Section 3.2 Graphing Equations
We have graphed points in a coordinate system, and now we will graph lines and curves.
A graph of an equation is a picture of that equationâs solution set. For example, the graph of \(y=-2x+3\) is shown in Figure 3.2.3.(c). The graph plots the ordered pairs whose coordinates make \(y=-2x+3\) true. Figure 2 shows a few points that make the equation true.
| \(y=-2x+3\) | \((x,y)\) |
| \(\substitute{5}\confirm{=}-2(\substitute{-1})+3\) | \((\substitute{-1},\substitute{5})\) |
| \(\substitute{3}\confirm{=}-2(\substitute{0})+3\) | \((\substitute{0},\substitute{3})\) |
| \(\substitute{1}\confirm{=}-2(\substitute{1})+3\) | \((\substitute{1},\substitute{1})\) |
| \(\substitute{-1}\confirm{=}-2(\substitute{2})+3\) | \((\substitute{2},\substitute{-1})\) |
| \(\substitute{-3}\confirm{=}-2(\substitute{3})+3\) | \((\substitute{3},\substitute{-3})\) |
| \(\substitute{-5}\confirm{=}-2(\substitute{4})+3\) | \((\substitute{4},\substitute{-5})\) |
Figure 2 tells us that the points \((-1,5)\text{,}\) \((0,3)\text{,}\) \((1,1)\text{,}\) \((2,-1)\text{,}\) \((3,-3)\text{,}\) and \((4,-5)\) are all solutions to the equation \(y=-2x+3\text{,}\) and so they should all be shaded as part of that equationâs graph. You can see them in Figure 3.2.3.(a). But there are many more points that make the equation true. More points are plotted in Figure 3.2.3.(b). Even more points are plotted in Figure 3.2.3.(c)âso many, that together the points look like a straight line.
The graph of an equation shades all the points \((x,y)\) that make the equation true once the \(x\)- and \(y\)-values are substituted in. Typically, there are so many points shaded, that the final graph appears to be a continuous line or curve that you could draw with one stroke of a pen.
Checkpoint 3.2.4.
The point \((4,-5)\) is on the graph in Figure 3.2.3.(c). What happens when you substitute these values into the equation \(y=-2x+3\text{?}\)
\(y\) |
\(=\) | \(-2x+3\) |
| \(\wonder{=}\) |
This equation is
- true
- false
.
Checkpoint 3.2.5.
Decide whether \((5,-2)\) and \((-10,-7)\) are on the graph of the equation \(y=-\frac{3}{5}x+1\text{.}\)
At \((5,-2)\text{:}\)
\(y\) |
\(=\) | \(-\frac{3}{5}x+1\) |
| \(\wonder{=}\) |
This equation is
- true
- false
and \((5,-2)\) is
- part of
- not part of
the graph of \(y=-\frac{3}{5}x+1\text{.}\)
At \((-10,-7)\text{:}\)
\(y\) |
\(=\) | \(-\frac{3}{5}x+1\) |
| \(\wonder{=}\) |
This equation is
- true
- false
and \((-10,-7)\) is
- part of
- not part of
the graph of \(y=-\frac{3}{5}x+1\text{.}\)
Explanation.
If the point \((5,-2)\) is on \(y=-\frac{3}{5}x+1\text{,}\) once we substitute \(x=5\) and \(y=-2\) into the lineâs equation, the equation should be true. Letâs try:
\begin{equation*}
\begin{aligned}
y\amp=-\frac{3}{5}x+1\\
-2\amp\wonder{=}-\frac{3}{5}(5)+1\\
-2\amp\confirm{=} -3+1
\end{aligned}
\end{equation*}
Because this last equation is true, we can say that \((5,-2)\) is on the graph of \(y=-\frac{3}{5}x+1\text{.}\)
However if we substitute \(x=-10\) and \(y=-7\) into the equation, it leads to \(-7=7\text{,}\) which is false. This tells us that \((-10,-7)\) is not on the graph.
To make our own graph of an equation with two variables \(x\) and \(y\text{,}\) we can choose some reasonable \(x\)-values, then calculate the corresponding \(y\)-values, and then plot the \((x,y)\)-pairs as points. For many algebraic equations, connecting those points with a smooth curve will produce an excellent graph.
Example 3.2.6.
