A graph, or coordinate plane, can be used to represent the results of equations by plotting their inputs and outputs. Basically, a value is input to an equation (with one variable), a series of mathematical actions occur, and a single output is created. A graph is made up of two perpendicular number lines, called the coordinate axes:
You may have talked about independent and dependent variables in your science class. In the case of linear equations, x is the independent variable (what is changed) and y is the dependent variable (what is measured as a result of x).
The coordinate axes break the graph into four sections, called quadrants. These quadrants help with the proper plotting of positive/negative inputs and positive/negative outputs:
Linear equations will generally come in three forms:
Any linear equation can be expressed in any of the three forms. Usually, different forms are used to emphasize different variables, but any form can be changed to any other form.
Standard form allows you to easily see the coefficients of each variable.
Slope-intercept form allows you to easily see the slope and y-intercept. This form is particularly useful for graphing linear equations.
Point-slope form allows you to easily see the slope as well as a point that the line passes through.
Though you need to know all three forms, you’ll use the slope‐intercept form to answer questions about graphs. If a linear equation is in standard form or point-slope form and you are dealing with graphing concepts, change the equation to slope‐intercept form. In slope‐intercept form, a value is input to the linear equation, multiplied by the slope, and added to the y‐intercept. The result is the output. This process is true for all real numbers. Inputting 0 for the x-value will always result in the y‐intercept for the equation and inputting 0 for the y-value will always result in the x‐intercept for the equation.
Now that the equation is in slope intercept form, it is easy to input values of x to find the output, y!
Notice where the line passes through the y-axis. This is the y-intercept, or “b” in the equation y = mx + b. Also notice that the points we plotted are evenly spaced, or change at a constant rate, defined by our slope!
Since this is a multiple‐choice test, you will not be asked to physically graph any equations, but you are expected to know what a linear equation’s graph will look like. You may be given a complete linear equation that has a known slope and y‐intercept, or you may be given two xy‐coordinates. If you are only given two xy-coordinates, you will need to determine what the linear equation is. To do this, you should first find the slope and then the y‐intercept.
It doesn’t matter which xy-coordinate you designate as the first or second coordinate.
To easily find the negative reciprocal slope:
Answers to Practice Problems