ORCCA Graphing Equations

Section4.2Graphing Equations

ObjectivesPCC Course Content and Outcome Guide

We have graphed points in a coordinate system, and now we will graph lines and curves.

Figure4.2.1Alternative Video Lesson

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 Figure 4.2.3.(c). The graph plots the ordered pairs whose coordinates make $y=-2x+3$ true. Table 4.2.2 shows a few points that make the equation true.

$y=-2x+3$	$(x,y)$
$\substitute{5}\stackrel{\checkmark}{=}-2(\substitute{-1})+3$	$(\substitute{-1},\substitute{5})$
$\substitute{3}\stackrel{\checkmark}{=}-2(\substitute{0})+3$	$(\substitute{0},\substitute{3})$
$\substitute{1}\stackrel{\checkmark}{=}-2(\substitute{1})+3$	$(\substitute{1},\substitute{1})$
$\substitute{-1}\stackrel{\checkmark}{=}-2(\substitute{2})+3$	$(\substitute{2},\substitute{-1})$
$\substitute{-3}\stackrel{\checkmark}{=}-2(\substitute{3})+3$	$(\substitute{3},\substitute{-3})$
$\substitute{-5}\stackrel{\checkmark}{=}-2(\substitute{4})+3$	$(\substitute{4},\substitute{-5})$

Table 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 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) — so many, that together the points look like a straight line.

Table4.2.2

a Cartesian graph of the points listed in the text that are solutions to the equation y=-2x+3; — (a)A few points…

a second Cartesian graph showing all of the points listed in the first graph and additional solutions to y=-2x+3 between those points — (a)A few points…

Generalizing…

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.

Checkpoint4.2.5

Checkpoint4.2.6

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
$-2$	$-2(-2)+5=\substitute{9}$	$(-2,9)$
$-1$	$-2(-1)+5=\substitute{7}$	$(-1,7)$
$0$	$-2(0)+5=\substitute{5}$	$(0,5)$
$1$	$-2(1)+5=\substitute{3}$	$(1,3)$
$2$	$-2(2)+5=\substitute{1}$	$(2,1)$

(a)Set up the table

(b)Complete the table

Figure4.2.8Making a table for

y = - 2 x + 5

$y=-2x+5$

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.

a Cartesian grid with points (-2,9),(-1,7),(0,5),(1,3),(2,1) — (a)Use points from the table

a Cartesian grid with points (-2,9),(-1,7),(0,5),(1,3),(2,1) connected with a solid line — (a)Use points from the table

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 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 example.

Example4.2.11

One car's gas tank holds 14 gal of fuel. Over the course of a long road trip, that car uses its fuel at an average rate of 0.032 ^gal⁄_mi. 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

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
$20$	$13.36$	$(20,13.36)$
$50$	$12.4$	$(50,12.4)$
$80$	$11.44$	$(80,11.44)$
$100$	$10.8$	$(100,10.8)$
$200$	$7.6$	$(200,7.6)$

Table4.2.12Make the table

Figure4.2.13Make the graph

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
$-3$	$(-3)^2=9$	$(-3,9)$
$-2$	$(-2)^2=4$	$(-2,4)$
$-1$	$(-1)^2=1$	$(-1,1)$
$0$	$(0)^2=0$	$(0,0)$
$1$	$(1)^2=1$	$(0,1)$
$2$	$(2)^2=4$	$(2,4)$
$3$	$(3)^2=9$	$(3,9)$

a Cartesian grid with points (-3,9),(-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9); the points are connected with a smooth curve with arrows on both ends

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.