Transforming Quadratic Functions

TransformingQuadratic Functions CONFIDENTIAL

Warm Up Graph each quadratic function. 1) y = 2x2 - 1 2) y = x2 - 2x - 2 3) y = -3x2 - x + 6 CONFIDENTIAL

y 4 f (x) = x2 2 x 0 2 -2 vertex(0, 0) -3 Axis of symmetry, x=0 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. CONFIDENTIAL

Compare the coefficients in the following functions. f (x) = x2 g (x) = 1 x2 2 h (x) = -3x2 f (x) = 1x2 + 0x + 0 g (x) = 1 x2 + 0x + 0 2 h (x) = -3x2 + 0x + 0 ax2 +bx + c CONFIDENTIAL

Compare the graphs of the same functions. The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola. CONFIDENTIAL

1 ? 1 4 Width of a Parabola The graph of f (x) = ax2 is narrower than the graph of f (x) = x2 if l a l > 1 and wider if l a l < 1. Compare the graphs of g (x) and h (x) with the graph of f (x) . l -2 l ? 1 2 > 1 1 < 1 4 narrower wider CONFIDENTIAL

l -2 l = 2 1 = 1 l 4l = 4 3 3 Width of a Parabola Order the functions from narrowest graph to widest. A) f (x) = -2x2 , g (x) = 1 x2 ,h (x) = 4x2 3 Step1: Find l al for each function. Step2: Order the functions. f (x) = -2x2 The function with the narrowest graph has the greatest |a|. g (x) = 1 x2 3 h (x) = 4x2 CONFIDENTIAL

g (x) = 1 x2 3 h (x) = 4x2 f (x) = -2x2 Check: Use a graphing calculator to compare the graphs. h (x) = 4x2 has the narrowest graph, and g (x) =1 x2 3 has the widest graph. CONFIDENTIAL

B) f (x) = 2x2 , g (x) - 2x2 Step1: Find l al for each function. l 2 l = 2 l -2 l = 2 Step2: Order the functions. f (x) = 2x2 g (x) = -2x2 Since the absolute values are equal, the graphs are the same width. CONFIDENTIAL

Now you try! Order the functions from narrowest graph to widest. 1) f (x) = -x2 , g (x) = 2 x2 3 2) f (x) = -4x2, g (x) = 6x2 , h (x) = 0.2x2 1) f (x) = -x2 , g (x) = 2 x2 3 2) g (x) = 6x2 , f (x) = -4x2, h (x) = 0.2x2 CONFIDENTIAL

Compare the coefficients in the following functions. f (x) = x2 g (x) = x2 - 4 h (x) = x2 + 3 f (x) = 1x2 + 0x + 0 g (x) = 1 x2 + 0x - 4 h (x) = 1x2 + 0x + 3 CONFIDENTIAL

Compare the graphs of the same 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. CONFIDENTIAL

Vertical Translations of a Parabola The graph of the function f (x) = x2+ c is the graph of f (x) = x2 translated vertically. • If c > 0, the graph of f (x) = x2 is translated c units up. • If c < 0, the graph of f (x) = x2 is translated c units down. CONFIDENTIAL

Comparing Graphs of Quadratic Functions Compare the graph of each function with the graph of f (x) = x2 . A) g (x) = -1 x2 + 2 3 Method1: Compare the graphs. • The graph of g (x) = (-1/3)x2 + 2 is wider than the graph of f (x) = x2. • The graph of g (x) = (-1/3)x2 + 2 opens downward, and the graph of f (x) = x2opens upward. • The axis of symmetry is the same. • The vertex of f (x) = x2 is (0, 0) . The vertex of g (x) = g (x) = (-1/3)x2 + 2 is translated 2 units up to (0, 2) . CONFIDENTIAL

f (x) = x2 g (x) = 2x2 - 3 B) g (x) = 2x2 - 3 Method 2: Use the functions. • Since l 2 l > l 1 l , the graph of g (x) = 2x2 - 3 is narrower than the graph of f (x) = x2. • Since –b = 0 for both functions, the axis of symmetry is 2a the same. • The vertex of f (x) = x2 is (0, 0) . The vertex of g (x) = 2x2- 3 is translated 3 units down to (0, -3) . CONFIDENTIAL

