13-6 Quadratic Functions Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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 Course Quadratic Functions

Problem of the Day The time in seconds t that it takes a penny to fall a certain distance d in feet can be modeled using the equation 2 s 2d2d 32 t = √. How much time will it take for a penny to fall 64 feet? Course Quadratic Functions

Learn to identify and graph quadratic functions. Course Quadratic Functions

Vocabulary quadratic function parabola Insert Lesson Title Here Course Quadratic Functions

A quadratic function contains a variable that is squared. In the quadratic function the y-intercept is c. The graphs of all quadratic functions have the same basic shape, called a parabola. The cross section of the large mirror in a telescope is a parabola. Because of a property of parabolas, starlight that hits the mirror is reflected toward a single point, called the focus. f(x) = ax 2 + bx + c Course Quadratic Functions

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

f(x) = x 2 – x + 1 Course Quadratic Functions 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– (–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

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

f(x) = x 2 + x + 1 Check It Out: Example 1B Course Quadratic Functions Plot the points and connect them with a smooth curve. xf(x) = x 2 + x + 1 –2–2 –1– (–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

A reflecting surface of a television antenna was formed by rotating the parabola f(x) = 0.1x 2 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: Application Course Quadratic Functions

Additional Example 2 Continued First, create a table of values and graph the cross section. 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 at x = 2, and the height f(2) = 0.1(2) 2 = 0.4 ft. The sides are 0.4 ft higher than the center. Course Quadratic Functions

A reflecting surface of a radio antenna was formed by rotating the parabola f(x) = x 2 – 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 Course Quadratic Functions

Course Quadratic Functions Check It Out: Example 2 Continued xf(x) = x 2 – x (–1) 2 – (–1) + 2 = 4 (0) 2 – (0) + 2 = 2 (1) 2 – (1) + 2 = 2 (2) 2 – (2) + 2 = 4 First, create a table of values and graph the cross section. 3 ft. The center of the antenna is at x = 0 and the height is 2 ft. If the diameter of the mirror is 3 ft, the highest point on the sides at x = 2, and the height f(2) = (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. f(x) = x 2 – 2 Insert Lesson Title Here Course Quadratic Functions

Lesson Quiz: Part II Create a table for each quadratic function, and use it to make a graph. 2. f(x) = x 2 + x – 6 Insert Lesson Title Here Course Quadratic Functions

Lesson Quiz: Part III 3. The function f(t) = 40t – 5t 2 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? Insert Lesson Title Here 75 m Course Quadratic Functions