1. Quadratic Functions 13-6 Warm Up Sandra is studying a bacteria colony that has a mass of 300 grams. If the mass of the colony doubles every 2 hours, what will its mass be after 20 hours? 307,200 grams

2. Quadratic Functions 13-6 Learn to identify and graph quadratic functions.

3. Quadratic Functions 13-6 Vocabulary quadratic function parabola

4. Quadratic Functions 13-6 A quadratic function can be written in the form f(x) = ax 2 + bx + c, where a ≠ 0. Quadratic functions always have a variable that is squared. The graphs of all quadratic functions have the same basic shape, called a parabola.

5. Quadratic Functions 13-6 Create a table for each quadratic function, and use it to make a graph. f(x) = x 2 + 1 Additional Example 1A: Graphing Quadratic Functions Plot the points and connect them with a smooth curve. xf(x) = x 2 + 1 –2–2 –1–1 0 1 2 (–2) 2 + 1 = 5 (–1) 2 + 1 = 2 (0) 2 + 1 = 1 (1) 2 + 1 = 2 (2) 2 + 1 = 5

6. Quadratic Functions 13-6 f(x) = x 2 – x + 1 Additional Example 1B: Graphing Quadratic Functions Plot the points and connect them with a smooth curve. xf(x) = x 2 – x + 1 –2–2 –1–1 0 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

7. Quadratic Functions 13-6

8. Quadratic Functions 13-6 The function f(x) = –16x 2 + 96x gives the height in feet of a model rocket x seconds after it is launched. What is the maximum height that the rocket reaches? How long does the rocket stay in the air? Additional Example 2: Application

9. Quadratic Functions 13-6 Additional Example 2 Continued First make a table of values. Then graph the function. The highest point of the parabola is (3, 144), so the maximum height of the rocket is 144 ft. After the rocket is launched, it stays in the air until its height is 0, or when f(x) = 0. The table and graph show that f(x) = 0 when x = 0 and when x = 6, so the rocket stays in the air for 6 seconds. xf(x) 0–16(0) 2 + 96(0) = 0 1–16(1) 2 + 96(1) = 80 2–16(2) 2 + 96(2) = 128 3–16(3) 2 + 96(3) = 144 4–16(4) 2 + 96(4) = 128 5 –16(5) 2 + 96(5) = 80 6–16(6) 2 + 96(6) = 0

10. Quadratic Functions 13-6 The function f(x) = –16x 2 + 48x gives the height in feet of a baseball x seconds after it is thrown. What is the maximum height that the baseball reaches? How long does the baseball stay in the air? Check It Out: Example 2

11. Quadratic Functions 13-6 Check It Out: Example 2 Continued xf(x) 0–16(0) 2 + 48(0) = 0 1–16(1) 2 + 48(1) = 32 2–16(2) 2 + 48(2) = 32 3–16(3) 2 + 48(3) = 0 1.5–16(1.5) 2 + 48(1.5) = 36 The highest point of the parabola is (1.5, 36), so the maximum height of the baseball is 36 ft. After the baseball is thrown, it stays in the air until its height is 0, or when f(x) = 0. The baseball stays in the air for 3 seconds.

12. Quadratic Functions 13-6 Lesson Quiz

13. Quadratic Functions 13-6 Lesson Quiz: Part I Create a table for the quadratic function, and use it to make a graph. 1. f(x) = x 2 – 2

14. Quadratic Functions 13-6 Lesson Quiz: Part II Create a table for each quadratic function, and use it to make a graph. 2. f(x) = x 2 + 4x – 2