Chapter 3.3, Problem 49HP

To determine

The function represented by the given table.

Answer to Problem 49HP

  $y=2.25x$

Explanation of Solution

Given:

The table,

  

Concept Used:

Equation of the line passing through the points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is given by $y-y_1=m\left(x-x_1\right)$ , where $\text{slope, }m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

The slope of any line is given by rise over run. That is, the ratio of the number of units it rise or fall over the number of units it move forward or backward from a fixed point.

Observe that the rate of change between week 4 to 6 and from week 6 to 8 is same, because

  $\begin{array}{l}\text{Rate of change }\left(\text{week 4 to 6}\right)=\frac{13.5-9.0}{6-4}=\frac{4.5}{2}=2.25\\ \text{Rate of change }\left(\text{week 6 to 8}\right)=\frac{18.0-13.5}{8-6}=\frac{4.5}{2}=2.25\end{array}$

So, the height of the plant is increasing at a constant rate of 2.25 inches per week.

So, the function that represents the height of the plant is a linear function with slope 2.25.

Now since function satisfy the point (4,9), so the function that describes the height of plant is

  $\begin{array}{c}y-y_1=m\left(x-x_1\right)\end{array}$

  $\begin{array}{c}y-9=2.25\left(x-4\right)\end{array}$

  $\begin{array}{c}y-9=2.25x-9\end{array}$

    $\begin{array}{c}y=2.25x-9+9\end{array}$

    $\begin{array}{c}y=2.25x\end{array}$

Thus, the function that describes the height of the plant is given by $y=2.25x$ .