Question

First, find the increasing functions. Then, classify each increasing function as having a larger or a smaller unit rate than the function represented in the graph.

Answer 1

To identify the increasing functions you need to identify the unit rate (rate of change) in each function, to be an increasing function the unit rate needs to be a possitive amount.

In a equation written in the form y=mx+b the unit rate is m

Then, for the given functions the increasing functions are:

$\begin{gathered} y=\frac{4}{3}x-\frac{5}{3} \\ \\ y=\frac{5}{4}x-3 \\ \\ y=\frac{7}{4}x-\frac{9}{4} \\ \\ y=\frac{6}{5}x-\frac{3}{5} \\ \\ y=\frac{8}{5}x-\frac{7}{5} \end{gathered}$

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To idenify the unit rate of the graphed function use two points (x,y) in the next formula:

$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \\ \text{Points: (0,-2)(2,1)} \\ \\ m=\frac{1-(-2)}{2-0}=\frac{1+2}{2}=\frac{3}{2} \end{gathered}$

Then, for the given increasing functions the next have a larger unit rate:

$\begin{gathered} m=\frac{3}{2}=1.5 \\ \\ \\ y=\frac{7}{4}x-\frac{9}{4}(m=\frac{7}{4}=1.75) \\ \\ y=\frac{8}{5}x-\frac{7}{5}(m=\frac{8}{5}=1.6) \end{gathered}$

And for the given increasing function the next have a smaller unit rate:

$\begin{gathered} m=\frac{3}{2}=1.5 \\ \\ y=\frac{4}{3}x-\frac{5}{3}(m=\frac{4}{3}=1.33) \\ \\ y=\frac{5}{4}x-3(m=\frac{5}{4}=1.25) \\ \\ y=\frac{6}{5}x-\frac{3}{5}(m=\frac{6}{5}=1.2) \end{gathered}$