Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions -: Points and Lines Sstem of Linear Equations: - two or more linear equations on the same coordinate grid. Solution of a Sstem of Linear Equations: - the intersecting point of two or more linear equations - on the Cartesian Coordinate Grid, the solution contains two parts: the -coordinate and the -coordinate Line Line (can be epressed as an Ordered Pair) - - - - - - - - - - - - - - - - - Intersecting Point (, ) Eample : Find the solution of the following sstem of equations b graphing manuall using + Solution of the Sstem of Linear Equations We can find the solution of a sstem of linear equation graphing manuall or using a graphing calculator. a. the slope and -intercept form For Line : For Line : + + + + -int (, ) Up + slope Right -int (, ) Down slope Right Line - - - - - - - - - (, ) - - - - - - - - (, ) (, ) (, ) Line Intersecting Point (, ) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus b. and -intercepts Line : For -int, let + () + (, ) For -int, let + + () (, ) Line : For -int, let () (, ) For -int, let () (, ) Line Intersecting Point (, ) (, ) (, ) (, ) - - - - - - - - - (, ) - - - - - - - - Line Eample : Find the solution of the following sstem of equations b using the graphing calculator. + Step : Rearrange to Slope Step : Set WINDOWS to Standard Mode Step : Graph and intercept form Y ZOOM GRAPH Choose Option will set : [,, ] : [,, ] Step : Run INTERSECT nd CALC TRACE ENTER ENTER ENTER Step : Verif using TABLE nd TABLE GRAPH Page. Choose Option Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions There are three tpes of solutions to a sstem of linear equations:. Intersecting Lines. Parallel Lines. Overlapping Lines One distinct Solution No Solution Man (Infinite) Solutions Different Slopes Identical Slopes Identical Slopes Different -Intercepts Different -Intercepts Identical -Intercepts - - - - - - - - - - - - - - - - - - - - - - - - - - Eample : Determine the number of solutions for the sstems of equations below. a. + b. + + + - - - - - - - - - - - - - - - - - - - - - - - - - Line : + + + + m, -int Line : + + + + m, -int Line : + + + + m, -int Line : + + + + m, -int Identical slopes, but different - intercepts mean parallel lines. Therefore, this sstem has NO SOLUTION. Identical slopes and -intercepts mean overlapping lines. Therefore, this sstem has MANY SOLUTIONS. When using the substitution method to solve a sstem of linear equations:. Isolate a variable from one equation. (Alwas pick the variable with as a coefficient.). Substitute the resulting epression into that variable of the other equation.. Solve for the other variable.. Substitute the result from the last step into one of the original equation and solve for the remaining variable. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Eample : Using the substitution method, algebraicall solve the sstem of equations below. ( ) ( ) ( ) ( + ) Epand each equation accordingl. Equation ( ) ( ) + Equation ( ) ( + ) Solve for both variables using the substitution method. Isolate from the first equation. + Substitute epression into in the second equation. ( ) Solve for the remaining variable. Pick the easier equation of the two. + + + Elimination b Multiplication: - most useful when neither equation has the same like terms. - b multipling different numbers (factors of their LCM) on each equation, we can change these equations into their equivalent form with the same like terms. + Eample : Solve the sstem of linear equations b elimination. First, rearrange the equations so and terms will line up. + + Eliminating Eliminate b addition. + LCM of and + ( + ) (Multipl Equation b to obtain ) (Multipl Equation b to obtain ) ( + ) + ( + ) + Net, substitute into one of the equations to solve for. + + () Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Review of Coordinate Geometr + + Midpoint M, Distance d ( ) + ( ) Eample : Find the distance (in eact value) between A (, ) and B (, ). A (, ) (, ) - - - - - - - - - - - - - - - - - - B (, ) (, ) d d AB AB ( ) + ( ) ( ) + ( ) + Eample : A circle has a diameter with endpoints (, ) and (, ). Find the eact length of the radius. Let A (, ) (, ) and B (, ) (, ) Length of Diameter AB Length of the Radius d d AB AB ( ) + ( ) + + Length of the Radius diameter AB Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Eample : Find the midpoints of the following line segments CD where C (, ) and D (, ). - - - - - - - - - - D (, ) + +,, Eample : Given that the midpoint of