Section4.2Graphing Equations
Ā¶We have learned how to plot points in a coordinate system. Next, we will learn how to graph lines and curves that represent solutions to equations with two variables. We will begin by graphing equations that are written in form \(y=\ldots\) with only \(x\)-variables on the right, like \(y=-2x+5\) and \(y=x^2\text{.}\) These equations are not difficult to graph, as you'll see.
A graph of an equation is a graphical representation of that equation's solution set. For example, the graph of \(y=-2x+3\) is shown in FigureĀ 4.2.3.(c). The graph plots the ordered pairs whose \(x\)- and \(y\)-coordinates will make \(y=-2x+3\) true. TableĀ 4.2.2 shows that many points on the graph 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 4.2.2 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 equation's graph. You can see them in FigureĀ 4.2.3.(a). But there are many more points that make the equation true. More points are plotted in FigureĀ 4.2.3.(b). Even more points are plotted in FigureĀ 4.2.3.(c) ā so many, that together the points look like a straight line.
Generalizingā¦
Remark4.2.4
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.
Exercise4.2.5
Exercise4.2.6
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.
Example4.2.7
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 |
| \(-2\) | \(\phantom{-2(-2)+5=\substitute{9}}\) | \(\phantom{(-2,9)}\) |
| \(-1\) | \(\phantom{-2(-1)+5=\substitute{7}}\) | \(\phantom{(-1,7)}\) |
| \(0\) | \(\phantom{-2(0)+5=\substitute{5}}\) | \(\phantom{(0,5)}\) |
| \(1\) | \(\phantom{-2(1)+5=\substitute{3}}\) | \(\phantom{(1,3)}\) |
| \(2\) | \(\phantom{-2(2)+5=\substitute{1}}\) | \(\phantom{(2,1)}\) |
| \(x\) | \(y=-2x+5\) | Point |
| \(-2\) | \(-2(-2)+5=\substitute{9}\) | \((-2,9)\) |
| \(-1\) | \(-2(-1)+5=\substitute{7}\) | \((-1,7)\) |
| \(0\) | \(-2(0)+5=\substitute{5}\) | \((0,5)\) |
| \(1\) | \(-2(1)+5=\substitute{3}\) | \((1,3)\) |
| \(2\) | \(-2(2)+5=\substitute{1}\) | \((2,1)\) |
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.
Remark4.2.10
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 caclulations, 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 example.
Example4.2.11
One car's gas tank holds 14āÆgal of fuel. Over the course of a long road trip, that car uses its fuel at an average rate of 0.032āÆgalāmi. If a driver 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.
Solution
Table4.2.12Make the table Figure4.2.13Make the graph
Choosing \(x\)-values from \(-2\) to \(2\text{,}\) as in our previous example, wouldn't make sense here. Driving a negative number of miles is not possible, and any long road trip is longer than \(2\) miles. So in this context, choose \(x\)-values that reflect the number of miles one might drive in a day.
| \(x\) | \(y=14-0.032x\) | Point |
| \(20\) | \(13.36\) | \((20,13.36)\) |
| \(50\) | \(12.4\) | \((50,12.4)\) |
| \(80\) | \(11.44\) | \((80,11.44)\) |
| \(100\) | \(10.8\) | \((100,10.8)\) |
| \(200\) | \(7.6\) | \((200,7.6)\) |
Units on axes labels
In the graph from ExampleĀ 4.2.11, 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.
Example4.2.14
Build a table and graph the equation \(y=x^2\text{.}\) Use \(x\)-values from \(-3\) to \(3\text{.}\)
Solution
| \(x\) | \(y=x^2\) | 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.
Subsection4.2.1Exercises
Determine if the given points are to be included in the graph of the equation.
Make a table for the equation with \(x\)-values suggested.
Make a table for the equation.
These exercises have Cartesian plots with some context.
13
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. Make a table of at least five values and plot a graph of this equation.
