Linear Functions: Learn It 4

Graphing a Linear Function from Point-Slope Form

Another way to graph a linear function is by using point-slope form. This form is useful when you know the slope and a single point on the line.

The point-slope form of a linear function is [latex]y - y_1 = m(x - x_1)[/latex] where [latex]m[/latex] is the slope and [latex](x_1,y_1)[/latex] is a point on the line.
Graph the function [latex]y - 2 = 3(x + 1)[/latex].

Method 1:

We can determine from the equation that the slope is [latex]m=3[/latex] and a point on the line is [latex](-1,2)[/latex].Step 1: Plot the point [latex](-1,2)[/latex].A graph with a single black dot at the point (–1, 2) on a coordinate plane.Step 2: Plot a second point using the slope. A slope of 3 means rise 3 and run 1.A coordinate plane showing two black dots: one at (–1, 2) and one at (0, 5).Step 3: Draw the line through the two points

A graph of a green line passing through the points (–1, 2) and (0, 5), both marked with black dots. The line slopes upward from left to right, showing a positive slope.

 

Method 2:

To verify, we can rewrite the equation in slope-intercept form:

[latex]\begin{align} y - 2 &= 3(x + 1) \\ y - 2 &= 3x + 3 \\ y &= 3x + 5 \end{align}[/latex]

This tells us the y-intercept is [latex](0, 5)[/latex] so now we can graph the line using the slope and y-intercept.

Graphing from point-slope form:

  1. Identify the point and the slope
  2. Graph the point
  3. Use the slope to graph a second point
  4. Draw the line through the two points

Graphing a Linear Function from Standard Form

The standard form of a linear equation is: [latex]Ax + By = C[/latex] where [latex]A[/latex] must be a nonnegative integer.

 You can graph a line from standard form in two ways: by finding intercepts or by rewriting the equation in slope-intercept form.

Method 1: Using Intercepts

Graph [latex]2x + 3y = 6[/latex].Method 1: InterceptsStep 1: Find the [latex]y[/latex]-intercept by setting [latex]x=0[/latex]:[latex]\begin{align}2(0)+3y&=6 \\ 3y &= 6 \\ y &= 2\end{align}[/latex]



Step 2: Find the [latex]x[/latex]-intercept by setting [latex]y=0[/latex]:


[latex]\begin{align} 2x + 3(0) &= 6 \\ 2x &= 6 \\ x &= 3 \end{align}[/latex]


Step 3: Plot the points [latex](0,2)[/latex] and [latex](0,3)[/latex]


A coordinate plane showing two black dots: one at the y-intercept (0, 2) and one at the x-intercept (3, 0).


then draw the line through them.


A graph of a purple line passing through the points (0, 2) and (3, 0), marked with black dots. The line slopes downward from left to right, indicating a negative slope.


Method 2: Slope-Intercept Form


You can also solve for [latex]y[/latex] and graph as usual.


[latex]\begin{array}{l}  2x + 3y = 6 \  3y = -2x + 6 \  y = -\frac{2}{3}x + 2  \end{array}[/latex]


Now plot the y-intercept [latex](0, 2)[/latex] and use the slope [latex]-\frac{2}{3}[/latex] (down 2, right 3) to plot the next points




Graphing from standard form using intercepts:


    

  1. Set [latex]x = 0[/latex] and solve for [latex]y[/latex] to find the [latex]y[/latex]-intercept
    2. 

  2. Set [latex]y = 0[/latex] and solve for [latex]x[/latex] to find the [latex]x[/latex]-intercept
    3. 

  3. Plot both intercepts
    4. 

  4. Draw the line through the two points
    5.