Let's break down and solve the questions based on the information provided in the image.

The function given is:

$M = 527 + 128.0\sqrt{t}, \quad 0 \leq t \leq 16$

where $t$ represents the year, with $t = 0$ corresponding to 1990.

(a) Describe the transformation of the parent function $f(x) = \sqrt{x}$:

• The base function is $\sqrt{x}$.
• In the equation, the constant $128.0$ scales the square root function vertically by a factor of 128.0.
• The addition of 527 translates the graph vertically upward by 527 units.

So, the correct description would include:

• Vertical stretch of 128.0
• Vertical shift of 527 units upward

(b) Find the average rate of change of the function from 1990 to 2006.

The average rate of change is given by:

$\frac{M(t=16) - M(t=0)}{16 - 0}$

Let's compute the values:

For $t = 16$: $M(16) = 527 + 128.0 \times \sqrt{16} = 527 + 128.0 \times 4 = 527 + 512 = 1039$

For $t = 0$: $M(0) = 527 + 128.0 \times \sqrt{0} = 527 + 0 = 527$

Now, compute the average rate of change: $\text{Average rate of change} = \frac{1039 - 527}{16} = \frac{512}{16} = 32$

Thus, the average rate of change is 32 billion miles per year.

(c) Rewrite the function so that $t = 0$ represents 2000.

Since $t = 0$ represents 1990, for 2000 we need to set $t = 10$ to correspond to 2000. So, the new function will be:

$M = 527 + 128.0\sqrt{t + 10}$

(d) Use the model from part (c) to predict the number of miles driven by vans, pickups, and SUVs in 2024.

For 2024, $t = 24$ (since 2024 is 24 years after 1990).

We need to use the equation from part (c), with $t = 24 - 10 = 14$:

$M(14) = 527 + 128.0 \times \sqrt{14}$

First, calculate $\sqrt{14}$:

$\sqrt{14} \approx 3.74$

Now calculate $M(14)$:

$M(14) = 527 + 128.0 \times 3.74 = 527 + 479.36 = 1006.36$

So, the predicted number of miles driven in 2024 is 1006.36 billion miles.

Let me know if you would like more details or further clarification!

5 related questions:

1. What is the transformation if we have a horizontal shift in the function $f(x) = \sqrt{x}$?
2. How do you find the instantaneous rate of change for non-linear functions like $\sqrt{x}$?
3. How does a vertical stretch differ from a horizontal stretch in transformations?
4. What are the units for the average rate of change in real-world applications like this one?
5. Can we model other types of vehicle mileage trends using similar functions?

Tip: Always start by identifying basic transformations such as shifts, stretches, and reflections when dealing with transformed parent functions like $\sqrt{x}$.