Preview Warm Up California Standards Lesson Presentation

Warm Up For each function, find the value of y for x = 0, x = 4, and x = –5. 1. y = 6x – 3 2. y = 3.8x – 12 3. y = 1.6x + 5.9 –3, 21, –33 –12, 3.2, –31 5.9, 12.3, –2.1

Standards California AF3.1 Graph functions of the form y = nx2 and y = nx3 and use in solving problems. California Standards

Vocabulary quadratic function parabola

A quadratic function is a function in which the greatest power of the variable is 2. The most basic quadratic function is y = nx2 where n ≠ 0. The graphs of all quadratic functions have the same basic shape, called a parabola.

Additional Example 1: Graphing Quadratic Functions Create a table for each quadratic function, and use it to graph the function. A. y = x2 + 1 Plot the points and connect them with a smooth curve. x x2 + 1 y –2 –1 1 2 (–2)2 + 1 5 (–1)2 + 1 2 (0)2 + 1 1 (1)2 + 1 2 (2)2 + 1 5

Additional Example 1: Graphing Quadratic Functions Plot the points and connect them with a smooth curve. B. y = x2 – x + 1 x x2 – x + 1 y –2 –1 1 2 (–2)2 – (–2) + 1 7 (–1)2 – (–1) + 1 3 (0)2 – (0) + 1 1 (1)2 – (1) + 1 1 (2)2 – (2) + 1 3

Check It Out! Example 1 Create a table for each quadratic function, and use it to make a graph. A. y = x2 – 1 Plot the points and connect them with a smooth curve. x x2 – 1 y –2 –1 1 2 (–2)2 – 1 3 (–1)2 – 1 0 (0)2 – 1 –1 (1)2 – 1 0 (2)2 – 1 3

Plot the points and connect them with a smooth curve. Check It Out! Example 1 Plot the points and connect them with a smooth curve. B. y = x2 + x + 1 x x2 + x + 1 y –2 –1 1 2 (–2)2 + (–2) + 1 3 (–1)2 + (–1) + 1 1 (0)2 + (0) + 1 1 (1)2 + (1) + 1 3 (2)2 + (2) + 1 7

Additional Example 2: Application A reflecting surface of a television antenna was formed by rotating the parabola y = 0.1x2 about its axis of symmetry. If the antenna has a diameter of 4 feet, about how much higher are the sides than the center?

Additional Example 2 Continued First, create a table of values. Then graph the cross section. y = 0.1x2 y The center of the antenna is at x = 0 and the height is 0 ft. If the diameter of the mirror is 4 ft, the highest point on the sides are at x = 2 and x = –2. The height of the sides at x = 0.1(2)2 = 0.4 ft. The sides are 0.4 ft higher than the center.

Check It Out! Example 2 A reflecting surface of a radio antenna was formed by rotating the parabola y = x2 – x + 2 about its axis of symmetry. If the antenna has a diameter of 3 feet, about how much higher are the sides than the center?

Check It Out! Example 2 Continued First, create a table of values and graph the cross section. 3 ft. x x2 – x + 2 y -1 1 2 (–1)2 – (–1) + 2 4 (0)2 – (0) + 2 2 (1)2 – (1) + 2 2 (2)2 – (2) + 2 4 The center of the antenna is at x = 0 and the height is 2 ft. If the diameter of the antenna is 3 ft, the highest point on the sides are at x = 2. The height of the antenna at x = (2)2 – (2) + 2 = 4 ft – 2 ft = 2 ft. The sides are 2 ft higher than the center.

Lesson Quiz: Part I Create a table for the quadratic function, and use it to make a graph. 1. y = x2 – 2

Lesson Quiz: Part II Create a table for each quadratic function, and use it to make a graph. 2. y = x2 + x – 6

Lesson Quiz: Part III 3. The function y = 40t – 5t2 gives the height of an arrow in meters t seconds after it is shot upward. What is the height of the arrow after 5 seconds? 75 m