Letâs plot a graph for the equation \(y=-2x+5\text{.}\) We use a table to organize our work:
| \(x\) | \(y=-2x+5\) | Point |
| \(-2\) | \(\phantom{-2(\substitute{-2})+5=\highlight{9}}\) | \(\phantom{(-2,9)}\) |
| \(-1\) | \(\phantom{-2(\substitute{-1})+5=\highlight{7}}\) | \(\phantom{(-1,7)}\) |
| \(0\) | \(\phantom{-2(\substitute{0})+5=\highlight{5}}\) | \(\phantom{(0,5)}\) |
| \(1\) | \(\phantom{-2(\substitute{1})+5=\highlight{3}}\) | \(\phantom{(1,3)}\) |
| \(2\) | \(\phantom{-2(\substitute{2})+5=\highlight{1}}\) | \(\phantom{(2,1)}\) |
| \(x\) | \(y=-2x+5\) | Point |
| \(-2\) | \(-2(\substitute{-2})+5=\highlight{9}\) | \((-2,9)\) |
| \(-1\) | \(-2(\substitute{-1})+5=\highlight{7}\) | \((-1,7)\) |
| \(0\) | \(-2(\substitute{0})+5=\highlight{5}\) | \((0,5)\) |
| \(1\) | \(-2(\substitute{1})+5=\highlight{3}\) | \((1,3)\) |
| \(2\) | \(-2(\substitute{2})+5=\highlight{1}\) | \((2,1)\) |
We use points from the table to graph the equation in Figure 8. First, we need a coordinate system to draw on. We will eventually need to see the \(x\)-values \(-2\text{,}\) \(-1\text{,}\) \(0\text{,}\) \(1\text{,}\) and \(2\text{.}\) So drawing an \(x\)-axis that runs from \(-5\) to \(5\) will be good enough. We will eventually need to see the \(y\)-values \(9\text{,}\) \(7\text{,}\) \(5\text{,}\) \(3\text{,}\) and \(1\text{.}\) So drawing a \(y\)-axis that runs from \(-1\) to \(10\) will be good enough.
Axes should always be labeled using the variable names they represent. In this case, with â\(x\)â and â\(y\)â. The axes should have tick marks that are spaced evenly. In this case it is fine to place the tick marks one unit apart (on both axes). Labeling at least some of the tick marks is necessary for a reader to understand the scale. Here we label the even-numbered ticks.
Then, connect the points with a smooth curve. Here, the curve is a straight line. Lastly, we can communicate that the graph extends further by sketching arrows on both ends of the line.
All that we have decided so far is drawn in Figure (a).
Next we carefully plot each point from the table in Figure (b). Itâs not necessary to label each pointâs coordinates, but we do so here.
Last we connect the points with a smooth curve. Here, the âcurveâ is a straight line. We communicate that the line extends further by putting arrowheads on both ends.
Remark 3.2.9.
Note that our choice of \(x\)-values in the table was arbitrary. As long as we determine the coordinates of enough points to indicate the behavior of the graph, we may choose whichever \(x\)-values we like. Having a few negative \(x\)-values will be good. For simpler calculations, itâs fine to choose \(-2\text{,}\) \(-1\text{,}\) \(0\text{,}\) \(1\text{,}\) and \(2\text{.}\) However sometimes the equation has context that suggests using other \(x\)-values, as in the next examples.
Example 3.2.10.
The gas tank in Sofiaâs car holds 14 gal of fuel. Over the course of a long road trip, her car uses fuel at an average rate of 0.032 galâmi. If Sofia fills the tank at the beginning of a long trip, then the amount of fuel remaining in the tank, \(y\text{,}\) after driving \(x\) miles is given by the equation \(y=14-0.032x\text{.}\) Make a suitable table of values and graph this equation.
Explanation.
Choosing \(x\)-values from \(-2\) to \(2\text{,}\) as in our previous example, wouldnât make sense here. Sofia cannot drive a negative number of miles, and any long road trip is longer than \(2\) miles. So in this context, choose \(x\)-values that reflect the number of miles Sofia might drive in a day.
| \(x\) | \(y=14-0.032x\) | Point |
| \(20\) | \(13.36\) | \((20,13.36)\) |
| \(50\) | \(12.4\) | \((50,12.4)\) |
| \(80\) | \(11.44\) | \((80,11.44)\) |
| \(100\) | \(10.8\) | \((100,10.8)\) |
| \(200\) | \(7.6\) | \((200,7.6)\) |
In Figure 12, notice how both axes are also labeled with units (âgallonsâ and âmilesâ). When the equation you plot has context like this example, including the units in the axes labels is very important to help anyone who reads your graph to understand it better.
