9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions

4/4/14 9-4 Functions Transforming Quadratic Quadratic Functions 9-4 Transforming Warm Up Lesson Presentation Lesson Quiz 9-4 Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y x2 3 x 0; (0, 3); opens upward 2. y 2x2 x 0; (0, 0); opens upward 3. y –0.5x2 – 4 x 0; (0, –4); opens downward Holt Algebra Holt Algebra 11 Holt Algebra 1 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Objective Graph and transform quadratic functions. Holt Algebra 1 Remember! You saw in Lesson 5-9 that the graphs of all linear functions are transformations of the linear parent function y x. Holt Algebra 1 1

4/4/14 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions The quadratic parent function is f(x) x2. The graph of all other quadratic functions are transformations of the graph of f(x) x2. For the parent function f(x) x2: The axis of symmetry is x 0, or the y-axis. The vertex is (0, 0) The function has only one zero, 0. Holt Algebra 1 9-4 Transforming Quadratic Functions Holt Algebra 1 9-4 Transforming Quadratic Functions Example 1A: Comparing Widths of Parabolas The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola. Order the functions from narrowest graph to widest. f(x) 3x2, g(x) 0.5x2 Step 1 Find a for each function. 3 3 0.05 0.05 Step 2 Order the functions. f(x) 3x2 g(x) Holt Algebra 1 0.5x2 The function with the narrowest graph has the greatest a . Holt Algebra 1 2

4/4/14 9-4 Transforming Quadratic Functions Example 1A Continued Order the functions from narrowest graph to widest. f(x) 3x2, g(x) 0.5x2 Check Use a graphing calculator to compare the graphs. 9-4 Transforming Quadratic Functions Example 1B: Comparing Widths of Parabolas Order the functions from narrowest graph to widest. f(x) x2, g(x) x2, h(x) –2x2 Step 1 Find a for each function. 1 1 Step 2 Order the functions. h(x) –2x2 f(x) 3x2 has the narrowest graph, and g(x) 0.5x2 has the widest graphü Holt Algebra 1 –2 2 f(x) x2 g(x) The function with the narrowest graph has the greatest a . x2 Holt Algebra 1 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Example 1B Continued Check It Out! Example 1a Order the functions from narrowest graph to widest. f(x) –x2, g(x) x2 Order the functions from narrowest graph to widest. f(x) x2, g(x) x2, h(x) –2x2 Check Use a graphing calculator to compare the graphs. h(x) –2x2 has the narrowest graph and g(x) x2 has the widest graph.ü Holt Algebra 1 Step 1 Find a for each function. –1 1 Step 2 Order the functions. f(x) –x2 g(x) x2 The function with the narrowest graph has the greatest a . Holt Algebra 1 3

4/4/14 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Check It Out! Example 1a Continued Order the functions from narrowest graph to widest. f(x) –x2, g(x) x2 Check It Out! Example 1b Order the functions from narrowest graph to widest. Check Use a graphing calculator to compare the graphs. f(x) –x2 has the narrowest graph and g(x) f(x) –4x2, g(x) 6x2, h(x) 0.2x2 Step 1 Find a for each function. –4 4 Step 2 Order the functions. x2 has the g(x) 6x2 f(x) –4x2 widest graph.ü Holt Algebra 1 9-4 Transforming Quadratic Functions 6 6 0.2 0.2 h(x) 0.2x2 The function with the narrowest graph has the greatest a . Holt Algebra 1 9-4 Transforming Quadratic Functions Check It Out! Example 1b Continued Order the functions from narrowest graph to widest. f(x) –4x2, g(x) 6x2, h(x) 0.2x2 Check Use a graphing calculator to compare the graphs. g(x) 6x2 has the narrowest graph and h(x) 0.2x2 has the widest graph.ü Holt Algebra 1 Holt Algebra 1 4

4/4/14 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of f(x) ax2 up or down the y-axis. Holt Algebra 1 9-4 Transforming Quadratic Functions Holt Algebra 1 9-4 Transforming Quadratic Functions Example 2A: Comparing Graphs of Quadratic Functions Helpful Hint When comparing graphs, it is helpful to draw them on the same coordinate plane. Compare the graph of the function with the graph of f(x) x2. g(x) x2 3 Method 1 Compare the graphs. The graph of g(x) x2 3 is wider than the graph of f(x) x2. The graph of g(x) x2 3 opens downward and the graph of f(x) x2 opens upward. Holt Algebra 1 Holt Algebra 1 5

4/4/14 9-4 Transforming Quadratic Functions Example 2A Continued Compare the graph of the function with the graph of f(x) x2 g(x) x2 3 The axis of symmetry is the same. The vertex of f(x) x2 is (0, 0). The vertex of g(x) x2 3 is translated 3 units up to (0, 3). 9-4 Transforming Quadratic Functions Example 2B: Comparing Graphs of Quadratic Functions Compare the graph of the function with the graph of f(x) x2 g(x) 3x2 Method 2 Use the functions. Since 3 1 , the graph of g(x) 3x2 is narrower than the graph of f(x) x2. Since for both functions, the axis of symmetry is the same. The vertex of f(x) x2 is (0, 0). The vertex of g(x) 3x2 is also (0, 0). Both graphs open upward. Holt Algebra 1 Holt Algebra 1 9-4 Transforming Quadratic Functions Example 2B Continued Compare the graph of the function with the graph of f(x) x2 g(x) 3x2 Check Use a graph to verify all comparisons. 9-4 Transforming Quadratic Functions Check It Out! Example 2a Compare the graph of each the graph of f(x) x2. g(x) –x2 – 4 Method 1 Compare the graphs. The graph of g(x) –x2 – 4 opens downward and the graph of f(x) x2 opens upward. The axis of symmetry is the same. The vertex of f(x) x2 is (0, 0). The vertex of g(x) –x2 – 4 is translated 4 units down to (0, –3). Holt Algebra 1 Holt Algebra 1 6

