To determine which statement is consistent with the table of values, we will first calculate the average rate of change of the function $f(x)$ between different points and then try to match it with the provided expressions.

The values provided in the table are:

$x = 2, 3, 4, 5 \quad \text{and} \quad f(x) = 4, 9, 18, 31$

Step 1: Calculate the average rate of change.

The average rate of change between two points $x_1$ and $x_2$ for a function $f(x)$ is given by:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Let’s compute the average rate of change between consecutive values:

1. Between $x = 2$ and $x = 3$:

$\frac{f(3) - f(2)}{3 - 2} = \frac{9 - 4}{3 - 2} = \frac{5}{1} = 5$

2. Between $x = 3$ and $x = 4$:

$\frac{f(4) - f(3)}{4 - 3} = \frac{18 - 9}{4 - 3} = \frac{9}{1} = 9$

3. Between $x = 4$ and $x = 5$:

$\frac{f(5) - f(4)}{5 - 4} = \frac{31 - 18}{5 - 4} = \frac{13}{1} = 13$

Step 2: Check if the function is quadratic or linear.

• A linear function has a constant rate of change. Since the rate of change (5, 9, 13) is not constant, the function cannot be linear.

• A quadratic function has a second difference that is constant. Let's check the second differences of the average rate of change:

First differences of the rates of change: $9 - 5 = 4, \quad 13 - 9 = 4$ The second difference is constant (4), which is a characteristic of a quadratic function.

Thus, $f(x)$ is quadratic.

Step 3: Compare the given expressions.

Now let's check the given options:

• The expression for the average rate of change is $4a - 4$ and $f$ is quadratic.
• The expression for the average rate of change is $5a - 6$ and $f$ is quadratic.

Since we know the function is quadratic, we can eliminate options that say $f$ is linear. This leaves us with two options to compare the expressions $4a - 4$ and $5a - 6$ to our calculations of average rates of change.

None of the options perfectly match our calculated rates of change, but based on the second difference and quadratic nature, the correct answer would be:

The expression for the average rate of change is $5a - 6$, and $f$ is quadratic.

Would you like more details on any step or further clarifications?

Here are 5 related questions:

1. How do you calculate the second difference for quadratic functions?
2. What are some examples of quadratic functions in real life?
3. How does the average rate of change differ between linear and quadratic functions?
4. How can you derive a quadratic function given points on a table?
5. What is the significance of the second difference being constant?

Tip: For quadratic functions, the second difference of consecutive outputs is always constant.