Now you try! Compare the graph of the function with the graph of f (x) = x2: 1) g (x) = x2 - 4 • Since l 1 l = l 1 l , the graph of g (x) = x2 - 4 is equally wider as the graph of f (x) = x2. • Since –b = 0 for both functions, the axis of symmetry is 2a 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, -4) . CONFIDENTIAL

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. CONFIDENTIAL

144 feet 64 feet Two identical water balloons are dropped from different heights as shown in the diagram. a.) Write the two height functions and compare their graphs. b.) Use the graphs to tell when each water balloon reaches the ground. a.) Step 1: Write the height functions. The y-intercept crepresents the original height. h1 (t) = -16t2 + 64 Dropped from 64 feet h2 (t) = -16t2 + 144 Dropped from 144 feet CONFIDENTIAL

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 balloon in h2 is dropped from 80 feet higher than the one in h1 , the y-intercept of h2 is 80 units higher. h2 (t) = -16t2 + 144 h1 (t) = -16t2 + 64 CONFIDENTIAL

b.) Use the graphs to tell when each water balloon reaches the ground. The zeros of each function are when the water balloons reach the ground. The water balloon dropped from 64 feet reaches the ground in 2 seconds. The water balloon dropped from 144 feet reaches the ground in 3 seconds. Check:These answers seem reasonable because the water balloon dropped from a greater height should take longer to reach the ground. CONFIDENTIAL

Now you try! 1) 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. b. Use the graphs to tell when each tennis ball reaches the ground. h1 (t) = -16t2 + 16 Dropped from 16 feet h2 (t) = -16t2 + 100 Dropped from 100 feet The tennis ball dropped from 16 feet reaches the ground in 1 seconds. The tennis balldropped from 100 feet reaches the ground in 3 seconds. CONFIDENTIAL

Assessment Order the functions from narrowest graph to widest. 1) f (x) = 3x2, g (x) = 2x2 1) f (x) = 3x2, g (x) = 2x2 2) f (x) = 5x2, g (x) = -5x2 2) Same width 3) f (x) =2x2, g ( x) = -2x2 3) Same width CONFIDENTIAL

Compare the graph of the function with the graph of f (x) = x2: 4) g (x) is narrower than f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, 9). 5) g (x) is wider than f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, -9). 4) g (x) = 3x2 + 9 5) g (x) = 1 x2 - 9 2 6) g (x) = x2 + 6 6) g (x) has same width as f (x). The vertex of f (x) is (0, 0). The vertex of g (x) is (0, 6). CONFIDENTIAL

7)Two baseballs are dropped, one from a height of 16 feet and the other from a height of 256 feet. a. Write the two height functions and compare their graphs. b. Use the graphs to tell when each baseball reaches the ground. h1 (t) = -16t2 + 16 Dropped from 16 feet h2 (t) = -16t2 + 256 Dropped from 256 feet The baseball dropped from 16 feet reaches the ground in 1 second. The baseball dropped from 100 feet reaches the ground in 4 seconds. CONFIDENTIAL

Tell whether each statement is sometimes, always, or never true. 8) The graphs of f (x) = ax2 and g (x) = -ax2 have the same width. 9) The function f (x) = ax2 + c has three zeros. 8) always true CONFIDENTIAL

y 4 f (x) = x2 2 x 0 2 -2 vertex(0, 0) -3 Axis of symmetry, x=0 Let’s review 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. CONFIDENTIAL

Compare the coefficients in the following functions. f (x) = x2 g (x) = 1 x2 2 h (x) = -3x2 f (x) = 1x2 + 0x + 0 g (x) = 1 x2 + 0x + 0 2 h (x) = -3x2 + 0x + 0 CONFIDENTIAL