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 = –0.5x2 – 4 x = 0; (0, 3); opens upward x = 0; (0, 0); opens upward x = 0; (0, –4); opens downward

9-4 Transforming Quadratic Functions Holt Algebra 1

The quadratic parent function is f(x) = x2 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.

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) = 0.5x2 The function with the narrowest graph has the greatest |a|.

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 |–2| = 2 Step 2 Order the functions. h(x) = –2x2 The function with the narrowest graph has the greatest |a|. f(x) = x2 g(x) = x2

Example 2A: Comparing Graphs of Quadratic Functions 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.

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.

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.

Example 3: Application 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. 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.

Step 2 Set the equation equal to zero to find the time and solve for t. 0= –16t2 + 400 Dropped from 400 feet. 0= –16t2 + 324 Dropped from 324 feet. 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

Lesson Quiz: Part I 1. Order the function f(x) = 4x2, g(x) = –5x2, and h(x) = 0.8x2 from narrowest graph to widest. 2. Compare the graph of g(x) =0.5x2 –2 with the graph of f(x) = x2. g(x) = –5x2, f(x) = 4x2, h(x) = 0.8x2 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).

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

Warm-Up 1. Order the function f(x) = 6x2, g(x) = –3x2, and h(x) = 0.2x2 from narrowest graph to widest. 2. Compare the graph of g(x) =x2 +2 with the graph of f(x) = x2.