To help with the given problem, I will analyze the graph and questions step by step. Let’s address each part of the problem:

26(a): Describe the function depicted in the graph by intervals of x-values as increasing, decreasing, or constant. Also, state if it is linear or nonlinear.

From the graph:

• From x = 0 to x = 4: The function is increasing. (Nonlinear)
• From x = 4 to x = 8: The function is decreasing. (Nonlinear)
• From x = 8 to x = 11: The function is constant. (Linear)
• From x = 11 to x = 14: The function is increasing. (Nonlinear)

26(b): Graph in the same grid a linear function that passes through the points (4, 25) and (8, 20).

1. The slope of the linear function passing through (4, 25) and (8, 20) is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 25}{8 - 4} = \frac{-5}{4} = -1.25$
2. Using the point-slope form of the equation: $y - y_1 = m(x - x_1)$ Substituting $m = -1.25$ and $(4, 25)$: $y - 25 = -1.25(x - 4)$ Simplify: $y = -1.25x + 30$

This is the equation of the linear function.

26(c): Find its equation.

Already found in part (b): $y = -1.25x + 30$.

26(d): List the rates of change for both functions from x = 0 to x = 4, x = 4 to x = 8, and x = 8 to x = 11.

For the given nonlinear function:

1. From x = 0 to x = 4: Rate of change is calculated as: $\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{40 - 10}{4 - 0} = \frac{30}{4} = 7.5$

2. From x = 4 to x = 8: Rate of change is: $\frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 40}{8 - 4} = \frac{-20}{4} = -5$

3. From x = 8 to x = 11: The function is constant, so: $\text{Rate of change} = 0$

For the linear function:

The rate of change is constant across all intervals, as calculated in part (b): $\text{Rate of change} = -1.25$

Let me know if you would like me to provide further visual explanations, plot the graph, or clarify specific details!

Here are 5 related questions to expand your understanding:

1. How do you differentiate between a linear and nonlinear function using its equation?
2. What is the significance of a function’s rate of change?
3. How can you verify whether a segment of a graph is increasing, decreasing, or constant using calculus?
4. Can a function be both linear and nonlinear on different intervals? Why or why not?
5. How does the slope of a linear equation relate to its graph’s steepness?

Tip: When finding the slope of a line, always carefully check the coordinates of the points you use to avoid small errors in calculations.