Section 5.2
Graphs of Lines
When we know the coordinates of a point on a line and we know the slope of the line, we can write an equation of the line and we can graph the line. Similarly, when we have the graph of a line, we identify the coordinates of one of the points on the line, and then figure out the slope. Then we can write the equation of the line.
Examples:
a. To graph the line $$y=\frac{2}{3}x-1$$
first notice the line has slope $m=\dfrac{2}{3}$ and a $y-$intercept of $-1$. With this information, click on the boxes in the sketch below to graph the line.
b. Write the equations of the two lines graphed below.
First we need to find the slope. The line $l$ passes throught the point $(2,1)$. Getting from that point to the other iindicated point involves going over 3 units and up 4 units; the slope is $\frac{4}{3}$. The equation of the line is $$y-1=\frac{4}{3}(x-2).$$
The line $m$ also passes through the point $(2,1)$. The slope of line $m$ is $-\frac{1}{3}$. The equation of the line is $$y-1=-\frac{1}{3}(x-2).$$
c. In the graph below the lines have the following equations.
A. $y-0=\dfrac{3}{2}(x-1)$. The line passes through the point $(1,0)$ and has slope $m=\dfrac{3}{2}.$
B. $y-2=-4(x+4)$. The line passes throught the point $(-4,2)$ and has slope $m=-4$.
C. $x=3$. This is a vertical line on which each point has an $x-$coordinate of 3.
D. $y=\dfrac{1}{3}x-3$. This line has a $y-$intercept of $-3$ and a slope of $m=\frac{1}{3}.$
E. $y=5$. This is a horizontal line on which each point has a $y-$coordinate of 5.
Graphing a line in slope intercept form: Graphing a line in slope intercept form
Converting to slope-intercept form: Converting to slope-intercept form
Exercises:
Sketch a graph of each of the following lines. Write an equation of each line.1. The line through the point $\left( 2,3 \right)$ with slope $m=\tfrac{2}{5}$.
2. The line through the point $\left( -1,2 \right)$ with slope $m=\tfrac{3}{2}$.
3. The line through the point $\left( 4,-3 \right)$ with slope $m=-\tfrac{1}{3}$.
4. The line through the point $\left( 2,0 \right)$ with slope $m=-\tfrac{3}{4}$.
5. The line through the point $\left( -2,-3 \right)$ with slope $m=0$.
6. The line through the two points $P\left( 3,-5 \right)\text{ and }Q\left( 3,5 \right).$
7. The line through the two points $P\left( -2,3 \right)\text{ and }Q\left( 4,4 \right).$
8. The line through the two points $P\left( 2,5 \right)\text{ and }Q\left( -4,2 \right).$
9. The line through the two points $P\left( 3,-5 \right)\text{ and }Q\left( 5,-3 \right).$
10. The line through the two points $P\left( 2,7 \right)\text{ and }Q\left( -1,4 \right).$
Determine an equation of each of the lines pictured below.
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17. Match each of the graphs (a. through f.) with its equation (one of those lettered A. through H.) Note that there are two extra equations.
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\begin{array}{l l}
A. \;\;y+3=\tfrac{1}{2}(x+2)\;\; & B. \;\;y-3=-\tfrac{2}{3}(x+2)\;\; \\
C. \;\;y=-2x\;\; & D. \;\;y+2=\tfrac{1}{2}(x+2)\;\; \\
E. \;\;y+4=-\tfrac{2}{3}(x-2)\;\; & F. \;\;y=\tfrac{1}{2}x+1\;\; \\
G. \;\;y=-\tfrac{2}{3}x\;\; & H. \;\;y=-\tfrac{1}{2}x-2\;\;
\end{array}
18. Sketch graphs of the two lines $y=3$ and $x=-2$ in a single coordinate system. Where do the two lines intersect?
19. If the graph of a line goes from the second quadrant to the fourth quadrant, what can you say about the line's slope? What if the line passes through both the first and third quadrants?