8.04 Review: Slopes of lines

Slope as a rate of change

The slope is the rate of change over an interval, or the ratio of vertical change to horizontal change.

$\text{slope }=\frac{\text{vertical change }}{\text{horizontal change }}$slope =vertical change horizontal change ​

We often call the horizontal measurement the "run" and the vertical measurement the "rise".

$\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​

To find the slope of a line, we can use any two ordered pairs $\left(x_1,y_1\right)$(x1​,y1​) and $\left(x_2,y_2\right)$(x2​,y2​) that lie on the line. The "run" is the horizontal measurement, which corresponds to the distance along the $x$x-axis, while the "rise" is the vertical measurement which corresponds to the distance along the $y$y-axis. That is, we can express the slope using two ordered pairs as:

$\text{slope }=\frac{y_2-y_1}{x_2-x_1}$slope =y2​−y1​x2​−x1​​

Worked examples

Question 1

Determine the slope of the line given:

 

 

Think: In this case, we have been shown a value for the "rise" of the line along with the corresponding valule for the "run". So we can calculate the slope by using the formula

$\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​

Do: We have that $\text{run }=3$run =3 and $\text{rise }=4$rise =4, and therefore

$\text{slope }=\frac{4}{3}$slope =43​

Reflect: The slope can be written as an improper fraction - why is this the case?

 

Increasing and decreasing

Slope is a measure of rate of change. A quantity that changes could have an increasing change or a decreasing change (or even have no change at all, i.e. remain constant). Let's think about how this can be represented by the slope value.

Below are a few line graphs. If we move along each line from left to right, what do these graphs all have in common?

The lines are all increasing - that is, as we move along the line from left to right, the $y$y-values get larger. We can also see that one end of each graph is in the top-right quadrant, and the other end of each graph is in the bottom-left quadrant.

The slopes of these lines are positive, which represents an increasing rate of change.

 

Here are a few more line graphs. If we again move along each line from left to right, what do the graphs all have in common this time?

The lines are all decreasing - that is, as we move along the line from left to right, the $y$y-values get smaller. We can also see that one end of each graph is in the top-left quadrant, and the other end of each graph is in the bottom-right quadrant.

The slopes of these lines are negative, which represents a decreasing rate of change.

 

Exploration

Use the applet below to explore positive and negative slopes. Use the sliders to change the value of the $slope$slope, indicated by $m$m.

 

When given the graph of a line, we can find the rise and run by drawing a right triangle created by any two points on the line. The segment of the line itself between these two points becomes the hypotenuse, while the other two legs lay in the $x$x- and $y$y-directions.

 

Use the applet below to explore how changing the rise and run impacts the slope and the graph. Use the $m$m slider to change the slope and use the endpoints to adjust the rise and run.

 

Worked examples

Question 2

Find the slope of the line that passes through $\left(3,6\right)$(3,6) and $\left(7,-2\right)$(7,−2)

Think: It is good mathematical practice to draw a quick sketch of the points. This helps to quickly identify what the line looks like (is it increasing or decreasing).

From this quick sketch, we can see that the slope will be decreasing, so we should expect to get a negative value.

Do:

We can calculate the slope by using the formula as follows:

$\text{slope }$slope	$=$=	$\frac{\text{rise }}{\text{run }}$rise run ​
	$=$=	$\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​
	$=$=	$\frac{-2-6}{7-3}$−2−67−3​
	$=$=	$\frac{-8}{4}$−84​
	$=$=	$-2$−2

Note that in this case we used point $A$A for the values of $\left(x_1,y_1\right)$(x1​,y1​), and point $B$B for the values of $\left(x_2,y_2\right)$(x2​,y2​). It is important to subtract the pairs in the same order, to make sure we get the right sign for the slope!

Reflect: Would the answer be different if we had used point $A$A for the values of $\left(x_2,y_2\right)$(x2​,y2​) and point $B$B for the values of $\left(x_1,y_1\right)$(x1​,y1​) instead? Similarly, would the answer be different if we calculated the slope using $\text{slope }=\frac{y_1-y_2}{x_1-x_2}$slope =y1​−y2​x1​−x2​​ instead?

 

Question 3

The slope of the line is $\frac{-1}{5}$−15​. The line passes through the point $M$M at $\left(-12,4\right)$(−12,4) and point $E$E. $E$E has a $y$y-coordinate of $y=-3$y=−3 . What is the $x$x -coordinate of point $E$E, denoted by $d$d?

Think: The rise is the difference in the $y$y-values of the points. The run is the difference in the $x$x-values of the points.

$\text{slope }=\frac{-1}{5}$slope =−15​

$M$M $\left(x_1,y_1\right)=\left(-12,4\right)$(x1​,y1​)=(−12,4)

$E$E $\left(x_2,y_2\right)=\left(d,-3\right)$(x2​,y2​)=(d,−3)

Do: We can use the slope formula to find the value of $d$d as follows:

$\text{slope }$slope	$=$=	$\frac{\text{rise }}{\text{run }}$rise run ​
$\frac{-1}{5}$−15​	$=$=	$\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​
$\frac{-1}{5}$−15​	$=$=	$\frac{-3-4}{d-\left(-12\right)}$−3−4d−(−12)​
$\frac{-1}{5}$−15​	$=$=	$\frac{-7}{d+12}$−7d+12​
$\frac{-1}{5}\left(d+12\right)$−15​(d+12)	$=$=	$\frac{-7}{d+12}\left(d+12\right)$−7d+12​(d+12)
$\frac{-1}{5}\left(d+12\right)$−15​(d+12)	$=$=	$-7$−7
$d+12$d+12	$=$=	$-7\left(\frac{-5}{1}\right)$−7(−51​)
$d+12$d+12	$=$=	$35$35
$d$d	$=$=	$35-12$35−12
$d$d	$=$=	$23$23

 

The answer is $d=23$d=23

 

Slope of horizontal and vertical lines

Horizontal lines

For a horizontal line, the $y$y-value is always the same for every point on the line.

Let's look at the coordinates for three points $A$A, $B$B, and $C$C on a horizontal line:

$A=\left(-4,4\right)$A=(−4,4)

$B=\left(2,4\right)$B=(2,4)

$C=\left(4,4\right)$C=(4,4)

All of the $y$y-coordinates are the same. Every point on the line has a $y$y-value equal to $4$4, regardless of the $x$x-value.

 

 

Horizontal lines have no "rise" - they are completely flat. That is, the rise of a horizontal line is $0$0. This means that the slope of a horizontal line is also $0$0, since the slope is calculated by $\frac{\text{rise }}{\text{run }}$rise run ​. This makes sense since a positive slope indicates an increase, while a negative slope indicates a decrease - but $0$0 is not positive or negative.

 

Vertical lines

For a vertical line, the $x$x-value is always the same for every point on the line.

Let's look at the coordinates for three points $A$A, $B$B, and $C$C on a vertical line:

$A=\left(5,-4\right)$A=(5,−4)

$B=\left(5,-2\right)$B=(5,−2)

$C=\left(5,4\right)$C=(5,4)

All of the $x$x-coordinates are the same. Every point on the line has an $x$x-value equal to $5$5, regardless of the $y$y-value.

 

Vertical lines have no "run" - i.e. the run is $0$0. If we substituted this into the slope formula, $\text{slope }=\frac{\text{rise }}{\text{run }}$slope =rise run ​, the fraction will have some value for rise in the numerator, with a value of $0$0 for the denominator. Division by $0$0 like this is undefined, and so we say that the slope of a vertical line is also undefined.

 

Practice questions

Question 4

Question 5

Question 6