Questions: The function g(x)=1.2 sqrt(x)+18.7 models the median height, g(x), in inches, of children who are x months of age. The graph of g is shown. According to the model, what is the median height of children who are 24 months, or 2 years, old? Use a calculator to The median height is 24.6 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? The model describes the actual height very well. Use the model to find the average rate of change, in inches per month, between birth and 8 months. The average rate of change is 0.4 inches per month. (Round to the nearest tenth.) Use the model to find the average rate of change, in inches per month, between 40 and 48 months.

The function g(x)=1.2 sqrt(x)+18.7 models the median height, g(x), in inches, of children who are x months of age. The graph of g is shown. According to the model, what is the median height of children who are 24 months, or 2 years, old? Use a calculator to

The median height is 24.6 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?

The model describes the actual height very well. Use the model to find the average rate of change, in inches per month, between birth and 8 months.

The average rate of change is 0.4 inches per month. (Round to the nearest tenth.) Use the model to find the average rate of change, in inches per month, between 40 and 48 months.

Transcript text: The function $g(x)=1.2 \sqrt{x}+18.7$ models the median height, $g(x)$, in inches, of children who are $x$ months of age. The graph of g is shown. According to the model, what is the median height of children who are 24 months, or 2 years, old? Use a calculator to The median height is 24.6 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? The model describes the actual height very well. Use the model to find the average rate of change, in inches per month, between birth and 8 months. The average rate of change is 0.4 inches per month. (Round to the nearest tenth.) Use the model to find the average rate of change, in inches per month, between 40 and 48 months.

Solution

Solution Steps

Step 1: Determine the transformations from f(x) = sqrt(x) to g(x) = 1.2sqrt(x) + 18.7

The function g(x) is obtained from f(x) by:

Vertically stretching f(x) by a factor of 1.2.
Shifting the resulting graph upwards by 18.7 units.

Step 2: Calculate the median height at 24 months.

Substitute x = 24 into the equation:

g(24) = 1.2 * sqrt(24) + 18.7 g(24) ≈ 1.2 * 4.899 + 18.7 g(24) ≈ 5.8788 + 18.7 g(24) ≈ 24.5788 Rounding to the nearest tenth gives 24.6 inches.

Since the actual median height is 25 inches and the model predicts 24.6 inches, the model describes the actual height very well.

Step 3: Calculate the average rate of change between birth (x=0) and 8 months (x=8).

g(0) = 1.2 * sqrt(0) + 18.7 = 18.7

g(8) = 1.2 * sqrt(8) + 18.7 ≈ 1.2 * 2.828 + 18.7 ≈ 3.394 + 18.7 ≈ 22.094

Average rate of change = (g(8) - g(0)) / (8 - 0) ≈ (22.094 - 18.7) / 8 ≈ 3.394 / 8 ≈ 0.42425

Rounding to the nearest tenth gives 0.4 inches per month.

Final Answer:

Vertically stretch f(x) by 1.2 and shift up by 18.7.
The median height at 24 months is approximately 24.6 inches. The model describes the actual height very well.
The average rate of change between birth and 8 months is approximately 0.4 inches per month.

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