Slope

The slope of a line, [latex]m[/latex], is a measure of the steepness of the line, and since the steepness is the same all along the line, the slope is constant.  The slope of a line is defined as the ratio of [latex]\frac{\text{rise}}{\text{run}}[/latex] where rise is the change in [latex]y[/latex]-values, platex]\Delta y[/latex], and run is the corresponding change in [latex]x[/latex]-values, [latex]\Delta x[/latex]. If [latex]y=f(x)[/latex], then slope = [latex]\frac{\Delta y}{\Delta x}=\frac{\text{change in function values}}{\text{change in }x\text{-values}}[/latex].

Consider the line in figure 1. It passes through the points (–4, –4) and (10, 3). The run is the distance between [latex]x=-4[/latex] and [latex]x=10[/latex]; a total of 14 units. Rather than counting units on the graph, we can subtract the two [latex]x[/latex]-coordinates: 10 – (–4) = 14. The rise is the distance between [latex]y=-4[/latex] and [latex]y=3[/latex]; a total of 7 units. We can find this number by subtracting the [latex]y[/latex]-coordinates: 3 – (–4) = 7. Consequently, the slope of the line is  [latex]m=\frac{7}{14}=\frac{1}{2}[/latex].

Let’s now consider two arbitrary points on a line. Suppose the first point is designated [latex](x_1, y_1)[/latex] and the second point is [latex](x_2,y_2)[/latex]. The run is the difference in [latex]x[/latex]-values, [latex]x_2-x_1[/latex], and the rise is the difference in [latex]y[/latex]-values, [latex]y_2-y_1[/latex]. So the slope of the line between these two points is [latex]m=\frac{y_2-y_1}{x_2-x_1}[/latex] (see figure 2). It does not matter which point we designate as the first point or the second point. We will always get the same slope as long as we subtract from one point to the other in the same order.

The notation [latex]\Delta[/latex] is often used in mathematics to represent “change in”, so “change in y” can be denoted by [latex]\Delta y[/latex], and “change in [latex]x[/latex]” can be denoted [latex]\Delta x[/latex].  So, slope = [latex]m=\frac{\Delta y}{\Delta x}[/latex].

The following video shows examples of how to find the slope of a line passing through two points and then determines whether the line is increasing, decreasing or neither.

The next video shows an example where the increase in cost for producing solar panels is determined by two data points.

Review of Fractions

Since a slope is a fraction, it is inevitable that there will be some operations with fractions involved when finding the equation of a linear function. Let’s have a quick review on how we add, subtract, multiply and divide two fractions.

Addition and Subtraction

To add or subtract two fractions, the fractions must have a common denominator. If they do not, we have to build two equivalent fractions so that they have the same denominator. We use the least common multiple (LCM) of the denominators to achieve this. Doing this makes the fractions look different from the originals but they still have the same value. The reason for using the LCM is that it could help avoid complicated computation or further simplification.

How do we find the LCM? The LCM must include all of the factors among the denominators because the LCM is a common multiple. This implies that we will need to factor each denominator into its prime factorization, so that we know what factors the LCM must include.

Multiplication and Division

To multiply fractions, we multiply the numerators and multiply the denominators. We can simplify by cancelling any common factors on the numerator and denominator to 1. To multiply a fraction by a whole number, we write the whole number as a fraction over 1.

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction.

Determining Linear Functions

A linear function may be described by the slope-intercept form [latex]f(x) = mx + b[/latex], where [latex]m[/latex] is the rate of change of the function or the slope of the graphed function, and [latex]b[/latex] is the initial value or [latex]y[/latex]-coordinate of the [latex]y[/latex]-intercept. There are several different situations and conditions we may be given to determine the equation of the linear function.

Given a Slope and the y-intercept

If we are given the slope and the [latex]y[/latex]-intercept, we are given the value of [latex]m[/latex] and [latex]b[/latex]. This allows us to jump right to slope-intercept form to find the function. For example, if we are told that the slope of a line is 2 and the [latex]y[/latex]-intercept is (0, 1), then using slope-intercept form, [latex]f(x)=x+b[/latex], the linear function will be [latex]f(x) = 2x + 1[/latex].

Given a Slope and a Point

If we are given a slope and a point, then we need to find the [latex]y[/latex]-intercept in order to determine the linear function [latex]f(x)=mx+b[/latex]. Since the given point [latex](x_1,y_1)[/latex] lies on the line, it is a solution of the linear function. We say that [latex](x_1,y_1)[/latex] satisfies the function. We can use this piece of information to solve for [latex]b[/latex] and determine the function.

For example, suppose we are told that the slope of a line is [latex]\dfrac{1}{3}[/latex] and the point (6, –1) lies on the line. We know that the function can be found using [latex]f(x)=mx+b[/latex]. Because [latex]m=\frac{1}{3}[/latex] the function must be:

[latex]f(x) = \frac{1}{3}x + b[/latex]

We now need to find the value of [latex]b[/latex] and we can use the given point (6, –1) to do this. Because (6, –1) satisfies the equation, [latex]f(6)=-1[/latex].

This means that:

[latex]\begin{equation}\begin{aligned} -1 &= \frac{1}{3} \cdot 6 + b\\-1 &= 2 + b\\-3 &= b \end{aligned}\end{equation}[/latex]

Now that we know [latex]m=\frac{1}{3}[/latex] and [latex]b=-3[/latex], we can write the linear function as [latex] f(x) = \frac{1}{3}x - 3[/latex].

