###### 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:

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.

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):

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

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.

in-context