Unit 10 - Graphing Quadratic Functions

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1 Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif and graph linear functions students should be able to appl transformations to functions VOCABULARY: 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. Dependent Variable It is a variable in a function that has its value determined b the choice of values of another variable. 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. Independent Variable A variable in a function that ma have its value freel chosen without considering values of an other variables. 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. SKILLS: 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 in different forms with and without technolog. Compare multiple representations of functions. Algebra I Unit Notes Graphing Quadratic Functions Page 1 of 9 5/17/01

2 STANDARDS: F.IF.C.7a F.IF.C.8a Graph linear and quadratic functions and show intercepts, maima, and minima. *(Modeling Standard) Use the process of factoring and completing the square in a quadratic function to show zeros, etreme values, and smmetr of the graph, and interpret these in terms of a contet. 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. F.IF.B.-1 F.IF.C.9-1 F.LE.B.5 F.LE.A.1 For a linear, eponential, or quadratic function that models a relationship between two quantities, interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maimums and minimums; smmetries; and end behavior. *(Modeling Standard) Compare properties of two linear, quadratic, and/or eponential functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). For eample, given a graph of one quadratic function and an algebraic epression for another, sa which has the larger maimum. Interpret the parameters in a linear or eponential function in terms of a contet. *(Modeling Standard) Distinguish between situations that can be modeled with linear functions and with eponential functions. *(Modeling Standard) LEARNING TARGETS:.1 To graph quadratic functions in standard form.. To graph quadratic functions in factored form..3 To graph quadratic functions in verte form.. To transform the graphs of quadratic equations..5 To compare properties of two or more functions represented in different was. 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? How can the zeros, verte, and ais of smmetr of a quadratic function be located? How are functions different and how are the similar? Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

3 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. Different forms of a quadratic function reveal different characteristics of the function. It is important to compare linear, quadratic and eponential functions in various forms to see how the are different or similar and assess their properties. It is necessar to recognize these differences and similarities to draw conclusions about these models in real-world situations. Notes, Eamples and Eam Questions Unit.1 To graph quadratic functions in standard form. Parent Function: The most basic quadratic function is 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( ). Graphing a Parabola Using a Table is the set of all points 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 5 (the vertical line that passes through the verte) is Algebra I Unit Notes Graphing Quadratic Functions Page 3 of 9 5/17/01

4 What is the domain (the input values) of f ( ) What is the range (the output values)? { 0}? {all real numbers} For what value of is f ( ) f ( ) is increasing for 0 increasing? For what values of is it decreasing? and decreasing for 0. Wh is (0,0) the verte of the graph of f ( )? It is the lowest point on the parabola. Graphing Quadratic Functions Using a Table E : Graph f ( ) 8b using a table. Make a table. Plot enough ordered pairs to see both sides of the curve. f ( ) 8 (, f( )) 1 f (1) 1 (1) 8 3 (1, 3) f () () 8 0 (,0) 3 f (3) 3 (3) 8 1 (3, 1) f () () 8 (,0) 5 f (5) 5 (5) 8 3 (5,3) 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 5 with a smooth curve to draw the parabola. Note: The verte is a maimum at 0,0, and the ais of smmetr is Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

5 What is the domain of f ( ) What is the range? { 0}? {all real numbers} For what value of is f ( ) f ( ) is increasing for 0 increasing? For what values of is it decreasing? and decreasing for 0. Wh is (0,0) the verte of the graph of f ( )? It is the highest point on the parabola. 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 5 Maimum Verte Minimum Ais of Smmetr D:{ } The domain is all real numbers D:{ } R :{ k} The range is all values greater R :{ k} than or equal to the minimum Ais of Smmetr The domain is all real numbers The range is all values less than or equal to the maimum. E : Graph the quadratic function 1. State the verte, ais of smmetr, domain, and range. Step One: Find the -coordinate of the verte. a 1, b 3 1 Find the -coordinate of the verte: Algebra I Unit Notes Graphing Quadratic Functions Page 5 of 9 5/17/01

6 Step Two: Make a table of values. When choosing -values for the table, use the verte, as well as a few values to the left of the verte, and a few values to the right of the verte Step Three: Plot the points from the table and draw the parabola. Verte: 3, Ais of Smmetr: 3 Note: When calculating the -coordinate of points to the right and left of the verte, notice the smmetr Domain: all real numbers - Range: E 5: Graph the quadratic function 5. 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, b 1 The ais of smmetr is the line 1 Step Three: Find the verte. The verte lies on the ais of smmetr, so the -coordinate is 1. The -coordinate is the value of the function at this - value, or f (1). f (1) (1) (1) 5 7 The verte is (1, 7) Step Four: Find the -intercept. Because c 5, the -intercept is 5. Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

