Linear Functions Overview

Linear Functions

• Linear functions are functions whose graphs are straight lines.
• A linear function is in the form f(x) = mx + b or y = mx + b, where m is the slope and b is the y-intercept.

Exercise #1

• The function f(x) = (2/3)x + 4 is a linear function.
• Finding f(9): f(9) = (2/3)(9) + 4 = 6 + 4 = 10. The point (9, 10) lies on the graph.
• A table of values for the function are included, plotting these points gives a straight line.
• The average rate of change between any two consecutive x-values is equal to the slope of the line. This value should be 2/3
• The y-intercept is the point where the line crosses the y-axis. It's the value of y when x = 0. In this case, the y-intercept is 4.

Linear Functions Definition

• A linear function has the form f(x) = mx + b (or y = mx + b)
• m represents the slope of the line.
• b represents the y-intercept, the point where the line crosses the y-axis.

Exercise #2

• For the function y = 5x + 2, choose two x-values, calculate their corresponding y-values, and then calculate the average rate of change.
• The average rate of change is equal to the slope of the line (m = 5).

Exercise #3 (Graphing a Linear Function)

• The given linear function is y = -x/2 + 7
• To graph, create a table of (x,y) values.
• Plot the points on the graph and draw a straight line through them.
• The slope of the line is -1/2
• The line is decreasing because the slope is negative.

Exercise #4 (Graphing Linear Equations)

• Provides several equations (y = 2x - 3, y = -x + 6 etc.)
• Find the slope and y-intercept for each equation.
• Graph these equations on a grid.

Exercise #1 (Multiple Choice Lines)

• Four lines are graphed on a coordinate plane.
• Match each graph to its correct equation.

Exercise #2 (Graphing and Identifying)

• Graph the equation y = (-3/2)x + 4 .
• Identify the slope and the y-intercept.

Exercise #3 (Average Rate of Change)

• Find two sets of coordinate points on the line.
• Calculate the average rate of change between the two coordinate pairs.
• The results is equal to the slope of the line.

Exercise #4 (Rewriting equations)

• The equation 2y - 6x = 12 is given.
• Steps are shown in the text to rewrite the equation in the slope-intercept form (y = mx + b).
• Identify the slope and y-intercept of the rewritten equation.

Exercise #5 (Rewriting equations)

• Several equations in various forms (e.g. 3y - 3x = 15) are provided
• Rewrite the equations in the slope-intercept form (y = mx + b)
• Identify the slope and y-intercept for each rewritten equation.

This quiz covers the fundamentals of linear functions, including their definitions and characteristics. You will explore the general form of a linear function, how to identify the slope and y-intercept, and practice calculating values through specific exercises. Ideal for students learning about mathematical functions in algebra.