Solution
Table4.2.15Make the table Figure4.2.16Make the graph
We'll include \(x=0\) representing the total expense right after buying and installing the equipment. Then we'll use some \(x\)-values several months and years into the future.
| \(x\) | \(y\) | Point |
| \(0\) | \(900\) | \((0,900)\) |
| \(1\) | \(931\) | \((1,931)\) |
| \(6\) | \(1086\) | \((6,1086)\) |
| \(12\) | \(1272\) | \((12,1272)\) |
| \(24\) | \(1644\) | \((24,1644)\) |
14
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. 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.
Solution
Table4.2.17Make the table Figure4.2.18Make the graph
We'll include \(x=0\) representing the loan balance at the time of purchase. Then we'll use \(x\)-values for each year during the life of the loan, all the way up to \(x=60\) when the loan is paid off.
| \(x\) | \(y\) | Point |
| \(0\) | \(18600\) | \((0,18600)\) |
| \(12\) | \(14880\) | \((12,14880)\) |
| \(24\) | \(11160\) | \((24,11160)\) |
| \(36\) | \(7440\) | \((36,7440)\) |
| \(48\) | \(3720\) | \((48,3720)\) |
| \(60\) | \(0\) | \((60,0)\) |
15
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 \(x\)-values that make sense in context.
Solution
Table4.2.19Make the table Figure4.2.20Make the graph
| \(T\) | \(P\) | Point |
| \(20\) | \(\approx94.16\) | \((20,94.16)\) |
| \(40\) | \(\approx98.09\) | \((40,98.09)\) |
| \(60\) | \(\approx102.0\) | \((60,102.0)\) |
| \(80\) | \(\approx105.9\) | \((80,105.9)\) |
| \(100\) | \(\approx109.9\) | \((100,109.9)\) |
16
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.
Solution
Table4.2.21Make the table Figure4.2.22Make the graph
| \(x\) | \(y\) | Point |
| \(1\) | \(1400\) | \((1,1400)\) |
| \(5\) | \(280\) | \((5,280)\) |
| \(10\) | \(140\) | \((10,140)\) |
| \(20\) | \(70\) | \((20,70)\) |
| \(30\) | \(46.67\) | \((30,46.67)\) |
17
Create a table of ordered pairs and then make a plot of the equation \(y=2x+3\text{.}\)
Solution
Table4.2.23A possible table
| \(x\) | \(y=2x+3\) | Point |
| \(-2\) | \(-1\) | \((-2,-1)\) |
| \(-1\) | \(1\) | \((-1,1)\) |
| \(0\) | \(3\) | \((0,3)\) |
| \(1\) | \(5\) | \((1,5)\) |
| \(2\) | \(7\) | \((2,7)\) |
18
Create a table of ordered pairs and then make a plot of the equation \(y=3x+5\text{.}\)
Solution
Table4.2.24A possible table
| \(x\) | \(y=3x+5\) | Point |
| \(-2\) | \(-1\) | \((-2,-1)\) |
| \(-1\) | \(2\) | \((-1,2)\) |
| \(0\) | \(5\) | \((0,5)\) |
| \(1\) | \(8\) | \((1,8)\) |
| \(2\) | \(11\) | \((2,11)\) |
19
Create a table of ordered pairs and then make a plot of the equation \(y=-4x+1\text{.}\)
Solution
Table4.2.25A possible table
| \(x\) | \(y=-4x+1\) | Point |
| \(-2\) | \(9\) | \((-2,9)\) |
| \(-1\) | \(5\) | \((-1,5)\) |
| \(0\) | \(1\) | \((0,1)\) |
| \(1\) | \(-3\) | \((1,-3)\) |
| \(2\) | \(-7\) | \((2,-7)\) |
20
