9 4 Transforming Quadratic Functions Warm Up For

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 = x 2 + 3 x = 0; (0, 3); opens upward 2. y = 2 x 2 x = 0; (0, 0); opens upward 3. y = – 0. 5 x 2 – 4 x = 0; (0, – 4); opens downward Holt Algebra 1

9 -4 Transforming Quadratic Functions Holt Algebra 11

9 -4 Transforming Quadratic Functions The quadratic parent function is f(x) = x 2. The graph of all other quadratic functions are transformations of the graph of f(x) = x 2. For the parent function f(x) = x 2: • 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 Order the functions from narrowest graph to widest. f(x) = 3 x 2, g(x) = 0. 5 x 2 Step 1 Find |a| for each function. |3| = 3 |0. 05| = 0. 05 Step 2 Order the functions. f(x) = 3 x 2 g(x) = 0. 5 x 2 Holt Algebra 1 The function with the narrowest graph has the greatest |a|.

9 -4 Transforming Quadratic Functions Example 1 B: Comparing Widths of Parabolas Order the functions from narrowest graph to widest. f(x) = x 2, g(x) = x 2, h(x) = – 2 x 2 Step 1 Find |a| for each function. |1| = 1 |– 2| = 2 Step 2 Order the functions. h(x) = – 2 x 2 f(x) = x 2 g(x) = Holt Algebra 1 x 2 The function with the narrowest graph has the greatest |a|.

9 -4 Transforming Quadratic Functions Holt Algebra 1

9 -4 Transforming Quadratic Functions Example 2 A: Comparing Graphs of Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2. g(x) = x 2 + 3 Method 1 Compare the graphs. • The graph of g(x) = x 2 + 3 is wider than the graph of f(x) = x 2. • The graph of g(x) = x 2 + 3 opens downward and the graph of f(x) = x 2 opens upward. Holt Algebra 1

9 -4 Transforming Quadratic Functions Example 2 B: Comparing Graphs of Quadratic Functions Compare the graph of the function with the graph of f(x) = x 2 g(x) = 3 x 2 Method 2 Use the functions. • Since |3| > |1|, the graph of g(x) = 3 x 2 is narrower than the graph of f(x) = x 2. Since for both functions, the axis of symmetry is the same. • The vertex of f(x) = x 2 is (0, 0). The vertex of g(x) = 3 x 2 is also (0, 0). • • Both graphs open upward. Holt Algebra 1

9 -4 Transforming Quadratic Functions The quadratic function h(t) = – 16 t 2 + 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. Holt Algebra 1

9 -4 Transforming Quadratic Functions 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. h 1(t) = – 16 t 2 + 400 Dropped from 400 feet. h 2(t) = – 16 t 2 + 324 Dropped from 324 feet. Holt Algebra 1

9 -4 Transforming Quadratic Functions Step 2 Set the equation equal to zero to find the time and solve for t. 0= – 16 t 2 + 400 Dropped from 400 feet. 0= – 16 t 2 + 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 Holt Algebra 1

9 -4 Transforming Quadratic Functions Holt Algebra 1

9 -4 Transforming Quadratic Functions Lesson Quiz: Part I 1. Order the function f(x) = 4 x 2, g(x) = – 5 x 2, and h(x) = 0. 8 x 2 from narrowest graph to widest. g(x) = – 5 x 2, f(x) = 4 x 2, h(x) = 0. 8 x 2 2. Compare the graph of g(x) =0. 5 x 2 – 2 with the graph of f(x) = x 2. • 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). Holt Algebra 1

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 h 1(t) = – 16 t 2 + 100 is a vertical translation of the graph of h 2(t) = – 16 t 2 + 196 the y-intercept of h 1 is 96 units lower than that of h 2. 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

9 -4 Transforming Quadratic Functions Warm-Up 1. Order the function f(x) = 6 x 2, g(x) = – 3 x 2, and h(x) = 0. 2 x 2 from narrowest graph to widest. 2. Compare the graph of g(x) =x 2 +2 with the graph of f(x) = x 2. Holt Algebra 1