Section4.2Graphing Equations
ΒΆWe 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 is shown in Figure 4.2.3.(c). The graph plots the ordered pairs whose coordinates make true. Table 4.2.2 shows a few points that make the equation true.
Table 4.2.2 tells us that the points and are all solutions to the equation and so they should all be shaded as part of that equation's graph. You can see them in Figure 4.2.3.(a). But there are many more points that make the equation true. More points are plotted in Figure 4.2.3.(b). Even more points are plotted in Figure 4.2.3.(c) β so many, that together the points look like a straight line.
Generalizingβ¦
Remark4.2.4
The graph of an equation shades all the points that make the equation true once the - and -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.
Checkpoint4.2.5
Checkpoint4.2.6
So to make our own graph of an equation with two variables and we can choose some reasonable -values, then calculate the corresponding -values, and then plot the -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 We use a table to organize our work:
Point | ||
Point | ||
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 -values is arbitrary. As long as we determine the coordinates of enough points to indicate the behavior of the graph, we may choose whichever -values we like. For simpler calculations, people often start with the integers from to However sometimes the equation has context that suggests using other -values, as in the next example.
Example4.2.11
One car's gas tank holds 14β―gal of fuel. Over the course of a long road trip, that car uses its fuel at an average rate of 0.032β―galβmi. If a driver fills the tank at the beginning of a long trip, then the amount of fuel remaining in the tank, after driving miles is given by the equation 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. 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 |
\(20\) | \(13.36\) | \((20,13.36)\) |
\(50\) | \(12.4\) | \((50,12.4)\) |
\(80\) | \(11.44\) | \((80,11.44)\) |
\(100\) | \(10.8\) | \((100,10.8)\) |
\(200\) | \(7.6\) | \((200,7.6)\) |
Example4.2.14
Build a table and graph the equation Use -values from to
\(x\) | \(y=x^2\) | 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.
SubsectionExercises
Determine if the given points are to be included in the graph of the equation.
1
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
2
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
3
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
4
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
5
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
6
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
7
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
8
Consider the equation
Which of the following ordered pairs are solutions to the given equation? There may be more than one correct answer.
Make a table for the equation with -values suggested.
9
Fill out this table for the equation The first row is an example.
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10
Fill out this table for the equation The first row is an example.
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11
Fill out this table for the equation The first row is an example.
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12
Fill out this table for the equation The first row is an example.
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13
Fill out this table for the equation The first row is an example.
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14
Fill out this table for the equation The first row is an example.
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15
Fill out this table for the equation The first row is an example.
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16
Fill out this table for the equation The first row is an example.
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17
Fill out this table for the equation The first row is an example.
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18
Fill out this table for the equation The first row is an example.
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Make a table for the equation.
19
Make a table of solutions for the equation
20
Make a table of solutions for the equation
21
Make a table of solutions for the equation
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Make a table of solutions for the equation
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Make a table of solutions for the equation
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Make a table of solutions for the equation
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Make a table of solutions for the equation
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Make a table of solutions for the equation
These exercises have Cartesian plots with some context.
27
A certain water heater will cost you to buy and have installed. This water heater claims that its operating expense (money spent on electricity or gas) will be about per month. According to this information, the equation models the total cost of the water heater after months, where 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 with a zero interest loan over a five-year period. That means you'll have to pay each month for the next five years (sixty months) to pay it off. According to this information, the equation models the loan balance after months, where 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 models this relationship, where the temperature is measured in Β°F and the pressure and the pressure is measured in lbβin2. Make a table of at least five values and plot a graph of this equation. Make sure to use -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 for the week. You might end up paying for it yourself, but you ask around to see if the other office coworkers want to split the cost evenly. The equation models this situation, where people contribute to the gift, and 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 -values that make sense in context.
Plot some graphs.
31
Create a table of ordered pairs and then make a plot of the equation
32
Create a table of ordered pairs and then make a plot of the equation
33
Create a table of ordered pairs and then make a plot of the equation
34
Create a table of ordered pairs and then make a plot of the equation
35
Create a table of ordered pairs and then make a plot of the equation
36
Create a table of ordered pairs and then make a plot of the equation
37
Create a table of ordered pairs and then make a plot of the equation
38
Create a table of ordered pairs and then make a plot of the equation
39
Create a table of ordered pairs and then make a plot of the equation
40
Create a table of ordered pairs and then make a plot of the equation Use -values from to
41
Create a table of ordered pairs and then make a plot of the equation
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Create a table of ordered pairs and then make a plot of the equation