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Graphing Linear Equations

Graphing Linear Equations

Author: Sophia Tutorial

This lesson will demonstrate how to graph linear equations.

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What's Covered

  • Using Slope-Intercept Form
  • Using Point-Slope Form
  • Using Standard Form

Graphing linear equations

Using Slope-Intercept Form

Slope-intercept form of a linear equation provides two useful pieces of information about the graph of the line: the slope of the line, and the location of the y-intercept.  Slope-intercept form is y = mx + b, where m is the slope of the line, and b is the y-coordinate of the y-intercept.  We know that all y-intercepts have an x-coordinate of 0, so the coordinate to the y-intercept is (0, b) in this form. 

Recall that we only need two points in order to graph a line.  Once we have two points plotted, we can connect the two points and extend the line in both directions.  When graphing a line using its equation in slope-intercept form, plot the y-intercept first.  This gives us one of our two points needed.  To find the other point, we'll use the slope.  The slope is the ratio between rise and run, or vertical distance over horizontal distance from one point to another.  Starting at the y-intercept, the slope will tell us how to move vertically and horizontally to another point on the line.  We won't have to go far until we find another point on our line. 

Graph the line y space equals space minus 3 over 4 x space plus space 4

Big Idea

To graph a line given its equation in slope-intercept form, plot the y-intercept first, using the b-value in the equation.  To find a second point on the line, use the slope, m, to move vertically and horizontally from the y-intercept.  Finally, connect the two points and extend them in both directions to plot the line. 


Using Point-Slope Form

When given the equation of a line in point-slope form, we follow a similar process as above: we can easily plot one point on the line from the equation.  From there, we can use the slope to move vertically and horizontally to another point on the line, and then draw the line on the graph. 

Graph the line left parenthesis y space plus space 3 right parenthesis space equals space 2 left parenthesis x space plus space 6 right parenthesis


Be careful with positive and negatives when working in this form.  The general equation is left parenthesis y minus y subscript 1 right parenthesis space equals space m left parenthesis x minus x subscript 1 right parenthesis.  This means that if you see a positive value after y or x, the coordinate is actually a negative value. 

Big Idea

When given an equation in point-slope form, use the x– and y–coordinates from the equation to plot one point on the line.  Next, use the information about slope to move vertically and horizontally from one point to find the location of another point.  Once you have two points plotted, connect them and extend the line to complete the graph. 


Using Standard Form

While the other two forms instantly provide information about the slope of a line, and either the y–intercept or some other point on the graph, equations written in standard form can seem unhelpful at first.  However, standard form allows us to easily find the line's x– and y–intercepts.  Recall that at our intercepts, one of x or y will have a value of zero; and because we have both the x–term and the y–term on one of our equations, when one of x or y is zero, the entire term has a value of zero.  This makes calculating intercepts rather easy, and if we can easily find both intercepts, we can easily graph the line. 

Graph the line 3 x space plus space 2 y space equals space 12

Recall that the x-intercept has the coordinate (x, 0).  To find the x-intercept, plug in zero for y and solve for x.  Similarly, the y-intercept has the coordinate (0, y).  Plug in zero for x and solve for y to find the y-intercept. 

Now that we have two points for our line, we can plot them and graph the line. 

Big Idea

Standard Form is ideal for finding both the x– and y–intercepts to a line.  Plug in 0 for y and solve for x in order to find the x-intercept, and plug in 0 for x and solve for y to find the y-intercept.  Once both intercepts are found, we can plot them on the graph and complete the graph of the line.