Calculating and Interpreting Slope

In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope given input and output values. Given two values for the input, (x_1) and (x_2), and two corresponding values for the output, (y_1) and (y_2)—which can be represented by a set of points, ((x_1,y_1)) and ((x_2,y_2))—we can calculate the slope (m), as follows

[begin{align*} m &= dfrac{text{change in output (rise)}}{ text{change in input (run)}} [4pt] &= dfrac{{Delta}y}{ {Delta}x} = dfrac{y_2−y_1}{x_2−x_1} end{align*}]

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where ({Delta}y) is the vertical displacement and ({Delta}x) is the horizontal displacement. Note in function notation two corresponding values for the output (y_1) and (y_2) for the function (f), (y_1=f(x_1)) and (y_2=f(x_2)), so we could equivalently write

[m=dfrac{f(x_2)-f(x_1)}{x_2-x_1} nonumber]

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Figure (PageIndex{6}) indicates how the slope of the line between the points, ((x_1,y_1)) and ((x_2,y_2)), is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form.

[y-y_1=m(x-x_1)]

The point-slope form is derived from the slope formula.

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[ begin{align*} &m=dfrac{y-y_1}{x-x_1} &text{assuming }x{neq}x_1 &m(x-x_1)=dfrac{y-y_1}{x-x_1}(x-x_1) &text{Multiply both sides by }(x-x_1). &m(x-x_1)=y-y_1 &text{Simplify} &y-y_1=m(x-x_1) &text{Rearrange} end{align*}]

Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form, (y−4=− frac{1}{2}(x−6)). We can convert it to the slope-intercept form as shown.

[begin{align*} y-4&=-dfrac{1}{2}(x-6) y-4&=-dfrac{1}{2}x+3 &text{Distribute the }-dfrac{1}{2}. y&=-dfrac{1}{2}x+7 &text{Add 4 to each side.}end{align*}]

Therefore, the same line can be described in slope-intercept form as (y=dfrac{1}{2}x+7).

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