7 Step Five: Graph b sketching the ais of smmetr and then plotting the verte and the intercept point(0, 5). Use the ais of smmetr to find another point on the parabola. Notice that (0, 5) is 1 unit left of the ais of smmetr. The point on the parabola smmetrical to (0, 5) is 1 unit to the right of the ais at (, 5). Connect points with a smooth curve to draw the parabola. Verte: 1, 7 9 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 : Find the minimum or maimum value of f( ) 5. State the domain and range and graph the function. 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. Step Two: Find the -value of the verte. b ( ) 1 a () Step Three: Then find the -value of the verte, f b f a. Minimum Value: 1 or.5. Algebra I Unit Notes Graphing Quadratic Functions Page 7 of 9 5/17/01

8 Domain: all real numbers,. Range: all real numbers greater than or equal to 1 {.5} or [.5, ) 1 9 Check with a Graphing Calculator. Graph f ( ) 5. 3 The graph and table should support the answer E 7: Graph the quadratic function 1 1. State the verte and ais of smmetr. 3 Step One: Find the -coordinate of the verte. 1 a, b Step Two: Make a table of values. When choosing -values, 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.) Step Three: Plot the points from the table and draw the parabola. Verte: 3, Ais of Smmetr: Using a Quadratic Model E 8: A basketball s path can be modeled b 1 15, where represents time (in seconds) and represents the height of the basketball (in feet). What is the maimum height that the basketball reaches? Graph the function 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 9.5ft. Algebra I Unit Notes Graphing Quadratic Functions Page 8 of 9 5/17/01

9 Using a Quadratic Model E 9: A baseball is thrown with a vertical velocit of 50 ft/sec from an initial height of ft. The height h in feet of the baseball can be modeled b ht ( ) 1t 50t, 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. seconds What is the maimum height that the ball reaches? About 5 ft. You Tr: Find the verte and ais of smmetr for the following quadratic function. Determine if the parabola will open up or down. Then graph the parabola. QOD: How man points does it take to determine a unique parabola? Sample Eam Questions 1. What is the equation of the parabola shown? (A) (B) (C) 1 (D) 1 Ans: B Algebra I Unit Notes Graphing Quadratic Functions Page 9 of 9 5/17/01

10 . Find the verte of 3 and state if it is a maimum or a minimum. (A) (B) (C) (D) (-1, -); maimum (-1, -); minimum (-, -1); maimum (-, -1); minimum Ans: B 3. What is the maimum of the quadratic function A. f 1 C. f 3 B. f D. f 8 f ( )? Ans: D. Which graph represents f 1? Ans: C 5. 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, ) Ans: A Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

11 . Which equation best represents the following graph? Wh? A. B. C. D Ans: B 7. Which of the following are true statements about the graph of I. Opens Up II. Opens Down III. Ais of smmetr = IV. Ais of smmetr = A. I and III onl B. I and IV onl C. II and III onl D. II and IV onl 8? Ans: D 8. Find the verte of the parabola: 1 7 A., 7 C. 3, 11 B. 3, 1 D., 151 Ans: B Algebra I Unit Notes Graphing Quadratic Functions Page 11 of 9 5/17/01

12 9. What is the domain and range of the function shown in the graph below? A. Domain: all real numbers C. Domain: all real numbers Range: all real numbers Range:.5 B. Domain: D. Domain: Range:.5 Range: all real numbers Ans: C. A quadratic function is given b ha b c, where a and c are negative real numbers. Which of these could be the graph of h? A. B. C. D. Ans: C Algebra I Unit Notes Graphing Quadratic Functions Page 1 of 9 5/17/01

13 Unit. To graph quadratic functions in factored form. Zero(s) of Quadratic Functions: the -value(s) where the function intersects the -ais To find the zero(s), factor the quadratic and set each factor equal to 0. **Note: We can graph quadratic functions when the function is in factored form. Step One: First factor the quadratic, if not alread factored Step Two: Find the zeros of the function. Plot these values on the -ais. Step Three: Find the verte of the function. Since the -coordinate of the verte of a parabola is eactl the midpoint of the -intercepts, use the midpoint formula to find the -coordinate of the verte. After finding the -coordinate, substitute in to find the -coordinate of the verte. Step Four: Note the value of a to decide whether the graph opens up or down. Connect the three points to sketch the parabola. E : Find the zero(s) of the quadratic function Step One: Factor the quadratic polnomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. 3and graph the parabola Step Four: Plot the points and sketch the parabola Algebra I Unit Notes Graphing Quadratic Functions Page 13 of 9 5/17/01

