DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Name Class Date 19.1Understanding Quadratic Functions Essential Question: What is the effect of the constant a on the graph of f(x)= ax 2? Resource Locker Explore Understanding the Parent Quadratic Function A function that can be represented in the form of ƒ(x) = ax 2 + bx + c is called a quadratic function. The terms a, b, and c, are constants where a ≠ 0. The greatest exponent of the variable x is 2. The most basic quadratic function is ƒ(x)= x 2, which is the parent quadratic function. A Here is an incomplete table of values for the parent quadratic function. Complete it. © Houghton Mifflin Harcourt Publishing Company x B Plot the ordered pairs as points on the graph, and connect the points to sketch a curve. y f(x)= x 2 −3 ƒ(x)= x2 = ( −3) = 9 8 −2 4 6 −1 1 4 0 0 1 1 2 4 3 9 2 2 x -4 -2 0 2 4 The curve is called a parabola. The point through which the parabola turns direction is called its vertex. The vertex occurs at ( 0, 0)for this function. A vertical line that passes through the vertex and divides the parabola into two symmetrical halves is called the axis of symmetry. For this function, the axis of symmetry is the y-axis. Reflect 1. Discussion What is the domain of ƒ(x)= x2 ? The domain is the set of all real numbers. 2. Discussion What is the range of ƒ(x)= x2 ? The range is the set y ≥ 0. Module 19 A1_MNLESE368187_U8M19L1.indd 889 889 Lesson 1 24/03/14 11:12 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti Graphing g(x) = ax 2 when a > 0 Explain 1 The graph g(x) = ax 2 , is a vertical stretch or compression of its parent functionƒ(x) = x2. The graph opens upward when a > 0. Vertical Compression Vertical Stretch g(x) = ax 2 with |a| > 1. g(x) = ax 2 with 0< |a| < 1. The graph of g(x) is narrower than the parent function ƒ(x). The graph of g(x) is wider than the parent function ƒ(x). y 8 g(x) 8 6 f(x) y 6 f(x) 4 4 g(x) 2 x -4 -2 0 2 x -4 4 -2 0 2 4 The domain of a quadratic function is all real numbers. When a > 0, the graph of g(x) = ax 2opens upward, and the function has a minimum value that occurs at the vertex of the parabola. So, the range of g(x) = ax 2,where a > 0, is the set of real numbers greater than or equal to the minimum value. Example 1 Graph each quadratic function by plotting points and sketching the curve. State the domain and range. g(x) = 2x2 y x g(x) = 2x2 -3 18 -2 8 6 -1 2 4 0 0 2 1 2 8 3 18 Domain: all real numbers © Houghton Mifflin Harcourt Publishing Company 2 8 x -4 -2 0 2 4 x Range: y ≥ 0 Module19 A1_MNLESE368187_U8M19L1.indd 890 890 Lesson1 24/03/14 11:12 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A B g(x)= __ 12 x 2 x g(x)= __ 12 x2 -3 1 4 __ 2 -2 2 0 0 2 2 3 1 4 _ 2 y 8 6 4 Domain: all real numbers 2 x -4 0 -2 2 Range: y ≥ 0 4 Reflect 3. For a graph that has a vertical compression or stretch, does the axis of symmetry change? No, the axis of symmetry does not change. Your Turn Graph each quadratic function. State the domain and range. 4. g(x)= 3x 2 D: all real; R: y ≥ 0 5. g(x)= _ 13 x 2 D: all real; R: y ≥ 0 4 y 4 2 y 2 x -4 © Houghton Mifflin Harcourt Publishing Company -2 0 2 x -4 4 -2 0 2 4 Explain 2 Graphing g(x) = ax 2when a < 0 The graph of y = −x 2 opens downward. It is a reflection of the graph of y = x 2 across the x-axis. So, When a < 0, the graph of g(x)= ax 2 opens downward, and the function has a maximum value that occurs at the vertex of the parabola. In this case, the range is the set of real numbers less than or equal to the maximum value. Vertical Stretch Vertical Compression (x)= ax 2 with |a| > 1. g (x)= ax 2 with 0 < |a| < 1. g The graph of g(x)is narrower than the parent function f(x). The graph of g(x)is wider than the parent function f (x). 4 2 y 4 f(x) 2 y f(x) x -4 -2 0 A1_MNLESE368187_U8M19L1.indd 891 -4 4 -2 -4 Module 19 2 x g(x) g(x) -2 0 2 4 -2 -4 891 Lesson 1 24/03/14 11:12 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti Example 2 Graph each quadratic function by plotting points and sketching the curve. State the domain and range. g(x) = −2x 2 y x g(x) = 2x2 -3 -18 -2 -8 -1 -2 0 0 -6 1 -2 -8 2 -8 3 -18 1 x2 g(x) = −_ 2 x 1 2 g(x) = − __ x 2 1 −4 _ 2 -2 −2 -1 1 −_ 2 0 −0 1 1 −_ 2 2 −2 3 1 −4 _ 2 0 -2 2 4 -4 Domain: all real numbers y≤0 Range: y -4 -2 0 x 2 4 -2 -4 -6 -8 Domain: all real numbers y≤0 Range: Reflect 6. Does reflecting the parabola across the x-axis (a < 0) change the axis of symmetry? No, the axis of symmetry is a line that extends both up and down and does not change © Houghton Mifflin Harcourt Publishing Company -3 -4 x upon reflection. Module19 A1_MNLESE368187_U8M19L1.indd 892 892 Lesson1 24/03/14 11:12 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Your Turn Graph each function. State the domain and range. 