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Let's break down the questions from the table and calculate the answers step by step.

The table provides selected values of the function \( f(x) \):

\[
\begin{array}{c|c|c|c|c}
x & 3 & 4 & 5 & 6 & 7 \\
\hline
f(x) & 0 & -3 & -4 & -3 & 0 \\
\end{array}
\]

### Part A:

(i) **Determine the rate of change for function \( f \) on the interval [3, 5]:**

The rate of change is calculated using the formula for the slope between two points:

\[
\text{Rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

For the interval [3, 5]:
- \( x_1 = 3 \), \( f(3) = 0 \)
- \( x_2 = 5 \), \( f(5) = -4 \)

\[
\text{Rate of change} = \frac{-4 - 0}{5 - 3} = \frac{-4}{2} = -2
\]

So, the rate of change is **-2**.

(ii) **How is the rate of change in part (i) displayed on the graph of the function?**

Since the rate of change is negative, the function is **decreasing** on the interval [3, 5], meaning that the graph would slope downward.

### Part B:

(i) **Determine the rate of change for function \( f \) on the interval [3, 6]:**

Using the same formula for the rate of change:

For the interval [3, 6]:
- \( x_1 = 3 \), \( f(3) = 0 \)
- \( x_2 = 6 \), \( f(6) = -3 \)

\[
\text{Rate of change} = \frac{-3 - 0}{6 - 3} = \frac{-3}{3} = -1
\]

So, the rate of change is **-1**.

(ii) **How is the rate of change in part (i) displayed on the graph of a function?**

Since the rate of change is negative but less steep than in part A, the function is still **decreasing**, but more gradually than in the interval [3, 5].

### Part C:

**Compare the average rates of change for \( f \) on the intervals [3, 5] and [3, 6]:**

- The rate of change on [3, 5] is **-2**.
- The rate of change on [3, 6] is **-1**.

The interval [3, 5] has a **ste

The table gives selected values of the function f(x). Part A: (i) Determine the rate of change for function f on the interval [3, 5]. (ii) How is the rate of change in part (i) displayed on the graph of a function? Part B: (i) Determine the rate of change for function f on the interval [3, 6]. (ii) How is the rate of change in part (i) displayed on the graph of a function? Part C: Compare the average rates of change for f on the intervals [3, 5] and [3, 6].

Learn how to determine the rate of change for the function f(x) based on selected table values in different intervals. This problem uses the slope formula to calculate changes and analyze function behavior on the graph. Suitable for students in Grades 8-10.

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