14 E 11: Find the zero(s) of the quadratic function Step One: Factor the quadratic polnomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. 1and graph the parabola Step Four: Plot the points and sketch the parabola. **a is positive, so the parabola opens up Using a Quadratic Model E 1: A science class designed a ball launcher and tested it b shooting a tennis ball straight up from the top of a 15-stor building. The determined that the motion of the ball could be described the function: ht ( ) 1t 1t 10, where t represents the time the ball is in the air in seconds and h(t) represents the height, in feet, of the ball above the ground at time, t. What is the maimum height of the ball? At what time will the ball hit the ground? a. Find the time it takes for the ball to hit the ground. Factor: ht t t ( ) 1( 9 ) ht ( ) 1( t1)( t) Solve for t: t = seconds as time cannot be negative. It takes seconds to hit the ground. Find the verte: 1.5 h 0 ft (.5) 1(.5) 1(.5) Algebra I Unit Notes Graphing Quadratic Functions Page 1 of 9 5/17/01

15 Sample Eam Questions 1. Where is the ais of smmetr in the quadratic f 3 9 5? A. = C. = B. = D. = Ans: C In questions and 3, consider a quadratic -intercept at (0, c).. The function f has an ais of smmetr at f that has -intercepts at (r, 0) and (s, 0), and a r s. A. True B. False Ans: A 3. The function f has -intercepts at (r +, 0) and (s +, 0). A. True B. False Ans: B. Look at the graph of the quadratic f below. The graph of intercepts. What is the value of b? A. B. C. 1 D. 1 g 3 b has the same - Ans: A Algebra I Unit Notes Graphing Quadratic Functions Page 15 of 9 5/17/01

16 5. Use the quadratic function f() = ( + 3)( 1), to graph the function. A. C B. D Ans: B Unit.3 To graph quadratic functions in verte form. If the quadratic is written in the form a ( h) k, then it is written in what is called verte form. In this form, the verte coordinates can be easil found because the verte is ( hk, ). This point is ver helpful when graphing the parabola. The a tells us whether it opens up (positive) or down (negative) and also indicates whether it will be skinn (value larger than one) or fat (value less than one). To find other points for the graph, substitute zero into the equation for and solve for. Using smmetr, find another point on the opposite side of the ais of smmetr 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. E 13: Graph the quadratic function Algebra I Unit Notes Graphing Quadratic Functions Page 1 of 9 5/17/01

17 Step Two: Identif the verte and ais of smmetr. Note: Another wa of writing the function is 1 3. So the verte is 3, and the ais of smmetr is Step Three: Find the -intercept. (0 3) 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 State the verte and ais of smmetr. Step One: Determine if the graph opens up or opens down. Because a is the graph opens up. Step Two: Identif the verte and ais of smmetr. 8 The verte is, 3 and the ais of smmetr is. E 1: Graph the quadratic function Step Three: Find the -intercept. (0 ) 3 5 Step Four: Using smmetr, another point is (,5) E 15: Writing the Equation of a Quadratic Function in Verte Form Write an equation for the parabola in verte form given the graph Algebra I Unit Notes Graphing Quadratic Functions Page 17 of 9 5/17/01

18 Step One: Find the verte from the graph. The verte is at 1,. So, the verte form of the equation a 1. is Step Two: To solve for a, we will choose a point on the parabola and substitute it into the equation for,. Choose3, 0. 0a 31 0a a a 1 1. Step Three: Substitute a and the verte form of the equation is E 1: Automotive Application The minimum braking distance d in feet for a vehicle on dr concrete is approimated b the function dv ( ).035v, 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 ( ).05 p v v. What kind of transformation describes this change and what does the transformation mean? Eamine both functions in verte form. dv ( ).035( v0) 0 d v v p ( ).05( 0) 0 The value a has increased from.035 to.05. 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 Graph: Graph both functions on a graphing calculator. The graph of verticall stretched compared with the graph of d. d p appears to be Algebra I Unit Notes Graphing Quadratic Functions Page 18 of 9 5/17/01

