Key Concepts

• The ordered pairs given by a linear function represent points on a line.
• Linear functions can be represented in words, function notation, tabular form, and graphical form.
• The rate of change of a linear function is also known as the slope.
• An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
• The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point at which the line crosses the y-axis.
• An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
• A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
• A constant linear function results in a graph that is a horizontal line.
• Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.
• The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line.
• The slope and initial value can be determined given a graph or any two points on the line.
• One type of function notation is the slope-intercept form of an equation.
• The point-slope form is useful for finding a linear equation when given the slope of a line and one point.
• The point-slope form is also convenient for finding a linear equation when given two points through which a line passes.
• The equation for a linear function can be written if the slope m and initial value b are known.
• A linear function can be used to solve real-world problems.
• A linear function can be written from tabular form.

Glossary

decreasing linear function

a function with a negative slope: If [latex]f\left(x\right)=mx+b, \text{then} m<0\\[/latex].

increasing linear function

a function with a positive slope: If [latex]f\left(x\right)=mx+b, \text{then} m>0\\[/latex].

linear function

a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line

point-slope form

the equation for a line that represents a linear function of the form [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)\\[/latex]

slope

the ratio of the change in output values to the change in input values; a measure of the steepness of a line

slope-intercept form

the equation for a line that represents a linear function in the form [latex]f\left(x\right)=mx+b\\[/latex]

y-intercept

the value of a function when the input value is zero; also known as initial value