4/4/14 9-4 Transforming Quadratic Functions Check It Out! Example 2b Compare the graph of the function with the graph of f(x) x2. 9-4 Transforming Quadratic Functions Check It Out! Example 2b Continued Compare the graph of the function with the graph of f(x) x2. g(x) 3x2 9 g(x) 3x2 9 Method 2 Use the functions. Since 3 1 , the graph of g(x) 3x2 9 is narrower than the graph of f(x) x2. Check Use a graph to verify all comparisons. Since for both functions, the axis of symmetry is the same. The vertex of f(x) x2 is (0, 0). The vertex of g(x) 3x2 9 is translated 9 units up to (0, 9). Both graphs open upward. Holt Algebra 1 Holt Algebra 1 9-4 Transforming Quadratic Functions Check It Out! Example 2c Compare the graph of the function with the graph of f(x) x2. g(x) x2 2 9-4 Transforming Quadratic Functions Check It Out! Example 2c Continued Compare the graph of the function with the graph of f(x) x2. g(x) x2 2 Method 1 Compare the graphs. The graph of g(x) is wider than the graph of f(x) The graph of g(x) x2 2 opens upward and the graph of f(x) x2 opens upward. Holt Algebra 1 The axis of symmetry is the same. x2 2 x2. The vertex of f(x) x2 is (0, 0). The vertex of g(x) x2 2 is translated 2 units up to (0, 2). Holt Algebra 1 7

4/4/14 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions The quadratic function h(t) –16t2 c can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height of c feet. This model is used only to approximate the height of falling objects because it does not account for air resistance, wind, and other real-world factors. Two identical softballs are dropped. The first is dropped from a height of 400 feet and the second is dropped from a height of 324 feet. Example 3: Application a. Write the two height functions and compare their graphs. Step 1 Write the height functions. The y-intercept c represents the original height. h1(t) –16t2 400 Dropped from 400 feet. h2(t) –16t2 324 Dropped from 324 feet. Holt Algebra 1 Holt Algebra 1 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Example 3 Continued Step 2 Use a graphing calculator. Since time and height cannot be negative, set the window for nonnegative values. The graph of h2 is a vertical translation of the graph of h1. Since the softball in h1 is dropped from 76 feet higher than the one in h2, the yintercept of h1 is 76 units higher. Holt Algebra 1 Example 3 Continued b. Use the graphs to tell when each softball reaches the ground. The zeros of each function are when the softballs reach the ground. The softball dropped from 400 feet reaches the ground in 5 seconds. The ball dropped from 324 feet reaches the ground in 4.5 seconds Check These answers seem reasonable because the softball dropped from a greater height should take longer to reach the ground. Holt Algebra 1 8

4/4/14 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Check It Out! Example 3 Caution! Remember that the graphs show here represent the height of the objects over time, not the paths of the objects. Two tennis balls are dropped, one from a height of 16 feet and the other from a height of 100 feet. a. Write the two height functions and compare their graphs. Step 1 Write the height functions. The y-intercept c represents the original height. h1(t) –16t2 16 Dropped from 16 feet. h2(t) –16t2 100 Dropped from 100 feet. Holt Algebra 1 Holt Algebra 1 9-4 Transforming Quadratic Functions 9-4 Transforming Quadratic Functions Check It Out! Example 3 Continued Step 2 Use a graphing calculator. Since time and height cannot be negative, set the window for nonnegative values. The graph of h2 is a vertical translation of the graph of h1. Since the ball in h2 is dropped from 84 feet higher than the one in h1, the y-intercept of h2 is 84 units higher. Holt Algebra 1 Check It Out! Example 3 Continued b. Use the graphs to tell when each tennis ball reaches the ground. The zeros of each function are when the tennis balls reach the ground. The tennis ball dropped from 16 feet reaches the ground in 1 second. The ball dropped from 100 feet reaches the ground in 2.5 seconds. Check These answers seem reasonable because the tennis ball dropped from a greater height should take longer to reach the ground. Holt Algebra 1 9

4/4/14 9-4 Transforming Quadratic Functions Lesson Quiz: Part I 1. Order the function f(x) 4x2, g(x) –5x2, and h(x) 0.8x2 from narrowest graph to widest. g(x) –5x2, f(x) 4x2, h(x) 0.8x2 2. Compare the graph of g(x) 0.5x2 –2 with the graph of f(x) x2. The graph of g(x) is wider. Both graphs open upward. Both have the axis of symmetry x 0. The vertex of g(x) is (0, –2); the vertex of f(x) is (0, 0). 9-4 Transforming Quadratic Functions Lesson Quiz: Part II Two identical soccer balls are dropped. The first is dropped from a height of 100 feet and the second is dropped from a height of 196 feet. 3. Write the two height functions and compare their graphs. The graph of h1(t) –16t2 100 is a vertical translation of the graph of h2(t) –16t2 196 the y-intercept of h1 is 96 units lower than that of h2. 4. Use the graphs to tell when each soccer ball reaches the ground. 2.5 s from 100 ft; 3.5 from 196 ft Holt Algebra 1 Holt Algebra 1 10

9-4 Transforming Quadratic Functions9-4 Transforming Quadratic Functions Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 9-4 Transforming Quadratic Functions Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y x2 3 2. y 2x2 3. y .