Learning Objectives
- Write the equation and draw the graph of a line using slope and y-intercept
- Write the equation of a line using slope and y-intercept
- Rearrange a linear equation so it is in slope-intercept form.
- Graph a line using slope and y-intercept
Slope-Intercept Form of a Line
When graphing a line we found one method we could use is to make a table of values. However, if we can identify some properties of the line, we may be able to make a graph much quicker and easier. One such method is finding the slope and the y-intercept of the equation. The slope can be represented by m and the y-intercept, where it crosses the axis and [latex]x=0[/latex], can be represented by [latex](0,b)[/latex] where b is the value where the graph crosses the vertical y-axis. Any other point on the line can be represented by [latex](x,y)[/latex].
In the equation,
[latex]y = mx + b[/latex]
[latex]\begin{array}{l}\,\,\,\,\,m\,\,\,\,=\,\,\,\text{slope}\\(x,y)=\,\,\,\text{a point on the line}\\\,\,\,\,\,\,\,b\,\,\,\,=\,\,\,\text{the y value of the y-intercept}\end{array}[/latex]
This formula is known as the slope-intercept equation. If we know the slope and the y-intercept we can easily find the equation that represents the line.
Example
Write the equation of the line that has a slope of [latex] \displaystyle \frac{1}{2}[/latex] and a y-intercept of [latex]−5[/latex].
We can also find the equation by looking at a graph and finding the slope and y-intercept.
Example
Write the equation of the line in the graph by identifying the slope and y-intercept.
We can also move the opposite direction, using the equation identify the slope and y-intercept and graph the equation from this information. However, it will be important for the equation to first be in slope intercept form. If it is not, we will have to solve it for y so we can identify the slope and the y-intercept.
Example
Write the following equation in slope-intercept form.
[latex]2x+4y=6[/latex]
Graphing Using Slope-Intercept
Once we have an equation in slope-intercept form we can graph it by first plotting the y-intercept, then using the slope, find a second point and connecting the dots.
Example
Graph [latex]y=\frac{1}{2}x-4[/latex] using the slope-intercept equation.