Create a table of ordered pairs and then make a plot of the equation \(y=-x-4\text{.}\)
Solution
Table4.2.26A possible table
| \(x\) | \(y=-x-4\) | Point |
| \(-2\) | \(-2\) | \((-2,-2)\) |
| \(-1\) | \(-3\) | \((-1,-3)\) |
| \(0\) | \(-4\) | \((0,-4)\) |
| \(1\) | \(-5\) | \((1,-5)\) |
| \(2\) | \(-6\) | \((2,-6)\) |
21
Create a table of ordered pairs and then make a plot of the equation \(y=\frac{5}{2}x\text{.}\)
Solution
Table4.2.27A possible table
| \(x\) | \(y=\frac{5}{2}x\) | Point |
| \(-2\) | \(-5\) | \((-2,-5)\) |
| \(-1\) | \(-\frac{5}{2}\) | \((-1,-\frac{5}{2})\) |
| \(0\) | \(0\) | \((0,0)\) |
| \(1\) | \(\frac{5}{2}\) | \((1,\frac{5}{2})\) |
| \(2\) | \(5\) | \((2,5)\) |
22
Create a table of ordered pairs and then make a plot of the equation \(y=\frac{4}{3}x\text{.}\)
Solution
Table4.2.28A possible table
| \(x\) | \(y=\frac{4}{3}x\) | Point |
| \(-3\) | \(-4\) | \((-3,-4)\) |
| \(0\) | \(0\) | \((0,0)\) |
| \(3\) | \(4\) | \((3,4)\) |
23
Create a table of ordered pairs and then make a plot of the equation \(y=-\frac{2}{5}x-3\text{.}\)
Solution
Table4.2.29A possible table
| \(x\) | \(y=-\frac{2}{5}x-3\) | Point |
| \(-5\) | \(-1\) | \((-5,-1)\) |
| \(0\) | \(-3\) | \((0,-3)\) |
| \(5\) | \(-5\) | \((5,-5)\) |
24
Create a table of ordered pairs and then make a plot of the equation \(y=-\frac{3}{4}x+2\text{.}\)
Solution
Table4.2.30A possible table
| \(x\) | \(y=-\frac{3}{4}x+2\) | Point |
| \(-4\) | \(5\) | \((-4,5)\) |
| \(0\) | \(2\) | \((0,2)\) |
| \(4\) | \(-1\) | \((4,-1)\) |
25
Create a table of ordered pairs and then make a plot of the equation \(y=x^2+1\text{.}\)
Solution
Table4.2.31A possible table
| \(x\) | \(y=x^2+1\) | Point |
| \(-2\) | \(5\) | \((-2,5)\) |
| \(-1\) | \(2\) | \((-1,2)\) |
| \(0\) | \(1\) | \((0,1)\) |
| \(1\) | \(2\) | \((1,2)\) |
| \(2\) | \(5\) | \((2,5)\) |
26
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 \(4\text{.}\)
Solution
Table4.2.32A possible table
| \(x\) | \(y=(x-2)^2\) | Point |
| \(0\) | \(4\) | \((0,4)\) |
| \(1\) | \(1\) | \((1,1)\) |
| \(2\) | \(0\) | \((2,0)\) |
| \(3\) | \(1\) | \((3,1)\) |
| \(4\) | \(4\) | \((4,4)\) |
27
Create a table of ordered pairs and then make a plot of the equation \(y=-3x^2\text{.}\)
Solution
Table4.2.33A possible table
| \(x\) | \(y=-3x^2\) | Point |
| \(-2\) | \(-12\) | \((-2,-12)\) |
| \(-1\) | \(-3\) | \((-1,3)\) |
| \(0\) | \(0\) | \((0,0)\) |
| \(1\) | \(-3\) | \((1,-3)\) |
| \(2\) | \(-12\) | \((2,-12)\) |
28
Create a table of ordered pairs and then make a plot of the equation \(y=-x^2-2x-3\text{.}\)
Solution
Table4.2.34A possible table
| \(x\) | \(y=-3x^2\) | Point |
| \(-2\) | \(-3\) | \((-2,-3)\) |
| \(-1\) | \(-2\) | \((-1,-2)\) |
| \(0\) | \(-3\) | \((0,-3)\) |
| \(1\) | \(-6\) | \((1,-6)\) |
| \(2\) | \(-11\) | \((2,-11)\) |