Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we'll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables, we graphed the line of the equation \(y=\frac{1}{2} x+3\) by plotting points. See Figure 4.24. Let's find the slope of this line.

This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.

Figure 4.24

The red lines show us the rise is \(1\) and the run is \(2\). Substituting into the slope formula:

\( \begin{align} \begin{array}{l} m=\frac{\text { rise }}{\operatorname{run}} \\ m=\frac{1}{2} \end{array} \end{align}\)

What is the \(y\)-intercept of the line? The \(y\)-intercept is where the line crosses the \(y\)-axis, so \(y\)-intercept is \(( 0,3)\). The equation of this line is:

\(y=\frac{1}{2} x+3\)

Notice, the line has:

\( \text { slope } m=\frac{1}{2} \text { and } y \text { -intercept }(0,3)\)

When a linear equation is solved for y , the coefficient of the x term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation y =1 2 x +3 is in slope–intercept form.

\(m=\frac{1}{2} ; y \text { -intercept is }(0,3)\)

\(\begin{align} \begin{array}{l} y=\frac{1}{2} x+3 \\ y=m x+ b \end{array} \end{align}\)

Slope-Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope \(m\) and \(y\)-intercept, \(( 0,b)\) is,

\(y=m x+b\)

Sometimes the slope–intercept form is called the "y-form".

Example 4.40

Use the graph to find the slope and y-intercept of the line, \(y=2x+1\).

Compare these values to the equation \(y=mx+b\).

Solution

To find the slope of the line, we need to choose two points on the line. We’ll use the points \(( 0,1)\)  and \((1,3)\).

  to choose two points on the line. We’ll use the points (0,1) and (1,3).
Find the rise and run. \(m=\frac{\text { rise }}{\text { run }}\)
  \(m=\frac{2}{1}\)
  \(m=2\)
Find the y-intercept of the line. The \(y\)-intercept is the point \((0,1)\).
We found slope \(m=2\) and \(y\) -intercept \((0,1)\).
 \(y=2 x+1\)
\(y=m x+b\)


The slope is the same as the coefficient of x  and the y-coordinate of the y-intercept is the same as the constant term.

Try It 4.79

Use the graph to find the slope and y-intercept of the line \(y=\frac{2}{3} x-1\). Compare these values to the equation \(y=m x+b\).

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of

Try It 4.80

Use the graph to find the slope and y-intercept of the line \(y=\frac{1}{2} x+3\). Compare these values to the equation \(y =m x +b\).

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of

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