AB is M (, ). If one of the endpoint of AB is A (, ), find the coordinate of endpoint B. + + M AB, + + (, ), A (, ) - -------- - - - - - - - - - - - - - - - - - M M (, ) C (, ) B + M CD + + + M CD (, ) B (, ) + + + - Assignment: pg. # to (ever other odd) (Omit #ce, ) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions - and -: Slopes of Lines and Finding Equations of Lines - - - - - - - - - B (, ) - - - - - - - - - A (, ) change in (run) ( ) Slope: - a measure on the steepness of the line segment. change in (rise) ( ) m Slope of a Line Segment rise run m change in change in Eample : Find the slope of the following line segments. a. CD where C (, ) and D (, ) b. PQ where P (, ) and Q (, ) C (, ) - - - - - - - - - - - - - - - - D (, ) - m CD ( ) ( ) rise ( ( )) run ( ()) - m CD P (, ) - - - - - - - - - - - Q (, ) - - - - - m PQ ( ) m PQ Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus c. AB where A (, ) and B (, ) d. RS where R (, ) and S (, ) A (, ) B (, ) - - - - - - - - - - - - - - - - - m AB ( ) m AB R (, ) - - - - - - - - - - - - - - S (, ) - - - m RS ( ) m RS undefined In general, slopes can be classified as follows: Positive Slope (m > ) Line goes UP from left to right. Negative Slope (m < ) Line goes DOWN from left to right. Zero Slope (m ) Horizontal (Flat) Line [Rise ] Page. Undefined Slope Vertical Line [Run ] Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : If the slope of a line is, and it passes through A (t, t ) and B (, ), find the value of t and point A. m t t t ( t ) ( t) ( t ) t t t + t + t t t A ((), () ) A (, ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Parallel Lines Perpendicular Lines slope of line slope of line slope of line negative reciprocal slope of line m l m l m l m l Eample : Given the slope of two lines below, determine whether the lines are parallel or perpendicular. a. m and m b. m and m c. m and m m and m (negative reciprocal slopes) m and m (same slopes) m and m (neither the same nor negative reciprocal) Perpendicular Lines Parallel Lines Neither Parallel nor Perpendicular Lines Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Eample : Find the slope of the line given and its perpendicular slope. a. + b. c. m ( means perpendicular) m (negative reciprocal) Eample : Find the equation in point-slope form and general form given the followings. a. (, ) and m b. (, ) and m + m m (negative reciprocal) (vertical line) m undefined m (negative reciprocal) m When given a slope (m) and a point (, ) on the line, we can find the equation of the line using the point-slope form: m (slope formula) m ( ) (Point-Slope form) If we rearrange the equations so that all terms are on one side, it will be in general form: - - - - - - - - - A + B + C (General form) (A, the leading coefficient for the term must be positive) - - - - - - - - - (, ) (, ) right For Slope-Point form: m ( ) ( ) ( ( ) ) + ( + ) up m up right For General form: + ( + ) ( + ) ( + ) + + + + - - - - - - - - - (, ) down (, ) right m - - - - - - - - - For Slope-Point form: m ( ) ( ( ) ) ( + ) down right For General form: ( + ) + + + + Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions When given a slope (m) and the -intercept (, b) of the line, we can find the equation of the line using the slope and -intercept form: m + b where m slope and b -intercept Eample : Given the -intercept and slope, write the equation of the line in slope and -intercept form, and standard form. Sketch a graph of the resulting equation. a. (, ) and m - - - - - - - - - (, ) - - - - - - - - m right -intercept b Multipl both sides b. (, ) up up right b. b and m right (, ) down (, ) - - - - - - - - - - - - - - - - - m -intercept b + Rearrange to the left side. + down right Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus c. b and m d. -intercept and m undefined right up (, ) (, ) - - - - - - - - - - - - - - - - - m -intercept b + up right (, ) right (, ) up - - - - - - - - - - - - - - - - - -intercept and undefined slope (Vertical Line). Rearrange to the left side. Rearrange to the left side. Vertical Line NO term. Horizontal Line NO term. Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : Find the equation of a line parallel to + and passes through (, ). (, ) (, ) (, ) right - - - - - - - - - Line : Eample : Find the equation of a line perpendicular to + and having the same - intercept as the line. Line Line up (, ) down - - - - - - - - - - - - - - - - right (, ) - - - - - - - - - right (, ) up Line Line (, ) right (, ) up m + Line : m (parallel lines same slope as m ) Using (, ) as (, ) and the form m + b, we have: + + b + b () ( ) Line : b + + b b + + + m Line : m (perpendicular lines negative reciprocal of m ) To find -intercept of, we let. () -int means (, ) Using (, ) as (, ) and the form m + b, we have: ( ) () + b b + b - Assignment: pg. # to (ever other odd) - Assignment: pg. # to (odd) (Omit #,,,,, ) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus -: Linear Functions and Models Linear Function: - a set ordered pairs that ehibit a straight line when plotted on a graph. Mathematical Model: - one or more functions, graphs, tables, equations, or inequalities that describe a real-world situation. There are two tpes of data a. Discrete Data: - a graph with a series of separated ordered pairs or broken lines. Eample: Cost of Parking is $. ever hour with a Dail Maimum of $.. Time (hr) Cost ($) Cost ($) Parking Cost Tim e (hour) b. Continuous Data: - a graph with an unbroken line that connects a series of ordered pairs. Eample: Distance versus Time of a car with a constant speed of km/h. Time (hr) Distance (km) Distance (km) Distance versus Time Time (hour) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Direct Variation: - a variable that varies directl (b a rate of change) with another variable. ( is directl proportional to ) or k where k rate of change Eample : Gasoline at one time costs $. per Litre. a. Write out the cost of gasoline as a function of volume bought. b. What is the rate of change? c. Set up a table of values from V L to V L with a scale of L. d. Graph the function. a. C(V). V b. Rate of Change $./L (unit price of gasoline) c. d. V in L C(V) in $. ().. ().. ().. ().. ().. ().. (). Cost ($) Cost of Gasoline vs Volume Bought..... Volume (L) Eample : The amount of fuel used b a vehicle varies directl with the distance travelled. On a particular trip,. L of gasoline is used for a distance of. km. a. Calculate the rate of change. b. Find the function of volume of gasoline used in terms of distance travelled. c. What is the distance travelled if L of gasoline is used? d. How much would it cost to fuel up the car when the price of gasoline was $. / L if the entire trip was km? a. V(d) kd b. V(d) kd c. V L, d? d. d km, V?. k (.). d V. ().L k V(d).d d.km. V. L $. L. L k. L/km d. km $. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Partial Variation: - a variable that varies partiall (b a constant rate of change and a fied amount) with another variable. k + b where k rate of change and b fied amount (initial amount when ) Eample : The cost of a school dance organized b the student council $. per person and $ to hire the DJ. a. What is the rate of change and the fied amount? b. Epress the cost as a function of the number of people attending. c. Set up a table of values from n to n people with a scale of people. d. Graph the function. e. Eplain whether the graph should be discrete or continuous. f. How man people attended if the cost was $? a. Rate of Change $. / person (cost per person) b. C(n).n + Fied Amount $ (fied cost) c. d. n C(n) in $. () +. () +. () +. () +. () +. () + Cost ($) $, $ $ $ $ $ Cost of a School Dance n (people) e. The graph should be discrete because ou can not have a decimal number to describe the number of people attending. f. C(n) $, n? C(n).n +.n +.n.n n. n people Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : A tai ride costs ou $. for km travelled, and $. if ou travelled km. a. Epress the equation as a function of Cost in terms of distance travelled. b. What is the rate of change and the fied amount? c. How much would it cost for a km ride to the airport? d. If the ride costs $., what is the distance travelled? a. There are two order pairs ( km, $.) and ( km, $.) m k $. $. km km k $./km C kd + b (Substitute an order pair into d and C) $. ($./km)( km) + b $. $. + b b $. Therefore, C(d).d + b. Rate of Change $./km (cost per km travelled) Fied Amount $. (Flat Rate) c. d km, C? d. C $., d? C(d).d + C(d).d + C(d).