Example 4.4.20

Your neighbor planted a sapling from Portland Nursery in his front yard. Ever since, for several years now, it has been growing at a constant rate. By the end of the third year, the tree was 15 ft tall; by the end of the sixth year, the tree was 27 ft tall. What's the tree's rate of growth (i.e. the slope)?

We could sketch a graph for this scenario, and include a slope triangle. If we did that, it would look like:

By the slope triangle and Equation (4.4.1) we have:

\begin{align*} \text{slope}=m\amp =\frac{\Delta y}{\Delta x}\\ \amp =\frac{12}{3}\\ \amp =4 \end{align*}

So the tree is growing at a rate of 4 ftyr.

Figure 4.4.21 Height of a Tree

But hold on. Did we really need this picture? The “rise” of \(12\) came from a subtraction of two \(y\)-values: \(27-15\text{.}\) And the “run” of \(3\) came from a subtraction of two \(x\)-values: \(6-3\text{.}\)

Here is a picture-free approach. We know that after 3 yr, the height is 15 ft. As an ordered pair, that information gives us the point \((3,15)\) which we can label as \((\overset{x_1}{3},\overset{y_1}{15})\text{.}\) Similarly, the background information tells us to consider \((6,27)\text{,}\) which we label as \((\overset{x_2}{6},\overset{y_2}{27})\text{.}\) Here, \(x_1\) and \(y_1\) represent the first point's \(x\)-value and \(y\)-value, and \(x_2\) and \(y_2\) represent the second point's \(x\)-value and \(y\)-value.

Now we can write an alternative to Equation (4.4.1):

\begin{equation} \text{slope}=m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\tag{4.4.3} \end{equation}

This is known as the slope formula. The following graph will help you understand why this formula works. Basically, we are still using a slope triangle to calculate the slope.

Figure 4.4.22 Understanding the slope formula
in-context