a maimum or a minimum. (A) (B) (C) (D) (-1, -4); maimum (-1, -4); minimum (-4, -1); maimum (-4, -1); minimum B Alg II Notes Unit Graphing Quadratic Functions Page 1 of 17 8/7/014

13 Algebra II Notes Quadratic Functions Unit Create a table for the quadratic function f() = , and use it to graph the function. a. c. b. d. C 4. Graph the function. Label the verte and ais of smmetr. Alg II Notes Unit Graphing Quadratic Functions Page 13 of 17 8/7/014

14 Algebra II Notes Quadratic Functions Unit Use a table of values to graph = Alg II Notes Unit Graphing Quadratic Functions Page 14 of 17 8/7/014

15 Algebra II Notes Quadratic Functions Unit How would the graph of the function be affected if the function were changed to? a. The graph would shift units up. b. The graph would shift units up. c. The graph would shift units to the right. d. The graph would shift unit down. 7. How would ou translate the graph of to produce the graph of a. translate the graph of down 4 units b. translate the graph of up 4 units c. translate the graph of left 4 units d. translate the graph of right 4 units B A 8. Which transformation from the graph of a function f() describes the graph of? a. horizontal shift left unit c. vertical compression b a factor of b. vertical shift up unit d. vertical shift down unit C 9. Identif the verte of. a. (, ) c. (14, ) b. (, 8) d. (14, 8) A 10. Use this description to write the quadratic function in verte form: The parent function is verticall stretched b a factor of 3 and translated 8 units right and 1 unit down. a. c. b. d. C 11. What is the maimum of the quadratic function A. f ( ) = 1 C. f ( ) = 3 B. f ( ) = 6 D. f ( ) = 8 f( ) = ? D Alg II Notes Unit Graphing Quadratic Functions Page 1 of 17 8/7/014

16 Algebra II Notes Quadratic Functions Unit Use this description to write the quadratic function in verte form: The parent function is verticall compressed b a factor of and translated 11 units left and units down. a. c. b. d. f = + 1? 13. Which graph represents ( ) B C 14. Consider. What are its verte and -intercept? a. verte: (, ), -intercept: (0, ) c. verte: (1, 1), -intercept: (0, ) b. verte: (, ), -intercept: (0, ) d. verte: (, 1), -intercept: (0, ) Sample Essa Question: A 1. Rick uses 1800 feet of fencing to build a rectangular pen. He divides the pen into two sections that have the same area. Let be the width (in feet) of the pen, as shown in the drawing. Part A: Write an epression to represent the length of the pen in terms of. Justif our work. Part B: Write an equation for the area of the pen in terms of. Graph the equation using a graphing calculator. Part C: Does the function in Part B have a maimum or a minimum value? Eplain. Part D: Rick wants the pen to have the largest possible area. What width should he use? What is the area of the pen with the largest area? Alg II Notes Unit Graphing Quadratic Functions Page 16 of 17 8/7/014

17 Algebra II Notes Quadratic Functions Unit Part A: ; Rick has 1800 feet of fencing. He uses 3 feet to make two sides and a divider for the pen. So, the amount of fencing he has to make the other two sides of the fence is Divide b to get the length of one of the two sides: Part B: ; see graph below. ; use -scale with intervals of 0, and -scale with intervals of 0,000.] Part C: The function has a maimum value because the value of the coefficient of is negative. Part D: 300 feet; 13,000 square feet Alg II Notes Unit Graphing Quadratic Functions Page 17 of 17 8/7/014