() +..d + C(d) $...d..d. d. d. km - Assignment: pg. # to (odd),, and (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus -: The Comple Numbers Comple Numbers: - non-real numbers that cannot be represented on a number line or a Cartesian plane. - also called imaginar number. Real Numbers (R) If i, then i (imaginar unit) Comple Numbers ( R) Q I W N Q Eample: or or i Rational Numbers (Q): - numbers that can be turned into a fraction b a, where a, b I, and b. - include all Terminating or Repeating Decimals. - include all Natural Numbers, Whole Numbers and Integers. - include an perfect roots (radicals). Irrational Numbers (Q ): - numbers that CANNOT be turned into a fraction b a, where a, b I, and b. - include all non-terminating, non-repeating decimals. - include an non-perfect roots (radicals). Real Numbers (R): - an numbers that can be put on a number line. - include all natural numbers, whole numbers, integers, rational and irrational numbers. Eample : Simplif the following. a. b. c. i i i d. ( ) e. ( ) Page. ( ) ( ) i Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : Simplif the following. a. ( )( ) b. ( )( ) c. (i) i i i i i i i i () d. (i) e. i () Eample : Simplif the following. i i i i i i i i i f. i i i i i i i i i i i i i a. ( i) + ( i) b. ( i) ( i) i i i i i i i i (Pattern continues) i + i i i + i i Subtacting a Bracket, change signs in the barcket Solving Equations with Comple Numbers:. Rewrite the equation into two separate equations. One with Real Numbers, the other with imaginar numbers.. Solve the two equations separatel and combine the results.. Verif the answers b substituting each part into the original equation. Eample : Solve for and for ( + )i i. ( + )i i (Epand) i i i (Separate Real from Imaginar) i i i i + i i i i Verif: (() + )i i i i i ()i Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Multipling Comple Numbers: - multipl numerical coefficients and recall that i. Eample : Simplif the following. a. (i)(i) b. (i) c. ( + i)( i) d. ( i) i i Conjugate of Comple Number (a + bi) is a conjugate of (a bi) Eample : Simplif the following. + i + i i (Epand) + i () + i Product of Conjugate Comple Numbers (a + bi)(a bi) a bi a b () (a + bi)(a bi) a + b a. ( i)( + i) b. ( + i)( i) c. ( i) ( + i) + i i i () + i i i () ( i)( i) i i + i i + () i [( i)( + i)] [ + i i i ] [ ()] [] Rationalizing Comple Numbers: - similar to rationalizing radical binomial epression, we multipl b a fraction consists of the conjugate of the denominator over itself. Eample : Simplif the following. a. i ( i ( + i ) ) ( + i) ( + i) + i i i ( + i) ( ) ( + i) ( + i) + i b. i + i ( i ) ( i ) ( + i) ( i) i i + i i + i i i + ( ) ( ) i ( i) i + i c. i ( + i ) ( + i ) ( i) ( + i) i i + i i + i i i + ( ) ( ) i - Assignment: pg. # to (Omit #,, ) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions -: Solving Quadratic Equations Roots: - solution to an algebraic equation; commonl called zeros or -intercepts. There are several was of solving quadratic equations Solve Quadratic Equations b Factoring:. Epand, Simplif, Cross-Multipl, and /or Rearrange the quadratic equation into the general form a + b + c (with the preference for a > ).. FACTOR the resulting general quadratic equation.. EQUATE each factor to and solve for. Eample : Solve the following quadratic equations. a. b. + A B means A and / or B m m + m + + + (Rearrange Terms) ( + )( + ) + + + + ( + )( + ) m(m + ) (m + ) m + m m + m m (m + )(m ) m m Sometimes, when the quadratic equation is alread in standard form, ou ma find that it cannot be factored b conventional approach. In those cases, we might need to solve b completing the square. Epand and Simplif a ( h) + k a + b + c Standard Form General Form Complete the Square To Complete the Squares from General Form to Standard Form. Factor out the leading coefficient if a is NOT out of the a and b terms.. Group the first two terms in a bracket.. Complete the Square in the bracket b adding another constant. Be sure to subtract this constant at the end of the equation. Also be aware that sometimes we have to subtract the PRODUCT of this constant and a (when a is NOT ).. Factor the perfect trinomial and combine the like terms at the end of the equation.. Solve for. Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus Eample : Solve b completing the square. All solutions must be in the simplest radical form. a. b. + ( ) ( + ) ( ) ( ) ( ) ( + ) ( ) ( )( + ) + + ( )( + ) + + + ( + ) ( + + ) ( + ) ( ) ± ± ( + ) ( + ) ± + ± When we need to find the solution to a non-factorable quadratic equation algebraicall, we need to complete the square. As we have seen before, the algebra can get quite long and tedious. In order to speed up this process, we need to find a better wa to solve for non-factorable quadratic equations. To determine EXACT SOLUTIONS of a NON-Factorable Quadratic Equation, use the b ± b ac Quadratic Formula: (MEMORIZE!!) a Eample : Use the quadratic formula to find the solutions of the following quadratic equations. a. + b. + + a