In the table, \(x\)-values ran from \(20\) to \(200\text{,}\) while \(y\)-values ran from \(13.36\) down to \(7.6\text{.}\) So it was appropriate to make the \(x\)-axis cover something like from \(0\) to \(250\text{,}\) and make the \(y\)-axis cover something like from \(0\) to \(20\text{.}\) The scales on the two axes ended up being different.
Example 3.2.13.
Plot a graph for the equation \(y=\frac{4}{3}x-4\text{.}\)
Explanation.
This equation doesnât have any context to help us choose \(x\)-values for a table. We could use \(x\)-values like \(-2\text{,}\) \(-1\text{,}\) and so on. But note the fraction in the equation. If we use an \(x\)-value like \(-2\text{,}\) we will have to multiply by the fraction \(\frac{4}{3}\) which will leave us still holding a fraction. And then we will have to subtract \(4\) from that fraction. Since we know that everyone can make mistakes with that kind of arithmetic, maybe we can avoid it with a more wise selection of \(x\)-values.
If we use only multiples of \(3\) for the \(x\)-values, then multiplying by \(\frac{4}{3}\) will leave us with an integer, which will be easy to subtract \(4\) from. So we decide to use \(-6\text{,}\) \(-3\text{,}\) \(0\text{,}\) \(3\text{,}\) and \(6\) for \(x\text{.}\)
| \(x\) | \(y=\frac{4}{3}x-4\) | Point |
| \(-6\) | \(\phantom{\frac{4}{3}(\substitute{-6})-4=\highlight{-12}}\) | \(\phantom{(-6,-12)}\) |
| \(-3\) | \(\phantom{\frac{4}{3}(\substitute{-3})-4=\highlight{-8}}\) | \(\phantom{(-3,-8)}\) |
| \(0\) | \(\phantom{\frac{4}{3}(\substitute{0})-4=\highlight{-4}}\) | \(\phantom{(0,-4)}\) |
| \(3\) | \(\phantom{\frac{4}{3}(\substitute{3})-4=\highlight{0}}\) | \(\phantom{(3,0)}\) |
| \(6\) | \(\phantom{\frac{4}{3}(\substitute{6})-4=\highlight{4}}\) | \(\phantom{(6,4)}\) |
| \(x\) | \(y=\frac{4}{3}x-4\) | Point |
| \(-6\) | \(\frac{4}{3}(\substitute{-6})-2=\highlight{-12}\) | \((-6,-12)\) |
| \(-3\) | \(\frac{4}{3}(\substitute{-3})-2=\highlight{-8}\) | \((-3,-8)\) |
| \(0\) | \(\frac{4}{3}(\substitute{0})-2=\highlight{-4}\) | \((0,-4)\) |
| \(3\) | \(\frac{4}{3}(\substitute{3})-2=\highlight{0}\) | \((3,0)\) |
| \(6\) | \(\frac{4}{3}(\substitute{6})-2=\highlight{4}\) | \((6,4)\) |
We use points from the table to graph the equation. First, plot each point carefully. Then, connect the points with a smooth curve. Here, the curve is a straight line. Lastly, we can communicate that the graph extends further by sketching arrows on both ends of the line.
Not all equations make a straight line once they are plotted.
Example 3.2.16.
Build a table and graph the equation \(y=x^2\text{.}\) Use \(x\)-values from \(-3\) to \(3\text{.}\)
Explanation.
| \(x\) | \(y=x^2\) | Point |
| \(-3\) | \((-3)^2=9\) | \((-3,9)\) |
| \(-2\) | \((-2)^2=4\) | \((-2,4)\) |
| \(-1\) | \((-1)^2=1\) | \((-1,1)\) |
| \(0\) | \((0)^2=0\) | \((0,0)\) |
| \(1\) | \((1)^2=1\) | \((0,1)\) |
| \(2\) | \((2)^2=4\) | \((2,4)\) |
| \(3\) | \((3)^2=9\) | \((3,9)\) |
In this example, the points do not fall on a straight line. Many algebraic equations have graphs that are non-linear, where the points do not fall on a straight line. We connected the points with a smooth curve, sketching from left to right.
