Question

Question was provided via image upload.

Explanation:

Identify the visible questions and given information

The image contains parts of a multi-part question about a model function \(g(x)\) representing the median height (in inches) of children who are \(x\) months of age.

• Part (a) is partially cut off at the top, showing only option D: "Stretch \(f(x)\) vertically by a factor of 3.2. Shift \(f(x)\) left by 21.2 units."
• Part (b) asks: "According to the model, what is the median height of children who are 48 months, or 4 years, old? Use a calculator to find the median height."
• The student has entered "49" in the box: "The median height is 49 inches. (Round to the nearest tenth of an inch)".
• It then asks: "The actual median height for children at 48 months is 49 inches. How well does the model describe the actual height?" with options A. very well and B. poorly.
• Part (c) asks: "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 [ ] inches per month. (Round to the nearest tenth)".

Since the algebraic formula for the model \(g(x)\) is not visible in the cropped image, we must estimate the values from the provided graph of \(g(x)\) shown at the top:

• The horizontal axis represents age \(x\) in months (from 0 to 60).
• The vertical axis represents height in inches (from 0 to 40+).
• At birth (\(x = 0\)), the graph starts at approximately \(y = 20\) inches.
• At \(x = 8\) months, the graph shows a height of approximately \(y = 28\) inches.

Evaluate the model at 48 months

Let's look at the graph at \(x = 48\):

• At \(x = 48\), the curve is slightly above 40, near 41 or 42 inches.
• The actual median height is given as 49 inches.
• Comparing the model's value (around 41-42 inches) to the actual height of 49 inches shows a significant difference of about 7 to 8 inches.
• Therefore, the model describes the actual height poorly.

Calculate the average rate of change

The average rate of change of a function \(g(x)\) between \(x = a\) and \(x = b\) is given by the formula:

$$ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} $$

For the interval between birth (\(x = 0\)) and 8 months (\(x = 8\)):

• At \(x = 0\), the height \(g(0) \approx 20\) inches.
• At \(x = 8\), the height \(g(8) \approx 28\) inches.
• Substitute these values into the formula:

$$ \frac{g(8) - g(0)}{8 - 0} \approx \frac{28 - 20}{8} = \frac{8}{8} = 1.0 $$

Thus, the average rate of change is approximately 1.0 inch per month.

Answer:

Question b

The model describes the actual height

• A. very well
• B. poorly (Correct answer)

Question c

The average rate of change is <blank>1.0</blank> inches per month.