Algebra II Quadratic Functions and Equations - Extrema Unit 05b

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1 Big Idea: Quadratic Functions can be used to find the maximum or minimum that relates to real world application such as determining the maximum height of a ball thrown into the air or solving problems about free fall, for example, the height of launched objects. Objectives: (Common Core) A.CED.A. Create equations that describe numbers or relationships. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.C.7a Analyze functions using different representations. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph linear and quadratic functions and show intercepts, maxima, and minima. F.IF.C.8a Analyze functions using different representations. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.IF.B.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Skills: Define, Identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Vocabulary: axis of symmetry: Is a vertical line through the vertex of the function s graph. The quadratic function y a x h k in vertex form has the axis of symmetry x h. Ex. Identify the axis of symmetry for the graph of f ( x) ( x ) Because h, the axis of symmetry is the vertical line x. Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 1 of 1 6/4/013

2 Standard Form of a Quadratic Function y ax bx c, when a 0 ; a, b, and c are real numbers b b Vertex: the vertex is the point, f a a Axis of Symmetry: b x y-intercept: c a Minimum Value: When a parabola opens upward, the y-value of the vertex is the minimum value. Maximum Value: When a parabola opens downward the y-value of the vertex is the maximum value. Axis of Symmetry: the vertical line that passes through the vertex of a quadratic function. Vertex Maximum Axis of Symmetry Vertex Minimum Axis of Symmetry D :{ x / x } R :{ y / y k} The domain is all real numbers The range is all values greater than or equal to the minimum. D :{ x / x } R :{ y / y k} The domain is all real numbers The range is all values less than or equal to the maximum. Graphing a Quadratic Function in Standard Form Ex 1: Graph the quadratic function y x x 6 1. State the vertex and axis of symmetry. 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 axis of symmetry. ( the x-coordinate of the vertex) x b a a 1, b 6 6 x 3 1 The axis of symmetry is the line x 3 Alg II Unit 05b Quadratic Functions - Extrema CCSS Page of 1 6/4/013

3 Step Three: Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is. The y-coordinate is the value of the function at this x-value, or f (3). f (3) (3) 6(3) 1 10 The vertex is (3, 10) Step Four: Find the y-intercept. Because c 1, the y-intercept is 1. Step Five: Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 1). Use the axis of symmetry to find another point on the parabola. Notice that (0, 1) is 3 units left of the axis of symmetry. The point on the parabola symmetrical to (0, 1) is 3 units to the right of the axis at (6,0). Connect points with a smooth curve to draw the parabola. Vertex: 3, 10 Axis of Symmetry: x 3 Optional: To check make a table of values. When choosing x-values, use the vertex, a few values to the left of the vertex, and a few values to the right of the vertex. x y Note: When calculating the y-coordinate of points to the right and left of the vertex, notice the symmetry. Ex : Find the minimum or maximum value of f ( x) x x 5. State the domain and range. Step One: Determine whether the function has a minimum or maximum value. Because a is positive, the graph opens upward and has a minimum value. Step Two: Find the x-value of the vertex. Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 3 of 1 6/4/013

4 b ( ) 1 x a () 4 Step Three: Then find the y-value of the vertex, b f a. f Minimum Value: 1 4 or 4.5. Domain: all real numbers,. Range: all real numbers greater than or equal to 1 4 { y/ y 4.5} Check with Graphing Calculator. Graph f ( x) x x 5. The graph and table support the answer. Using a Quadratic Model Ex: A basketball s path can be modeled by y x x , where x represents time (in seconds) and y represents the height of the basketball (in feet). What is the maximum height that the basketball reaches? Graph the function and find the maximum (in the Calc menu). The maximum is the vertex. The maximum height of the basketball is the y-coordinate of the vertex, which is approximately 9.5ft. Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 4 of 1 6/4/013

5 Ex 3. A baseball is thrown with a vertical velocity of 50 ft/sec from an initial height of 6 ft. The height h in feet of the baseball can be modeled by the ball was thrown. h( t) 16t 50t 6, where t is the time in seconds since Approximately how many seconds does it take the ball to reach it maximum height? About 1.6 seconds What is the maximum height that the ball reaches? About 45 ft. You Try: The height of an arrow shot into the air is h( t) 16t 36t where h(t) is the height (in feet) of the arrow above the ground, t, seconds after it is released. Find the maximum height the arrow reaches. Use a graphing calculator to verify your answer. Closure: Show equivalent quadratic functions in both standard form and vertex form. Review how to use each form to determine the y-intercept, axis of symmetry, vertex, and maximum/minimum value. Use a graph to check. f x x x ( ) 4 6 f x ( ) ( x 1) 4 y-intercept: 6 Axis of Symmetry: x 1 Vertex: (1,4) Minimum Value: 4 Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 5 of 1 6/4/013

6 Activity: Complete the graphic organizer in each box, write the criteria or equation to find each property of the parabola for f ( x) ax bx c. Opens upward or downward Axis of Symmetry Properties of Parabolas y-intercept Vertex Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 6 of 1 6/4/013

