Quadratic Functions 12-7 Warm Up Problem of the Day Course 3 Warm Up Problem of the Day Lesson Presentation

Quadratic Functions 12-7 Warm Up Course 3 12-7 Quadratic Functions 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

Quadratic Functions 12-7 Problem of the Day Course 3 12-7 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 2d 32 t = √ . How much time will it take for a penny to fall 64 feet? 2 s

Quadratic Functions Learn to identify and graph quadratic functions. Course 3 12-7 Quadratic Functions Learn to identify and graph quadratic functions.

Insert Lesson Title Here Course 3 12-7 Quadratic Functions Insert Lesson Title Here Vocabulary quadratic function parabola

Course 3 12-7 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) = ax2 + bx + c

Course 3 12-7 Quadratic Functions Additional Example 1A: Quadratic Functions of the Form f(x) = ax2 + bx + c Create a table for each quadratic function, and use it to make a graph. A. f(x) = x2 + 1 Plot the points and connect them with a smooth curve. x f(x) = x2 + 1 –2 –1 1 2 (–2)2 + 1 = 5 (–1)2 + 1 = 2 (0)2 + 1 = 1 (1)2 + 1 = 2 (2)2 + 1 = 5

Course 3 12-7 Quadratic Functions Additional Example 1B: Quadratic Functions of the Form f(x) = ax2 + bx + c B. f(x) = x2 – x + 1 Plot the points and connect them with a smooth curve. x f(x) = x2 – x + 1 –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

Quadratic Functions 12-7 Try This: Example 1A Course 3 12-7 Quadratic Functions Try This: Example 1A Create a table for each quadratic function, and use it to make a graph. A. f(x) = x2 – 1 Plot the points and connect them with a smooth curve. x f(x) = x2 – 1 –2 –1 1 2 (–2)2 – 1 = 3 (–1)2 – 1 = 0 (0)2 – 1 = –1 (1)2 – 1 = 0 (2)2 – 1 = 3

Quadratic Functions 12-7 Try This: Example 1B B. f(x) = x2 + x + 1 Course 3 12-7 Quadratic Functions Try This: Example 1B B. f(x) = x2 + x + 1 Plot the points and connect them with a smooth curve. x f(x) = x2 + x + 1 –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

Course 3 12-7 Quadratic Functions You may recall that when a product ab is 0, either a must be 0 or b must be zero. 0(–20) = 0 100(0) = 0 You can use this knowledge to find intercepts of functions.

Quadratic Functions 12-7 The product is 0 when x = 5 or when x = 8. Course 3 12-7 Quadratic Functions The product is 0 when x = 5 or when x = 8. Example: f(x) = (x – 5)(x – 8) (5 – 5)(5 – 8) = 0 (8 – 5)(8 – 8)= 0 Some quadratic functions can be written in the form f(x) = (x – r)(x – s). Although the variable does not appear to be squared in this form, the x is multiplied by itself when the expressions in parentheses are multiplied together.

Course 3 12-7 Quadratic Functions Additional Example 2A: Quadratic Functions of the Form f(x) = a(x – r)(x – s) Create a table for the quadratic function, and use it to make a graph. A. f(x) = (x – 2)(x + 3) Plot the points and connect them with a smooth curve. x f(x) = (x – 2)(x + 3) –2 –1 1 2 (–2 – 2)(–2 + 3) = –4 (–1 – 2)(–1 + 3) = –6 (0 – 2)(0 + 3) = –6 (1 – 2)(1 + 3) = –4 (2 – 2)(2 + 3) = 0 The parabola crosses the x-axis at x = 2 and x = –3.

Course 3 12-7 Quadratic Functions Additional Example 2B: Quadratic Functions of the Form f(x) = a(x – r)(x – s) Create a table for the quadratic function, and use it to make a graph. B. f(x) = (x – 1)(x + 4) Plot the points and connect them with a smooth curve. x f(x) = (x – 1)(x + 4) –2 –1 1 2 (–2 – 1)(–2 + 4) = –6 (–1 – 1)(–1 + 4) = –6 (0 – 1)(0 + 4) = –4 (1 – 1)(1 + 4) = 0 (2 – 1)(2 + 4) = 6 The parabola crosses the x-axis at x = 1 and x = –4.

Quadratic Functions 12-7 Remember! Course 3 12-7 Quadratic Functions The x-intercepts are where the graph crosses the x-axis. Remember!

Quadratic Functions 12-7 Try This: Example 2A Course 3 12-7 Quadratic Functions Try This: Example 2A Create a table for the quadratic function, and use it to make a graph. A. f(x) = (x – 1)(x + 1) Plot the points and connect them with a smooth curve. x f(x) = (x – 1)(x + 1) –2 –1 1 2 (–2 – 1)(–2 + 1) = 3 (–1 – 1)(–1 + 1) = 0 (0 – 1)(0 + 1) = –1 (1 – 1)(1 + 1) = 0 (2 – 1)(2 + 1) = 3 The parabola crosses the x-axis at x = 1 and x = –1.

Quadratic Functions 12-7 Try This: Example 2B Course 3 12-7 Quadratic Functions Try This: Example 2B Create a table for the quadratic function, and use it to make a graph. B. f(x) = (x – 1)(x + 2) Plot the points and connect them with a smooth curve. x f(x) = (x – 1)(x + 2) –2 –1 1 2 (–2 – 1)(–2 + 2) = 0 (–1 – 1)(–1 + 2) = –2 (0 – 1)(0 + 2) = –2 (1 – 1)(1 + 2) = 0 (2 – 1)(2 + 2) = 4 The parabola crosses the x-axis at x = 1 and x = –2.

Additional Example 3: Application Course 3 12-7 Quadratic Functions Additional Example 3: Application A reflecting surface of a television antenna was formed by rotating the parabola f(x) = 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 3 Continued Course 3 12-7 Quadratic Functions Additional Example 3 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.

Insert Lesson Title Here Course 3 12-7 Quadratic Functions Insert Lesson Title Here Lesson Quiz: Part 1 Create a table for the quadratic function, and use it to make a graph. 1. f(x) = x2 + 2x – 1 x –2 –1 1 2 y 7

Insert Lesson Title Here Course 3 12-7 Quadratic Functions Insert Lesson Title Here Lesson Quiz: Part 2 Create a table for each quadratic function, and use it to make a graph. 2. f(x) = (x – 2)(x + 2) x –2 –1 1 2 y –3 –4

Insert Lesson Title Here Course 3 12-7 Quadratic Functions Insert Lesson Title Here Lesson Quiz: Part 3 3. The function f(t) = 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