7. g(x) = −3x 2 D: all real; R: y ≤ 0 y -4 -2 0 8. g(x) = −__13 x 2 D: all real; R: y ≤ 0 x 2 y -4 4 -2 -2 x 0 2 4 -2 -4 -4 -6 -6 -8 -8 Writing a Quadratic Function Given a Graph Explain 3 You can determine a function rule for a parabola with its vertex at the origin by substitutingx and y values for any other point on the parabola into g(x) = ax 2 and solving fora. Example 3 Write the rule for the quadratic functions shown on the graph. 8 Use the point (2, 2). y Start with the functional form. 6 2 (2, 2) x © Houghton Mifflin Harcourt Publishing Company -2 0 2 -2 -4 (-2, -8) -6 -8 Module19 A1_MNLESE368187_U8M19L1.indd 893 x 2 -2 2 4 Evaluate x 2. 2 = 4a Divide both sides by 4 to isolate a. __1 = a ( 2 g(x) = __21 x 2 Write the function rule. 4 y -4 = a(2) Replace x and g(x) with point values. 2 4 -4 g(x) = ax 2 ) Use the point -2, −8 . Start with the functional form. g(x) = ax 2 Replace x and g(x) with point values. −8 = a −2 Evaluate x 2. ( ) −8 = 4 a Divide both sides by 4 to isolate a. −2 = a Write the function rule. g(x) = −2x 2 893 2 Lesson1 24/03/14 11:12 AM DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A DO NOT Correcti Your Turn 9. 10. y 8 y -4 6 4 -2 0 2 -2 4 (-1, 1) -4 (1, 4) x -6 x -4 0 -2 2 Use the point (1, 4). g(x) = ax -8 4 Use the point (1, −1). g(x) = ax 2 2 4 = a (1) −1 = a (1) 2 4=a g(x) = 4x 2 −1 = a g(x) = −x 2 2 Modeling with a Quadratic Function Explain 4 Real-world situations can be modeled by parabolas. Example 3 For each model, describe what the vertex, y-intercept, and endpoint(s) represent in the situation it models, and then determine the equation of the function. Depth (yards) -4 -4 -2 0 y x 2 -8 -16 -24 -32 (-4, -32) (4, -32) Time (seconds) Module 19 A1_MNLESE368187_U8M19L1.indd 894 4 The y-intercept occurs at the vertex of the parabola at (0, 0), where the dolphin is at the surface to breathe. The endpoint (-4, -32) represents a depth of 32 yards below the surface at 4 seconds before the dolphin reaches the surface to breathe. The endpoint (4, -32) represents a depth of 32 yards below the surface at 4 seconds after the dolphin reaches the surface to breathe. The graph is symmetric about the y-axis with the vertex at the origin, so the function will be of the form y = ax 2, or d(t) = at 2. Use a point to determine the equation. d(t) = at 2 2 -32 = a(4) -32 = a ∙ 16 -2 = a0 The function is d(t) = -2t 2. 894 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Malcolm Schuyl/Alamy This graph models the depth in yards below the water’s surface of a dolphin before and after it rises to take a breath and descends again. The depth d is relative to time t, in seconds, and t = 0 is when dolphin reaches a depth of 0 yards at the surface. Lesson 1 24/03/14 11:12 AM B Satellite dishes reflect radio waves onto a collector by using a reflector (the dish) shaped like a parabola. The graph shows the height h in feet of the reflector relative to the distance x in feet from the center of the satellite dish. Height (feet) DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A y (60, 12) 10 x 0 10 20 30 40 50 60 Distance from Center (feet) The y-intercept occurs at the vertex, which represents the distance x = 0 feet from the center of the dish. The left end-point represents the height h = 0 feet at the center of the dish. The right end-point represents the height h = 12 feet at the distance x = 60 feet from the center of the dish. h(x) = ax 2 The function will be of the form . Use 60 , 12 to determine the equation. h(x) = ax 2 ( ) 12 = a 60 12 = 2 3600 a 1 a =_ 300 1 ___ h(x) = 300 x 2 Your Turn 11. The graph shows the height h in feet of a rock dropped down a deep well as a function of time t in seconds. © Houghton Mifflin Harcourt Publishing Company Using the point (2, -64) to determine the equation: h(t) = at 2 -64 = a(2) 2 a = -16 0.5 1 1.5 2 x -16 -32 -48 (2, -64) Time Seconds d(t) = -16t 2 A1_MNLESE368187_U8M19L1.indd 895 y 0 -64 -64 = 4a Module 19 16 Height (feet) The y-intercept occurs at the left end-point, which is also the vertex, and represents the height h = 0 at which the rock was released at ground level. The right end point represents the height h = -64 feet at which the rock hits the bottom of the well 2 seconds after it was released. t= 895 Lesson 1 24/03/14 11:12 AM

© Copyright 2018