Reading Questions Reading Questions
1.
When a point like \((5,8)\) is on the graph of an equation, where the equation has variables \(x\) and \(y\text{,}\) what happens when you substitute in \(5\) for \(x\) and \(8\) for \(y\text{?}\)
2.
What are all the things to label when you set up a Cartesian coordinate system?
3.
When you start making a table for some equation, you have to choose some \(x\)-values. Explain three different ways to choose those \(x\)-values that were demonstrated in this section.
4.
What is an example of an equation that does not make a straight line once you make a graph of it?
Exercises Exercises
Testing Points as Solutions.
1.
Consider the equation
\(y=3 x+7\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (0,8)\)
- \(\displaystyle (-6,-11)\)
- \(\displaystyle (-8,-17)\)
- \(\displaystyle (4,24)\)
2.
Consider the equation
\(y=4 x+4\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (-8,-28)\)
- \(\displaystyle (-5,-16)\)
- \(\displaystyle (8,40)\)
- \(\displaystyle (0,8)\)
3.
Consider the equation
\(y=-7 x - 2\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (3,-23)\)
- \(\displaystyle (-8,55)\)
- \(\displaystyle (-7,47)\)
- \(\displaystyle (0,-2)\)
4.
Consider the equation
\(y=-6 x - 5\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (0,-5)\)
- \(\displaystyle (7,-47)\)
- \(\displaystyle (-9,49)\)
- \(\displaystyle (-8,47)\)
5.
Consider the equation
\(y=\frac{2}{7} x-5\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (0,0)\)
- \(\displaystyle (-7,-7)\)
- \(\displaystyle (-28,-12)\)
- \(\displaystyle (35,5)\)
6.
Consider the equation
\(y=\frac{6}{7} x-3\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (-28,-23)\)
- \(\displaystyle (21,15)\)
- \(\displaystyle (-7,-9)\)
- \(\displaystyle (0,0)\)
7.
Consider the equation
\(y=-\frac{5}{8} x-5\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (16,-13)\)
- \(\displaystyle (-24,13)\)
- \(\displaystyle (0,-5)\)
- \(\displaystyle (-40,20)\)
8.
Consider the equation
\(y=-\frac{3}{8} x-2\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
- \(\displaystyle (0,-2)\)
- \(\displaystyle (-16,4)\)
- \(\displaystyle (8,-4)\)
- \(\displaystyle (-40,16)\)
Tables for Equations.
9.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-x+1\) | Points |
| \(-3\) | \(4\) | \(\left(-3,4\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-x+1\) |
10.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-x+2\) | Points |
| \(-3\) | \(5\) | \(\left(-3,5\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-x+2\) |
11.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=3x+1\) | Points |
| \(-3\) | \(-8\) | \(\left(-3,-8\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=3x+1\) |
12.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=3x+8\) | Points |
| \(-3\) | \(-1\) | \(\left(-3,-1\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=3x+8\) |
13.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-3x+5\) | Points |
| \(-3\) | \(14\) | \(\left(-3,14\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-3x+5\) |
14.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-3x+1\) | Points |
| \(-3\) | \(10\) | \(\left(-3,10\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-3x+1\) |
15.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=\frac{7}{8} x - 7\) | Points |
| \(-24\) | \(-28\) | \(\left(-24,-28\right)\) |
| \(-16\) | ||
| \(-8\) | ||
| \(0\) | ||
| \(8\) | ||
| \(16\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=\frac{7}{8} x - 7\) |
16.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=\frac{9}{4} x +2\) | Points |
| \(-12\) | \(-25\) | \(\left(-12,-25\right)\) |
| \(-8\) | ||
| \(-4\) | ||
| \(0\) | ||
| \(4\) | ||
| \(8\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=\frac{9}{4} x +2\) |
17.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-\frac{9}{2} x +10\) | Points |
| \(-6\) | \(37\) | \(\left(-6,37\right)\) |
| \(-4\) | ||
| \(-2\) | ||
| \(0\) | ||
| \(2\) | ||
| \(4\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-\frac{9}{2} x +10\) |
18.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-\frac{1}{8} x - 2\) | Points |
| \(-24\) | \(1\) | \(\left(-24,1\right)\) |
| \(-16\) | ||
| \(-8\) | ||
| \(0\) | ||
| \(8\) | ||
| \(16\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-\frac{1}{8} x - 2\) |
19.