Given Two Points

If we are given two points that lie on the graph of the function, then we have to find both the slope and the [latex]y[/latex]-intercept in order to determine the linear function. We can find the slope by finding the rate of change  between the two points:

[latex]m=\dfrac{y_2 - y_1}{x_2 - x_1}[/latex]

After we find the slope, then we can use either one of the two points on the line to find the value of [latex]b[/latex] because both of the two points satisfy the linear function.

For example, given the two points (–1, 3) and (4, –7) that satisfy the function , the slope of the line representing the linear function will be:

Slope = [latex]m=\dfrac{-7-3}{4-(-1)} =\dfrac{-10}{5} = -2 [/latex]

Since both point (–1, 3) and (4, –7) satisfy the function, we may use either of the two points. Let’s use the point (4, –7):

[latex]\begin{equation}\begin{aligned}f(x)&=-2x+b\\-7&=-2\cdot 4+b\\-7&=-8+b\\1&=b\end{aligned}\end{equation}[/latex]

We now know that [latex]m=-2[/latex] and [latex]b=1[/latex] so the linear function is [latex] f(x) = -2x + 1[/latex].

Given a Graph

If we are given a line on the coordinate plane, we may use any of the methods discussed above to find the function of the line by identifying the slope and the [latex]y[/latex]-intercept of the line, or the slope and a point on the line, or two points on the line.

If we are given a horizontal line, its slope is 0 because the rise is 0 (i.e., [latex]\dfrac{0}{run} = 0[/latex]). Sometimes we can identify the [latex]y[/latex]-intercept on the graph, and the function is [latex]f(x)=0x +b[/latex] or [latex]f(x)=b[/latex]. For example, for the function of the red horizontal line in figure 3, the [latex]y[/latex]-intercept is (0, 3) and therefore the function is [latex]f(x)=3[/latex].

If we are given a vertical line, like the blue line in figure 3, it is not a function because a vertical line is a one-to-many mapping (i.e., one input with many different outputs). Therefore, we are not able to use the function form [latex]f(x)=mx+b[/latex] to express a vertical line. Also, the slope of a vertical line is undefined because the run is 0, and a number divided by 0 is undefined (i.e., [latex]\dfrac{rise}{0}[/latex] is undefined). The equation for a vertical line is [latex]x = a[/latex] where [latex]a[/latex] is the [latex]x[/latex]-coordinate of the [latex]x[/latex]-intercept. Indeed, for all of the points on a vertical line, the [latex]x[/latex]-coordinates are identical and the [latex]y[/latex]-coordinates are any real number on the vertical line. For example, the equation of the blue vertical line in figure 3 is [latex]x = -2[/latex].

Given a Point and a Parallel Linear Function

Finding the equation of a function whose graph is parallel to another line, still requires finding the slope and [latex]y[/latex]-intercept. Fortunately, the slope can be found from the parallel line since parallel lines have the same slope. Knowing the slope, we can then use another point on the line to determine the [latex]y[/latex]-intercept and consequently the function.

For example, suppose we are asked to find the linear function that passes through the point (8, –2), whose graph runs parallel to the graph of the function [latex]g(x)=\frac{3}{4}x+2[/latex]. The slope of the line that represents [latex]g(x)[/latex] is [latex]\frac{3}{4}[/latex]. This means that the line representing [latex]f(x)[/latex] also has a slope of [latex]\frac{3}{4}[/latex] since they are parallel. this means that [latex]f(x)=\frac{3}{4}x + b[/latex]. Now, we can find the value of [latex]b[/latex] by using the given point (8, –2):

[latex]\begin{equation}\begin{aligned}f(x)&=\frac{3}{4}x+b\\-2&=\frac{3}{4}\cdot 8+b\\-2&=6+b\\-8=b\end{aligned}\end{equation}[/latex]

Now we know that [latex]m=\frac{3}{4}[/latex] and [latex]b=-8[/latex]. Therefore, the function is [latex]f(x) = \frac{3}{4}x - 8[/latex].

Given a Point and a Perpendicular Linear Function

Determining the function whose graph passes through a given point and is perpendicular to the another function is found in a similar way to that of a given point and a parallel function. We must still determine the slope and [latex]y[/latex]-intercept of the function, Since the function is perpendicular to the given function, we know that the slopes are the negative reciprocals of each other.

For example, suppose we are asked to find the linear function whose graph passes through the point (–6, 1) and is perpendicular to the graph of the function [latex]g(x)=\frac{3}{4}x - 5[/latex]. The line represented by [latex]g(x)[/latex] has a slope of [latex]\frac{3}{4}[/latex]. This means that the line represented by [latex]f(x)[/latex] must have a slope of [latex]-\dfrac{4}{3}[/latex] because the lines are perpendicular.

Knowing the slope, we now have [latex]f(x) = -\frac{4}{3}x + b[/latex]. To find the value of [latex]b[/latex], we can use the point (–6, 1):

[latex]\begin{equation}\begin{aligned}f(x)&=-\frac{4}{3}x+b\\1&=-\frac{4}{3}\cdot (-6)+b\\1&=8+b\\-7&=b\end{aligned}\end{equation}[/latex]

Now that we know [latex]m=-\frac{4}{3}[/latex] and [latex]b=-7[/latex], we can write the linear function as [latex]f(x) = -\frac{4}{3}x - 7[/latex].