Suppose a company had the slope and \begin{align*}y\end{align*}-intercept of a line representing revenue based on units sold. How could the company graph this line? Does it have enough information? What if another company graphed its own revenue line, and it was parallel to the first company's line? Does this mean that the slopes of the two lines are the same, the \begin{align*}y\end{align*}-intercepts are the same, both are the same, or neither are the same? In this Concept, you'll learn to graph lines using slope-intercept form so that you can answer questions like these.
Guidance
Once we know the slope and the \begin{align*}y-\end{align*}intercept of an equation, it is quite easy to graph the solutions.
Example A
Graph the solutions to the equation \begin{align*}y=2x+5\end{align*}.
Solution:
The equation is in slope-intercept form. To graph the solutions to this equation, you should start at the \begin{align*}y-\end{align*}intercept. Then, using the slope, find a second coordinate. Finally, draw a line through the ordered pairs.
Example B
Graph the equation \begin{align*}y=-3x+5\end{align*}.
Solution:
Using the definition of slope-intercept form, this equation has a \begin{align*}y-\end{align*}intercept of (0, 5) and a slope of \begin{align*}\frac{-3}{1}\end{align*}.
Slopes of Parallel Lines
Parallel lines will never intersect, or cross. The only way for two lines never to cross is if the method of finding additional coordinates is the same.
Therefore, it's true that parallel lines have the same slope.
You will use this fact in later algebra lessons as well as in geometry.
Example C
Determine the slope of any line parallel to \begin{align*}y=-3x+5\end{align*}.
Solution:
Because parallel lines have the same slope, the slope of any line parallel to \begin{align*}y=-3x+5\end{align*} must also be –3.
Guided Practice
Graph \begin{align*}y=-\frac{2}{5}x\end{align*} by graphing the \begin{align*}y\end{align*}-intercept first, and then using the slope to find a second point to graph.
Solution:
First, graph the \begin{align*}y\end{align*}-intercept, which is \begin{align*}(0,0)\end{align*}.
Next, the slope is \begin{align*}-\frac{2}{5}\end{align*}. The negative can go along with the denominator or numerator. Either way, you will get a slope on the same line. Let the rise be 2 and the run be -5. This means, from the starting point, go to the left 5, and then go up 2:
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Graphs Using Slope-Intercept Form (11:11)
Plot the following functions on a graph.
- \begin{align*}y=2x+5\end{align*}
- \begin{align*}y=-0.2x+7\end{align*}
- \begin{align*}y=-x\end{align*}
- \begin{align*}y=3.75\end{align*}
- \begin{align*}\frac{2}{7} x-4=y\end{align*}
- \begin{align*}y=-4x+13\end{align*}
- \begin{align*}-2+\frac{3}{8} x=y\end{align*}
- \begin{align*}y=\frac{1}{2}+2x\end{align*}
In 9 – 16, state the slope of a line parallel to the line given.
- \begin{align*}y=2x+5\end{align*}
- \begin{align*}y=-0.2x+7\end{align*}
- \begin{align*}y=-x\end{align*}
- \begin{align*}y=3.75\end{align*}
- \begin{align*}y=-\frac{1}{5}x-11\end{align*}
- \begin{align*}y=-5x+5\end{align*}
- \begin{align*}y=-3x+11\end{align*}
- \begin{align*}y=3x+3.5\end{align*}
Mixed Review
- Graph \begin{align*}x = 4\end{align*} on a Cartesian plane.
- Solve for \begin{align*}g: |8-11|+4g=99\end{align*}.
- What is the order of operations? When is the order of operations used?
- Give an example of a negative irrational number.
- Give an example of a positive rational number.
- True or false: An integer will always be considered a rational number.