Section 4.2 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 equation's solution set. For example, the graph of \(y=-2x+3\) is shown in FigureĀ 4.2.2.(c). The graph plots the ordered pairs whose coordinates make \(y=-2x+3\) true. TableĀ 4.2.1 shows a few points that make the equation true.
| \(y=-2x+3\) | \((x,y)\) |
| \(\substitute{5}\stackrel{\checkmark}{=}-2(\substitute{-1})+3\) | \((\substitute{-1},\substitute{5})\) |
| \(\substitute{3}\stackrel{\checkmark}{=}-2(\substitute{0})+3\) | \((\substitute{0},\substitute{3})\) |
| \(\substitute{1}\stackrel{\checkmark}{=}-2(\substitute{1})+3\) | \((\substitute{1},\substitute{1})\) |
| \(\substitute{-1}\stackrel{\checkmark}{=}-2(\substitute{2})+3\) | \((\substitute{2},\substitute{-1})\) |
| \(\substitute{-3}\stackrel{\checkmark}{=}-2(\substitute{3})+3\) | \((\substitute{3},\substitute{-3})\) |
| \(\substitute{-5}\stackrel{\checkmark}{=}-2(\substitute{4})+3\) | \((\substitute{4},\substitute{-5})\) |
TableĀ 4.2.1 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 equation's graph. You can see them in FigureĀ 4.2.2.(a). But there are many more points that make the equation true. More points are plotted in FigureĀ 4.2.2.(b). Even more points are plotted in FigureĀ 4.2.2.(c) ā so many, that together the points look like a straight line.
Remark 4.2.3.
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 4.2.4.
Checkpoint 4.2.5.
So 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 (not-so-complicated) algebraic equations, connecting those points with a smooth curve will produce an excellent graph.
Example 4.2.6.
Let's 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. 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.
Remark 4.2.9.
Note that our choice of \(x\)-values is 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. For simpler calculations, people often start with the integers from \(-2\) to \(2\text{.}\) However, sometimes the equation has context that suggests using other \(x\)-values, as in the next examples.
Example 4.2.10.
The gas tank in Sofia's 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 galāmi. 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.
Choosing \(x\)-values from \(-2\) to \(2\text{,}\) as in our previous example, wouldn't 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 the graph from ExampleĀ 4.2.10, notice how both axes indicate units that help describe the meaning of each variable. Whenever a graph has real-world context, be sure to label both axes clearly with both variable name (like \(x\)) and units.
Example 4.2.13.
Plot a graph for the equation \(y=\frac{4}{3}x-4\text{.}\)
This equation doesn't 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 \(\frac{4}{3}\text{,}\) 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 4.2.16.
Build a table and graph the equation \(y=x^2\text{.}\) Use \(x\)-values from \(-3\) to \(3\text{.}\)
| \(x\) | \(y=x^2\) | 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. Since each \(x\)-value corresponds to a single \(y\)-value (the square of \(x\)) we connected the points with a smooth curve, sketching from left to right.
Exercises Exercises
Warmup
1.
Decide whether \((2,1)\) and \((-5,13)\) are on the graph of the equation \(y=-2x+3\text{.}\)
2.
Decide whether \((8,-7)\) and \((5,0)\) are on the graph of the equation \(y=-\frac{3}{2}x+5\text{.}\)
Testing Points as Solutions
3.
Consider the equation
\(y=6 x+4\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (0,7)\)
\(\displaystyle (4,30)\)
\(\displaystyle (-2,-8)\)
\(\displaystyle (-5,-26)\)
4.
Consider the equation
\(y=7 x+10\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (-4,-18)\)
\(\displaystyle (8,67)\)
\(\displaystyle (0,15)\)
\(\displaystyle (-2,-4)\)
5.
Consider the equation
\(y=-4 x - 5\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (0,-5)\)
\(\displaystyle (-4,11)\)
\(\displaystyle (-2,6)\)
\(\displaystyle (2,-13)\)
6.
Consider the equation
\(y=-3 x - 8\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (-6,10)\)
\(\displaystyle (6,-26)\)
\(\displaystyle (0,-8)\)
\(\displaystyle (-2,3)\)
7.
Consider the equation
\(y=\frac{8}{9} x-5\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (18,11)\)
\(\displaystyle (0,0)\)
\(\displaystyle (-9,-10)\)
\(\displaystyle (-27,-29)\)
8.
Consider the equation
\(y=\frac{2}{3} x-2\)
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
\(\displaystyle (0,0)\)
\(\displaystyle (-3,1)\)
\(\displaystyle (15,8)\)
\(\displaystyle (-15,-12)\)
Tables for Equations
9.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-x+3\) | Points |
| \(-3\) | \(6\) | \(\left(-3,6\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
10.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-x+4\) | Points |
| \(-3\) | \(7\) | \(\left(-3,7\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
11.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=4x+5\) | Points |
| \(-3\) | \(-7\) | \(\left(-3,-7\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
12.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=4x+2\) | Points |
| \(-3\) | \(-10\) | \(\left(-3,-10\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
13.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-3x+8\) | Points |
| \(-3\) | \(17\) | \(\left(-3,17\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
14.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-2x+5\) | Points |
| \(-3\) | \(11\) | \(\left(-3,11\right)\) |
| \(-2\) | ||
| \(-1\) | ||
| \(0\) | ||
| \(1\) | ||
| \(2\) |
15.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=\frac{9}{2} x - 5\) | Points |
| \(-6\) | \(-32\) | \(\left(-6,-32\right)\) |
| \(-4\) | ||
| \(-2\) | ||
| \(0\) | ||
| \(2\) | ||
| \(4\) |
16.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=\frac{9}{8} x +4\) | Points |
| \(-24\) | \(-23\) | \(\left(-24,-23\right)\) |
| \(-16\) | ||
| \(-8\) | ||
| \(0\) | ||
| \(8\) | ||
| \(16\) |
17.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-\frac{1}{6} x - 9\) | Points |
| \(-18\) | \(-6\) | \(\left(-18,-6\right)\) |
| \(-12\) | ||
| \(-6\) | ||
| \(0\) | ||
| \(6\) | ||
| \(12\) |
18.
Make a table for the equation.
The first row is an example.
| \(x\) | \(y=-\frac{3}{2} x - 7\) | Points |
| \(-6\) | \(2\) | \(\left(-6,2\right)\) |
| \(-4\) | ||
| \(-2\) | ||
| \(0\) | ||
| \(2\) | ||
| \(4\) |
19.
Make a table for the equation.
| \(x\) | \(y=-9x\) |
20.
Make a table for the equation.
| \(x\) | \(y=-4x\) |
21.
Make a table for the equation.
| \(x\) | \(y=2x-8\) |
22.
Make a table for the equation.
| \(x\) | \(y=4x+3\) |
23.
Make a table for the equation.
| \(x\) | \(y={\frac{8}{5}}x+5\) |
24.
Make a table for the equation.
| \(x\) | \(y={\frac{5}{4}}x+8\) |
25.
Make a table for the equation.
| \(x\) | \(y={\frac{1}{2}}x+10\) |
26.
Make a table for the equation.
| \(x\) | \(y=-{\frac{9}{17}}x - 9\) |
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 you'll 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 Ā°F and the pressure and the pressure \(P\) is measured in lbāin2. 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 \(y=-x^2-2x-3\text{.}\)