b c () b ± b ac ( ) ± ( ) ( )( ) a () + a b c ± ± ± b ± b ac ± a Page. ± ( ) ( ) ( )( ) ( ) No Real Solutions Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Discriminant: - the part of the quadratic formula that determine the tpe of solution(s) of the equation. - - - - - - - - - - - - - - - - - - b ac > (Discriminant is Positive) Two Distinct Real Solutions Sometimes, due to various algebraic steps involved in solving an equation, a root can be omitted or an etra root ma appeared. Gaining a Root: - when both sides of the equations are squared. - all roots must be verified against the original equation and all etraneous roots must be rejected. Eample : Solve Quadratic Formula: b ± + + + + + + + + + ( )( ) ( )( + ) ( )( ) ( ) ( ( ) ( + + + + ) ) + + ( )( + ) b ac a Determinant b ac - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - For : b ac (Discriminant ) Equal Real Solutions + + + Coprighted b Gabriel Tang B.Ed., B.Sc. Page. - b ac < (Discriminant is Negative) Imaginar Solutions (No Real Solutions) Verif b substitution into the original equation. For : + + + undefined undefined is an Etraneous Root
Chapter : Linear and Quadratic Functions PreCalculus Losing a Root: - commonl occur when dividing both sides of the equation b a common factor - when the same variable common factor appears on both sides of the equation, one must epand each sides before solving. - the onl eception to the above rule is if the common factor was numerical onl (without variable). Eample : Solve ( + )( ) ( ) The Correct Method: ( + )( ) ( ) (Epand both sides) + (Quadratic Equation) + (Collect all terms on one side) ( )( ) (Factor resulting equation) Verif Both Roots: For, (() + ) (() ) () (() ) ()() ()() The Incorrect Method: For, + ( ) ( ) ( + )( ) ( ) (Cancel Variable Common Factor on both sides WRONG!) + (Missing a Root!) - Assignment: pg. # to (odd) and (Omit #) Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions -: Quadratic Functions and their Graphs Verte at (p, q) g() is opening DOWN (a < ) (Verte is a Maimum) Maimum at q g () a ( p) + q Equation for Ais of Smmetr p Domain: R Range: g() q Eample : Using a graphing calculator, find the coordinate of the verte of. Determine the direction of the opening. State the maimum or minimum value of the parabola and the range. Indicate the equation of the ais of smmetr. Find the - and -intercepts. Using the graphing calculator to find the verte. - - - - - - - - - - - - - - - - -. Graph the Equation. Run Minimum function (if a <, then run the Maimum function). GRAPH nd CALC a. Take cursor left b. Take cursor right of the ENTER of the verte. verte. ENTER TRACE f () a ( p) + q Verte at (p, q) f() is opening UP (a > ) (Verte is a Minimum) Minimum at q Equation for Ais of Smmetr p Domain: R Range: g() q The process of completing the square from the general form, a + b + c, to the standard form, a ( p) + q, has the following generalization. a ( b a ) + ac b a p b a and q ac b a Be sure the verte is on the screen. ENTER c. Press again ENTER d. Read the Result. V(,.) Range:. Minimum at. opens up (a > ) Ais of Smmetr: Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions Using the graphing calculator to find -intercepts.. Graph the Equation. Run Zero (-intercept) function GRAPH nd CALC a. Take cursor left of the desired -intercept. TRACE PreCalculus b. Take cursor right of the desired -intercept. Be sure all -intercepts are on the screen. c. Press ENTER again ENTER d. Read the Result. ENTER. Redo the same steps for the other -intercept. ENTER ver near zero -intercept (let ) () () -int Eample : Turn the general equation,, into the standard form, a( p) + q. Find the coordinate of the verte. Determine the direction of the opening. State the maimum or minimum value of the parabola and the range. Indicate the equation of the ais of smmetr. Find the - and -intercepts. ( + ) ( + + ) + ( + ) + V, opens down (a < ) Maimum at Range: Ais of Smmetr: -intercept (let ) () () -intercepts (let ) b ± b ac a -int V, ( ) ± ( ) ( )( ) ( ) - - - - - - - - - - ± ± - - - - - - - - -ints ± ± Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : Find the intersecting points between + and + using a. an algebraic method. b. a graphing calculator. a. Using the substituting method b equating (since both equations have alread isolated ). + and + + + + ( + ) ( + )( ) For : () + For : () + and Intersecting Points: (, ) and (, ) b. Using a graphing calculator,. Enter the equations. Graph Y GRAPH. Run INTERSECT nd CALC TRACE a. Bring Cursor near the left intersecting point b. Press three times ENTER. Run INTERSECT nd CALC TRACE a. Bring Cursor near the right of intersecting point b. Press three times ENTER - Assignment: pg. #a & b, to (odd) (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.
