3.2 Linear Functions

Vocabulary Linear Equation in two variables (x and y): An equation that can be written in the form y = mx + b, where m and b are constants. Linear Function: A function whose graph forms a non-vertical line. A linear functions has a constant rate of change. Nonlinear Function: A function that does not have a constant rate of change. It’s graph does not form a line.

Identifying Linear Functions Using Graphs Does the graph represent a linear or nonlinear function? Explain. Example 1: This is a nonlinear function since it’s graph does not form a line. This is a linear function since it’s graph forms a line.

Identifying Linear Functions Using Tables Does the table represent a linear or nonlinear function? Explain. Example 2: +3 +3 +3 a. x 3 6 9 12 y 36 30 24 18 As x increases by 3, y decreases by 6. The rate of change is constant. So, the function is linear. -6 -6 -6 +2 +2 +2 x 1 3 5 7 y 2 9 20 35 b. As x increases by 2, y increases by different amounts. The rate of change is not constant. So, the function is nonlinear. +7 +11 +15

You try! Are the following linear or nonlinear functions? Explain. Since this graph does not form a line it is a nonlinear function. Since this graph forms a line it is a linear function. As x increases by 1, y increases by 2. The rate of change is constant so this is a linear function. As x increases by 1, y decreases by different amounts. The rate of chance is not constant so this is a nonlinear function.

Identifying Linear Functions Using Equations Which of the following represents a linear function? Example 3:  

Graphing Linear Functions Solution to a Linear Equation in Two Variables The solution is an ordered pair (x, y) that makes the equation true. Discrete Domain A discrete domain is a set of input values that consist of only certain numbers in an interval Continuous Domain Continuous domain is a set of input values that consists of all numbers in an interval.

Graphing Discrete Data Example 4: The linear function y = 15.95x represents the cost (y) in dollars of (x) tickets for a museum. Each customer can buy a maximum of 4 tickets. Find the domain of the function. Since you can not buy part of a ticket the domain must be a whole number. The maximum tickets that can be purchased is 4. The domain would be: 0, 1, 2, 3, 4

Graphing Discrete Date Example 4: The linear function y = 15.95x represents the cost (y) in dollars of (x) tickets for a museum. Each customer can buy a maximum of 4 tickets. b) Graph the function using the domain.

Graphing Continuous Data Example 5: A cereal bar contains 130 calories. The number c of calories consumed is a function of the number b of bars eaten. Does this represent a linear function? Yes, as b increases by 1, c will increase by 130. The rate of change is constant. b) Find the domain of the function. Is the domain discrete or continuous? Explain. You can eat part of a cereal bar. The number b can be any value greater than or equal to 0. The domain is b>0 and it is continuous. c) Graph the function using its domain.

Graphing Continuous Data Example 5: A cereal bar contains 130 calories. The number c of calories consumed is a function of the number b of bars eaten. c) Graph the function using its domain.

You try! Is the domain discrete or continuous? Explain. Input Number of stories, x 1 2 3 Output Height of building (feet), y 12 24 36 The domain is the number of stories. Since there can not be partial stories in a building, the domain must be a whole number. The domain is discrete.

Writing Real-Life Problems Example 6: Write a real life problem to fit the data shown in each graph. Is the domain in each function discrete or continuous? Explain.