Review Topics for Success

Learning Outcomes

By the end of this section, you will be able to:

  • Graph linear functions using tables
  • Define slope for a linear function
  • Calculate slope given two points
  • Interpret the slope of a linear function that models a real-world situation

When studying linear equations, you learned that if you can identify two points that lie on the line, you can calculate the slope and write the equation that describes the line. Since graphs of linear functions behave exactly like lines graphed in the plane, the same idea applies. Given two pairs of coordinates, inputs and outputs, you can calculate the rate of change of the function, and write its equation.

The reverse process holds as well; given a linear function in equation form, you can evaluate the function for chosen input values and use the resulting ordered pairs to sketch the graph of the function.

[latex]y=mx+b[/latex] is an equation that describes an infinite set of points in the plane contained in a line.

[latex]f(x)=mx+b[/latex] is a function that takes a value [latex]x[/latex] as an input and produces an output, [latex]f(x)[/latex].

When graphed, a linear function describes a line in the plane.

The formula for slope and equations of lines you’ve already learned are also applicable to linear functions.

  • The slope of a line indicates the direction it slants and its steepness. Slope is defined algebraically as: [latex]m=\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]
  • Given the slope and one point on a line, we can find the equation of the line using point-slope form, [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]
  • When simplified, the point-slope form of the equation of a line reveals the slope-intercept form, [latex]y = mx +b[/latex].

In this review section, you’ll prepare to write and graph linear functions and absolute value functions, build linear models, and fit linear models to data by reviewing and practicing the skills you learned to work with linear and absolute value equations. Just remember the important translation: [latex]y=f(x)[/latex]. This allows us to shift our perception from looking at lines as geometric objects containing an infinite set of points, [latex]\left(x, y\right)[/latex] to seeing them as descriptions of functions that take an input [latex]x[/latex] and produce an output [latex]f(x)[/latex]. Points on the line can be viewed as data points of the form [latex]\left(x, f(x)\right)[/latex]. And slope can be viewed as a rate of change from one input to the next. Keep this idea in mind as you work through the review section to graph a linear function using a table of values, then calculate and interpret the slope of such a graph.

Warm up for this module by refreshing important concepts and skills you’ll need for success. As you study these review topics, recall that you can also return to Algebra Essentials any time you need to refresh the basics.

Recall for success

Look for red boxes like this one throughout the text. They’ll show up just in time to give helpful reminders of the math you’ll need, right where you’ll need it.