Chapter : Linear and Quadratic Functions PreCalculus -: Quadratic Models Eample : The dimensions of a garden are m b m. It is to be surrounded b a brick path that has the same width on all sides of the garden. The area of the brick path itself has to be the same as the area of the garden. Determine the width of the garden. Let the width of the brick path Garden Area m m m Brick Path Area ( + )( + ) m m ( + ) Garden Area Brick Path Area ( + )( + ) ( + + + ) + (Solve for factor) ( + ) ( )( + ) (not possible negative width). m The width of the brick path is. m ( + ) Eample : A marketing firm for the Santa Clara Transit has determined that there will be, less people riding the light rail for ever five cents increased on the fare. There are currentl, people ride the light rail at $. on a dail basis. At what price should the Santa Clara Transit charge per fare to ield the maimum profit? Let the number of cents increase on the fare. (If, the new fare is $.; if, the new fare is $., and so on.) Revenue Fare Number of Riders Ma Revenue (. +.)(,,) Ma Revenue,, +, Ma Revenue + +, (Ma/Min Problem Complete the Square) Ma Revenue ( + ) +, Ma Revenue ( + ) +, + Ma Revenue ( ) +, V(, $,) Since, the amount of fare increase is ($.) $. Hence, the new fare is $. + $. $. Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions Eample : The height of an object thrown upward in metres can be epressed as a function of time in seconds b h(t) t + v i t + h i. The initial velocit is represented b v i in m/s and h i is the initial height in metres. A cannon ball was fired with a height of m and a initial velocit of m/s. a. State the function of height in terms of time and sketch the graph representing the function. b. Find the maimum height of the cannon ball and the time it happened. c. Determine the time at which the object landed on the ground. a. h(t) t + v i t + h i v i m/s h i m h(t) t + t + b. Maimum Height need to Complete the Square. h(t) t + t + h(t) (t t ) + h(t) (t t + ) + + h(t) (t ) + V( sec, m) The maimum height is at m in seconds. Time (s) Height (m) - Height (m) Cannon Ball Height vs. Time Time (s) Maimum Height V( sec, m) Landing Time (. sec, m) c. Time takes to land need to find t-intercept (at m, t?) h(t) t + t + (t t ) t t t b ± b ac a ( ) ± ( ) ( )( ) () Coprighted b Gabriel Tang B.Ed., B.Sc. Page. ± ± t. seconds and. seconds (time cannot be negative) The time to land is. seconds. ±
Chapter : Linear and Quadratic Functions PreCalculus Eample : Three-tiers wedding cakes are sold in sizes of inches, inches and inches in diameter. Their prices are $, $ and $ respectivel. Determine the quadratic function that models the prices of the wedding cakes in terms of their diameters b a. an algebraic approach. b. using the graphing calculator. a. using an algebraic approach. P(d) ad + bd + c (, ), (, ), (, ) a() + b() + c a + b + c a() + b() + c a + b + c a() + b() + c a + b + c Combining the first and second equations: Combining the second and third equations: a + b + c a + b + c ( a + b + c) ( a + b + c) a + b a + b a + b ( a + b) a a Substitute a : a + b () + b b b P(d) d + d + Substitute a and b : a + b + c () + () + c + + c c b. Using a Graphing Calculator,. To Enter a Table of values, use the STAT menu. STAT Choose Option ENTER Enter Coordinates When finish entering, nd QUIT MODE. Obtaining Equation with Correlation of Determination: a. Setting DiagnosticOn for Correlation of Determination nd ENTER CATALOG Again and Move Down using ENTER Select DiagnosticOn Note: After DiagnosticOn is selected; it will remain ON even when the calculator is turned Off. However, resetting the calculator will turn the Diagnostic Off (factor setting). Page. Coprighted b Gabriel Tang B.Ed., B.Sc.
Chapter : Linear and Quadratic Functions b. Use Quadratic Regression to obtain equation STAT Select CALC, use Choose Option QuadReg ENTER ENTER Again The quadratic equation is + +. The function that models the price of a wedding cake in terms of its diameter is P(d) d + d + Correlation of Determination (r ) is, which means the equation is an ecellent fit for the coordinates. (The closer it is to, the better the fit.) - Assignment: pg. # to (odd) (Omit #) Coprighted b Gabriel Tang B.Ed., B.Sc. Page.