19 Sample Eam Questions 1. Graph the function. Label the verte and ais of smmetr. Ans:. A quadratic function is defined as 7. Which statement is true? A. The parabola has a maimum value of 7. B. The parabola has a minimum value of 7. C. The parabola has a maimum value of. D. The parabola has a minimum value of. Ans: B 3. Use the graph below. Which equation could define the given parabola, where a is a positive real number? f a 3 A. f a 3 B. f a 3 C. f a 3 D. Ans: A Algebra I Unit Notes Graphing Quadratic Functions Page 19 of 9 5/17/01

20 . A parabola is defined as f a 3, where a is a positive real number. As a increases, what happens to the -coordinate of the parabola s verte? A. it decreases B. it increases C. it does not change Ans: C 5. A parabola is defined as f a 3, where a is a positive real number. As a increases, what happens to the -coordinate of the parabola s -intercept? A. it decreases B. it increases C. it does not change Ans: C Unit. To transform the graphs of quadratic equations. Reflection, Shrinking and Stretching, Vertical and Horizontal Translations Comparing and 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. : The verte is DISCOVERY: Calculator Eploration: Transformations with and. Use the graphing calculator or Desmos 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 Algebra I Unit Notes Graphing Quadratic Functions Page 0 of 9 5/17/01

21 3. 3 Compare to Verte: Up 3 0,3 Opens the same as. Compare to Verte: Down 0, Opens the same as 5. ( 3) Compare to Verte: left 3 3, 0 Opens the same as. ( 3) Compare to Verte: right 33, 0 Opens the same as 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. Algebra I Unit Notes Graphing Quadratic Functions Page 1 of 9 5/17/01

22 E 17: a. Describe the shift, reflect and stretch of the parent function. 1 f( ) ( 5) f ( ) Vertical Shrink b a factor of 1 (wider) Horizontal Shift 5 units left. b. c. f ( ) ( ) 1 f Horizontal Shift units right Vertical Shift 1 units down ( ) 3( 5) 7 Reflect across the ais Vertical Stretch b a factor of 3 Horizontal Shift 5 units right Vertical Shift 7 units up E 18: Graph the function below using knowledge of transformations. Graph the parent function on the same graph. f( ) ( 1) 1 Step One: Graph the parent function: f ( ) (dashed in blue) Step Two: Describe the transformations from the parent graph Vertical stretch b a factor of Horizontal shift 1 unit right Vertical shift units up Step Three: Graph the transformed function E 19: Graph the function using transformations. Find the verte, ais of smmetr, domain and range. f ( ) ( 3) Step One: Describe the transformations from the parent graph Reflect across the -ais Horizontal shift 3 units left Vertical shift units down Step Two: Graph using transformations verte: ( 3, ) ais: 3 Domain: (all real numbers) Range: Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

23 Eplore: The height h in feet of a baseball on Earth after t seconds can be modeled b the function ht ( ) 1( t 1.5) 3 where 1 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 59 feet higher and.5 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 ( ) 999,999( 5) 5 without graphing it. The graph would b a ver narrow parabola opening upward with its verte at (-5, 5) 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 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) -intercept: Ais of Smmetr: 1 Verte: (1, ) Minimum Value: SAMPLE EXAM QUESTIONS 1. 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 5 units up. C. The graph would shift units to the right. D. The graph would shift 5 units down. Ans: B Algebra I Unit Notes Graphing Quadratic Functions Page 3 of 9 5/17/01

24 . How would ou translate the graph of to produce the graph of A. translate the graph of down units B. translate the graph of up units C. translate the graph of left units D. translate the graph of right units Ans: A 3. 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 Ans: C. Identif the verte of. A. (, ) C. (1, ) B. (, 8) D. (1, 8) Ans: A 5. 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. Ans: C. 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 5 units down. A. C. B. D. Ans: B Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

25 7. 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? Ans: 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 50, 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; 135,000 square feet 8. Use the graph. Which equation defines this set of parabolas? A. k 1 1 B. 1 k C. k In questions 9-, consider a quadratic -intercept at (0, c). Ans: A f that has -intercepts at (r, 0) and (s, 0), and a 9. The function f has a -intercept at (0, c ). A. True B. False Ans: A Algebra I Unit Notes Graphing Quadratic Functions Page 5 of 9 5/17/01