Make a table for the equation.
| \(x\) | \(y=-14x\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-14x\) |
20.
Make a table for the equation.
| \(x\) | \(y=-10x\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-10x\) |
21.
Make a table for the equation.
| \(x\) | \(y=-3x+2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-3x+2\) |
22.
Make a table for the equation.
| \(x\) | \(y=2x-4\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=2x-4\) |
23.
Make a table for the equation.
| \(x\) | \(y={\frac{19}{6}}x+2\) |
| Values of \(x\) and \(y\) satisfying the equation \(y={\frac{19}{6}}x+2\) |
24.
Make a table for the equation.
| \(x\) | \(y={\frac{6}{5}}x+5\) |
| Values of \(x\) and \(y\) satisfying the equation \(y={\frac{6}{5}}x+5\) |
25.
Make a table for the equation.
| \(x\) | \(y=-{\frac{11}{8}}x+7\) |
| Values of \(x\) and \(y\) satisfying the equation \(y=-{\frac{11}{8}}x+7\) |
26.
Make a table for the equation.
| \(x\) | \(y={\frac{1}{5}}x+9\) |
| Values of \(x\) and \(y\) satisfying the equation \(y={\frac{1}{5}}x+9\) |
Cartesian Plots in Context.
27.
A certain water heater will cost you \(\$900\) to buy and have installed. This water heater claims that its operating expense (money spent on electricity or gas) will be about \(\$31\) per month. According to this information, the equation \(y=900+31x\) models the total cost of the water heater after \(x\) months, where \(y\) is in dollars. Make a table of at least five values and plot a graph of this equation.
28.
You bought a new Toyota Corolla for \(\$18{,}600\) with a zero interest loan over a five-year period. That means youâll have to pay \(\$310\) each month for the next five years (sixty months) to pay it off. According to this information, the equation \(y=18600-310x\) models the loan balance after \(x\) months, where \(y\) is in dollars. Make a table of at least five values and plot a graph of this equation. Make sure to include a data point representing when you will have paid off the loan.
29.
The pressure inside a full propane tank will rise and fall if the ambient temperature rises and falls. The equation \(P=0.1963(T+459.67)\) models this relationship, where the temperature \(T\) is measured in °F and the pressure and the pressure \(P\) is measured in lbâin2. Make a table of at least five values and plot a graph of this equation. Make sure to use \(T\)-values that make sense in context.
30.
A beloved coworker is retiring and you want to give her a gift of week-long vacation rental at the coast that costs \(\$1400\) for the week. You might end up paying for it yourself, but you ask around to see if the other \(29\) office coworkers want to split the cost evenly. The equation \(y=\frac{1400}{x}\) models this situation, where \(x\) people contribute to the gift, and \(y\) is the dollar amount everyone contributes. Make a table of at least five values and plot a graph of this equation. Make sure to use \(x\)-values that make sense in context.
Graphs of Equations.
31.
Create a table of ordered pairs and then make a plot of the equation \(y=2x+3\text{.}\)
32.
Create a table of ordered pairs and then make a plot of the equation \(y=3x+5\text{.}\)
33.
Create a table of ordered pairs and then make a plot of the equation \(y=-4x+1\text{.}\)
34.
Create a table of ordered pairs and then make a plot of the equation \(y=-x-4\text{.}\)
35.
Create a table of ordered pairs and then make a plot of the equation \(y=\frac{5}{2}x\text{.}\)
36.
Create a table of ordered pairs and then make a plot of the equation \(y=\frac{4}{3}x\text{.}\)
37.
Create a table of ordered pairs and then make a plot of the equation \(y=-\frac{2}{5}x-3\text{.}\)
38.
Create a table of ordered pairs and then make a plot of the equation \(y=-\frac{3}{4}x+2\text{.}\)
39.
Create a table of ordered pairs and then make a plot of the equation \(y=x^2+1\text{.}\)
40.
Create a table of ordered pairs and then make a plot of the equation \(y=(x-2)^2\text{.}\) Use \(x\)-values from \(0\) to \(4\text{.}\)
41.
Create a table of ordered pairs and then make a plot of the equation \(y=-3x^2\text{.}\)
42.
Create a table of ordered pairs and then make a plot of the equation \(y=-x^2-2x-3\text{.}\)
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