7 Sample CCSD Common Exam Practice Question(s): f x x x 1? 1. Which graph represents f x x 4x 6?. What is the maximum of the quadratic function A. f x 1 B. f x 3 C. f x 6 D. f x 8 Sample SAT Question(s): Taken from College Board online practice problems. The figure above shows the graph of a quadratic function in the xy-plane. Of all the points xy, on the graph, for what value of x is the value of y greatest? Grid-In Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 7 of 1 6/4/013

8 Sample Questions 1. Consider. What are its vertex and y-intercept? a. vertex: (, ), y-intercept: (0, ) c. vertex: (1, 1), y-intercept: (0, ) b. vertex: (, ), y-intercept: (0, ) d. vertex: (, 1), y-intercept: (0, ). Consider. What are its vertex and y-intercept? a. vertex: (, ), y-intercept: (0, 9) c. vertex: (, ), y-intercept: (0, 9) b. vertex: (3, 9), y-intercept: (0, ) d. vertex: (0, 9), y-intercept: (, ) 3. Consider. What are its vertex and y-intercept? a. vertex: (, 38), y-intercept: (0, 11) c. vertex: (, ), y-intercept: (0, 11) b. vertex: (3, ), y-intercept: (0, 11) d. vertex: (0, 11), y-intercept: (3, ) 4. Consider. What are its vertex and y-intercept? a. vertex: (, ), c. vertex: (, ), y-intercept: 4 y-intercept: 4 b. vertex: (, 0), y-intercept: d. vertex: (, ), y-intercept: 5. Consider. What are its vertex and y-intercept? a. vertex: (, ), y-intercept:3 c. vertex: (, ), y-intercept: 3 b. vertex: (, 3), y-intercept: d. vertex: (, ), y-intercept: 6. Consider. What is the vertex and y-intercept? a. vertex: (0, 9), c. vertex: (, 1), y-intercept: 9 y-intercept: 9 b. vertex: (, 3), d. vertex: (, 1), y-intercept: y-intercept: ANS: A ANS: A ANS: B ANS: C ANS: C ANS: C 7. Consider the function. Find its vertex and y-intercept. a. c. (, ); ; b. (, ); d. (, 0); ANS: B 8. What is the minimum or maximum of? a. c. minimum at (3, 0) minimum at b. maximum at d. maximum at (3, 0) Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 8 of 1 6/4/013

9 ANS: A 9. What is the minimum or maximum of? a. c. minimum at (3, 0) minimum at b. maximum at d. maximum at (3, 0) ANS: B 10. Consider the function. Find its vertex and y-intercept. ANS: V(, ); Find the axis of symmetry and the vertex of the graph of the function. ANS: ; 1. Graph. Find the axis of symmetry and the vertex. a. y (-1.5, 1.75) x 4 The axis of symmetry is. The vertex is. Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 9 of 1 6/4/013

10 13. If an object is dropped from a height of 68 feet, the function gives the height of the object after t seconds. Graph this function. Approximately how long does it take the object to reach the ground (h = 0)? ANS about.1 seconds 15. A rocket is launched from atop a 9 foot cliff with an initial vertical velocity of 11 feet per second. The height of the rocket t seconds after launch is given by the equation Graph the equation to find out how long after the rocket is launched it will hit the ground. Estimate your answer to the nearest tenth of a second. ANS: 7.7 seconds Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 10 of 1 6/4/013

11 Sample Essay Questions: 1. A ball is hit upward inside a baseball stadium. The equation models the ball's height y (in feet) x seconds after it is hit. Part A: Give the coordinates of the vertex of the graph of the equation. Show your work. Then use a graphing calculator to graph the equation. Part B: The height of the stadium wall is at least 60 ft tall. Is it possible that the ball was hit over the wall? Justify your answer. ANS: Part A: coordinates of vertex: (1.875, 56.5); see graph below. Part B: No; because the graph in part (a) opens down, the y-coordinate of the vertex represents the maximum height of the ball. That height is just over 56 ft, so it does not go over the 60-foot high stadium wall.. Rick uses 1800 feet of fencing to build a rectangular pen. He divides the pen into two sections that have the same area. Let x be the width (in feet) of the pen, as shown in the drawing. Part A: Write an expression to represent the length of the pen in terms of x. Justify your work. Part B: Write an equation for the area y of the pen in terms of x. Graph the equation using a graphing calculator. Part C: Does the function in Part B have a maximum or a minimum value? Explain. Part D: Rick wants the pen to have the largest possible area. What width x should he use? What is the area of the pen with the largest area? ANS: Part A: x; Rick has 1800 feet of fencing. He uses 3x 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 x. Divide x by to get the length of one of the two sides: x. Part B: ; see graph below. ; use x-scale with intervals of 50, and y- scale with intervals of 0,000.] Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 11 of 1 6/4/013

12 Part C: The function has a maximum value because the value of the coefficient of x is negative. Part D: 300 feet; 135,000 square feet Alg II Unit 05b Quadratic Functions - Extrema CCSS Page 1 of 1 6/4/013