Lane ORCCA (2020–2021): Open Resources for Community College Algebra

Example 4.9.1. Building a Table of \(x\)- and \(y\)-values.

First, we will graph \(y=-2x+1\) by building a table of values. In theory this method can be used for any type of equation, linear or not. Every student must feel comfortable with building a table of values based on an equation.

\(x\)-value	\(y\)-value	Point
\(\highlight{-2}\)	\(y=-2(\substitute{-2})+1=5\)	\((-2,5)\)
\(\highlight{-1}\)	\(y=-2(\substitute{-1})+1=3\)	\((-1,3)\)
\(\highlight{0}\)	\(y=-2(\substitute{0})+1=1\)	\((0,1)\)
\(\highlight{1}\)	\(y=-2(\substitute{1})+1=-1\)	\((1,-1)\)
\(\highlight{2}\)	\(y=-2(\substitute{2})+1=-3\)	\((2,-3)\)

Example 4.9.4. Using Slope Triangles.

Although making a table is straightforward, the slope triangle method is both faster and reinforces the true meaning of slope. In the slope triangle method, we first identify some point on the line. With a line in slope-intercept form, we know the \(y\)-intercept, which is \((0,1)\text{.}\) Then, we can draw slope triangles in both directions to find more points.

Compared to the table method, the slope triangle method:

• is less straightforward
• doesn't take the time and space to make a table
• doesn't involve lots of calculations where you might make a human error
• shows slope triangles, which reinforces the meaning of slope

Example 4.9.7. Using intercepts.

If we use the \(x\)- and \(y\)-intercepts to plot \(y=-2x+1\text{,}\) we have some calculation to do. While it is apparent that the \(y\)-intercept is at \((0,1)\text{,}\) where is the \(x\)-intercept? Here are two methods to find it.

So the \(x\)-intercept is at \(\left(\frac{1}{2},0\right)\text{.}\) Plotting both intercepts:

This worked, but here are some observations about why this method is not the greatest.

• We had to plot a point with fractional coordinates.
• We only plotted two points and they turned out very close to each other, so even the slightest inaccuracy in our drawing skills could result in a line that is way off.

Example 4.9.10. Building a Table of \(x\)- and \(y\)-values.

To make a table, we could substitute \(x\) for various numbers and use algebra to find the corresponding \(y\)-values. Let's start with \(x=-2\text{,}\) planning to move on to \(x=-1,0,1,2\text{.}\)

Example 4.9.11. Using intercepts.

Next, we will try graphing \(3x+4y=12\) using intercepts. We set up a small table to record the two intercepts:

So the line's \(x\)-intercept is at \((4,0)\) and its \(y\)-intercept is at \((0,3)\text{.}\) Now we can complete the table and then graph the line:

	\(x\)-value	\(y\)-value	Intercepts
\(x\)-intercept	\(4\)	\(0\)	\((4,0)\)
\(y\)-intercept	\(0\)	\(3\)	\((0,3)\)

Example 4.9.14. With Slope Triangles.

With the \(y\)-intercept at \((0,3)\) and slope \(-\frac{3}{4}\text{,}\) we can graph the line using slope triangles:

Compared with the intercepts method, the slope triangle method takes more time, but shows more points with slope triangles, and thus a more accurate graph. Also sometimes (as with Example 4.7.13) when we graph a standard form equation like \(2x-3y=0\text{,}\) the intercepts method doesn't work because both intercepts are actually at the same point, and we have to resort to something else like slope triangles anyway.