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

Objective Graph and transform quadratic functions.

You saw in Lesson 4-10 that the graphs of all linear functions are transformations of the linear parent function y = x. Remember!

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.

The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

Example 1A: Comparing Widths of Parabolas 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 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. f(x) = 3x2 has the narrowest graph, and g(x) = 0.5x2 has the widest graph 

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 1B Continued 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.

The value of c makes these graphs look different 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.

When comparing graphs, it is helpful to draw them on the same coordinate plane. Helpful Hint

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 2A Continued Compare the graph of the function with the graph of f(x) = x2 g(x) = x2 + 3 The vertex of f(x) = x2 is (0, 0). g(x) = x2 + 3 is translated 3 units up to (0, 3). The axis of symmetry is the same.

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.

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.

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 g(x) = –x2 – 4 f(x) = x2 is (0, 0). is translated 4 units down to (0, –4). The vertex of

Example 2c Continued Compare the graph of the function with the graph of f(x) = x2. g(x) = 3x2 + 9 Check Use a graph to verify all comparisons.

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.

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.

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 y-intercept of h1 is 76 units higher.

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.

Remember that the graphs shown here represent the height of the objects over time, not the paths of the objects. Caution!

Homework 8.4 Worksheet

Lesson Quiz: Part I 1. Order the functions 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