Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo species, is a linear function.

One well known form for writing linear functions is known as the slope-intercept form, where [latex]x[/latex] is the input value, [latex]m[/latex] is the rate of change, and [latex]b[/latex] is the initial value of the dependant variable.

[latex]\begin{array}{cc}\text{Equation form}\hfill & y=mx+b\hfill \\ \text{Function notation}\hfill & f\left(x\right)=mx+b\hfill \end{array}[/latex]

(4.1.1) – Define slope for a linear function

We often need to calculate the slope given input and output values. Given two values for the input, [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex], and two corresponding values for the output, [latex]{y}_{1}[/latex] and [latex]{y}_{2}[/latex] —which can be represented by a set of points, [latex]\left({x}_{1}\text{, }{y}_{1}\right)[/latex] and [latex]\left({x}_{2}\text{, }{y}_{2}\right)[/latex]—we can calculate the slope [latex]m[/latex], as follows

[latex]\displaystyle m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]

where [latex]\Delta y[/latex] is the vertical displacement and [latex]\Delta x[/latex] is the horizontal displacement. Note in function notation two corresponding values for the output [latex]{y}_{1}[/latex] and [latex]{y}_{2}[/latex] for the function [latex]f[/latex], [latex]{y}_{1}=f\left({x}_{1}\right)[/latex] and [latex]{y}_{2}=f\left({x}_{2}\right)[/latex], so we could equivalently write

[latex]\displaystyle m=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]

The graph in Figure 5 indicates how the slope of the line between the points, [latex]\left({x}_{1,}{y}_{1}\right)[/latex] and [latex]\left({x}_{2,}{y}_{2}\right)[/latex], is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.

The slope of a function is calculated by the change in [latex]y[/latex] divided by the change in [latex]x[/latex]. It does not matter which coordinate is used as the [latex]\left({x}_{2,\text{ }}{y}_{2}\right)[/latex] and which is the [latex]\left({x}_{1},\text{ }{y}_{1}\right)[/latex], as long as each calculation is started with the elements from the same coordinate pair.

The units for slope are always [latex]\frac{\text{units for the output}}{\text{units for the input}}[/latex] Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.

(4.1.2) – Calculate Slope Given Two Points

When the slope of a linear function is positive, the line is moving in an uphill direction across the coordinate axes. This is also called an increasing linear function. Likewise, a decreasing linear function is one whose slope is negative.  The graph of a decreasing linear function is a line moving in a downhill direction across the coordinate axes.

In mathematical terms,

For a linear function [latex]f(x)=mx+b[/latex] if [latex]m>0[/latex], then [latex]f(x)[/latex] is an increasing function.

For a linear function [latex]f(x)=mx+b[/latex] if [latex]m<0[/latex], then [latex]f(x)[/latex] is a decreasing function. For a linear function [latex]f(x)=mx+b[/latex] if [latex]m=0[/latex], then [latex]f(x)[/latex] is a constant function.  Sometimes we say this is neither increasing nor decreasing. In the following example we will first find the slope of a linear function through two points, then determine whether the line is increasing, decreasing, or neither.

In the following video we show examples of how to find the slope of a line passing between two points, then determine whether the line is increasing, decreasing or neither.

In the next video we show an example where we determine the increase in cost for producing solar panels given two data points.

The following video provides na example of how to write a function that will give the cost in dollars for a given number of credit hours taken, x.