Example4.4.18

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 this:

This is a grid with a line, passing the points (3,15) and (6,27). A slope triangle is drawn, starting at (3,15), passing (6,15), and ends at (6,27). The label from (3,15) to (6,15) is "6-3=3 years"; the label from (6,15) to (6,27) is "27-15=12 feet".

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.

Figure4.4.19Height 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.

It's important to use subscript instead of superscript here, because \(y^2\) means to take the number \(y\) and square it. Whereas \(y_2\) tells you that there are at least two \(y\)-values in the conversation, and \(y_2\) is the second of them.

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.

This is a grid with a line, passing the points (3,15) and (6,27). A slope triangle is drawn, starting at (3,15), passing (6,15), and ends at (6,27). The label from (3,15) to (6,15) is "6-3=3 years"; the label from (6,15) to (6,27) is "27-15=12 feet".
This is a grid with a line, passing the points (3,15) and (6,27). A slope triangle is drawn, starting at (3,15), passing (6,15), and ends at (6,27). The label from (3,15) to (6,15) is "x2-x1"; the label from (6,15) to (6,27) is "y2-y1".
Figure4.4.20Understanding the slope formula
in-context