26 . If f opens upward, then f opens downward. A. True B. False Ans: A 11. Define and sketch the three quadratic functions that have the following characteristics. (a) f has an ais of smmetr at = and no -intercepts. (b) g has a -intercept at 3 and opens downward. (c) h has a zero at = and a minimum value of. Answers will var. (a) f should be of the form, or equivalent to, f a k where a > 0 and k > 0. (b) g should be of the form, or equivalent to, g a b 3 where a < 0. (c) h should be of the form, or equivalent to, h a r where a r. f() g() g 3 h() f 1 h Unit.5 To compare properties of two or more functions represented in different was. In this unit, students should be able to compare quadratic functions that are in different forms table form, graph form, verte form, standard form and factored form. We can compare where the vertices of different functions are (where the highest maimum or lowest minimum occurs), compare which quadratic has the highest average rate of change within a given interval, or compare a certain output value of the functions. The average rate of change of a function over a specified interval gives us a sense of whether the function values are mostl increasing or mostl decreasing on the interval. The Average Rate of Change of a function f ( b) f( a) f( ) from a to b is:. Hopefull this formula will look familiar to the students! b a Algebra I Unit Notes Graphing Quadratic Functions Page of 9 5/17/01

27 E 0: Consider the quadratic function f ( ) ( 3) whose graph is shown. a) Find the average rate of change from 1 to 3. f( b) f( a) f(3) f(1) 0 ba 31 Since the average rate of change is negative, we know that the function values are mostl decreasing on this interval. b) Find the average rate of change from to 5. f(5) f() 0 ( 3) Since the average rate of change is positive, we know that the function values are mostl increasing on this interval. c) Find the average rate of change from to. f() f() 3 ( 3) 0 0 Since the average rate of change is equal to zero, the quadratic function increases and decreases over the interval but does so equall, as ou can see b looking at the graph. E 1: John is standing on a platform 300 ft in the air and throws a ball straight up at 0 feet per second. This situation can be modeled b: ht ( ) 1t 0t 300. John s friend, Frank, is standing on a different platform 00 ft in the air and throws a ball straight up at 0 feet per second. The situation can be modeled with the equation: ht ( ) 1t 0t 00. a) Which ball reaches the highest altitude? Find the maimum value b finding the -coordinate of the verte for John and Frank. John: Frank: h( t) ( 1) h( t) ( 1) John has the higher maimum value. Algebra I Unit Notes Graphing Quadratic Functions Page 7 of 9 5/17/01

28 b) Which ball will hit the ground first? Factor both equations to find the zeros. John: Frank: 0 (t 5t75) 0 (t15)( t5) t 5 or (t 5t5) 08(t5)( t5) t 5 or.5 Since time cannot be negative, both balls hit the ground at 5 seconds. It is a tie! E. Sall and Sam are testing out their new potato shooters from their tree houses which are at different heights. The chart to the right shows the time, t, in seconds and height, h, in meters of the potato pieces shot from Sam s shooter. The time, T, and height, H, of Sall s potato shooter can be represented b the following equation: Sam s Shooter Sall s Shooter: H t t 5 t h Use the information provided to answer the following questions: a) Whose potato pieces went higher? Find the verte for Sam and Sall s Shooter Sall: () () 5 9 (,9). Since Sall s starts at 5 meters 3 ( 1) (when = 0), Sall s travels meters up. Sam: Looking at the smmetr in the table, the verte occurs at 1.5 seconds. Using the rate of change for this parabola, the verte would be at (1.5,.5). It would travel to.5 meters. However, Sam s shooter starts at 7m, so it onl travels.5 meters up. Sall s potato pieces went higher. b) Whose potato pieces staed in the air longer, Sall s or Sam s? Looking at the table under the time column, Sam s pieces hit the ground (h=0) at 3.5 seconds. For Sall, we can factor the equation to find the zeros: ( t5)( t 1). At t = 5 seconds, Sall s pieces will hit the ground, so again Sall wins and her pieces staed in the air longer. Algebra I Unit Notes Graphing Quadratic Functions Page 8 of 9 5/17/01

29 Sample Eam Questions 1. The table below is of a quadratic function, g, where is measured in seconds and measured in meters g g is What is the approimate rate of change over the interval 0? (A).8 m/s (B) 8.7 m/s (C).3 m/s (D) 5.7 m/s. The table below is of the quadratic f f Ans: D g 5. A second quadratic is defined as Which is true about the two functions minimum values? (A) f has a smaller minimum value. (B) g has a smaller minimum value. (C) The minimum values of f and g are equal. (D) Which function has the smaller minimum cannot be determined from the information given. Ans: B Algebra I Unit Notes Graphing Quadratic Functions Page 9 of 9 5/17/01

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