Question

the function $g(x) = 1.7\sqrt{x} + 17.0$ models the median height, $g(x)$, in inches, of children who are $x$ months of age. the graph of $g$ is shown.
a. describe how the graph can be obtained using transformations of the square root function $f(x) = \sqrt{x}$.
\bigcirc a. shrink $f(x)$ horizontally by a factor of 1.7. shift $f(x)$ down by 17.0 units.
\bigcirc b. stretch $f(x)$ horizontally by a factor of 1.7. shift $f(x)$ down by 17.0 units.
\bigcirc c. stretch $f(x)$ vertically by a factor of 1.7. shift $f(x)$ left by 17.0 units.
\bigcirc d. stretch $f(x)$ vertically by a factor of 1.7. shift $f(x)$ up by 17.0 units.
b. according to the model, what is the median height of children who are 24 months, or 2 years, old? use a calculator to find the median height.
the median height is 25.3 inches. (round to the nearest tenth of an inch.)
the actual median height for children at 24 months is 25 inches. how well does the model describe the actual height?
\bigcirc a. very well.
\bigcirc d. poorly.
the model describes the actual height a or d.
c. use the model to find the average rate of change, in inches per month, between birth and 12 months.
the average rate of change is \square inches per month. (round to the nearest tenth.)

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( g(x) \) between \( x = a \) and \( x = b \) is given by \( \frac{g(b)-g(a)}{b - a} \). Here, \( a = 0 \) (birth, 0 months) and \( b = 12 \) (12 months).

Step2: Calculate \( g(0) \)

Substitute \( x = 0 \) into \( g(x)=1.7\sqrt{x}+17.0 \):
\( g(0)=1.7\sqrt{0}+17.0 = 0 + 17.0=17.0 \)

Step3: Calculate \( g(12) \)

Substitute \( x = 12 \) into \( g(x)=1.7\sqrt{x}+17.0 \):
\( g(12)=1.7\sqrt{12}+17.0 \). First, \( \sqrt{12}\approx3.464 \), then \( 1.7\times3.464\approx5.89 \), so \( g(12)\approx5.89 + 17.0 = 22.89 \)

Step4: Compute average rate of change

Using the formula \( \frac{g(12)-g(0)}{12 - 0}=\frac{22.89 - 17.0}{12}=\frac{5.89}{12}\approx0.5 \) (rounded to the nearest tenth)

Answer:

0.5