Question

x | -1 | 0 | 1 | 2 | 3 | 4
f(x) | -2 | 1 | 4 | 7 | 10 | 13
h(x) = x² + x - 6
place the tiles in order from least to greatest according to the average rate of change of the functions over the interval 0, 3.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). We will apply this formula to each of the three functions: \( f(x) \) (from the table), \( g(x) \) (from the graph), and \( h(x) = x^2 + x - 6 \) over the interval \([0, 3]\).

Step2: Calculate average rate of change for \( f(x) \)

For \( f(x) \), \( a = 0 \), \( b = 3 \). From the table, \( f(0) = 1 \) and \( f(3) = 10 \).
Using the formula: \(\frac{f(3) - f(0)}{3 - 0} = \frac{10 - 1}{3} = \frac{9}{3} = 3\).

Step3: Calculate average rate of change for \( g(x) \)

First, we need to find the coordinates of \( g(x) \) at \( x = 0 \) and \( x = 3 \) from the graph.

• At \( x = 0 \), the graph of \( g(x) \) passes through \( (0, 1) \) (since when \( x = 0 \), \( y = 1 \)).
• At \( x = 3 \), looking at the graph, when \( x = 3 \), \( y = 7 \) (since the point \( (3, 7) \) is on the graph of \( g(x) \)).

Now, apply the average rate of change formula: \(\frac{g(3) - g(0)}{3 - 0} = \frac{7 - 1}{3} = \frac{6}{3} = 2\).

Step4: Calculate average rate of change for \( h(x) \)

For \( h(x) = x^2 + x - 6 \), \( a = 0 \), \( b = 3 \).
First, find \( h(0) \): \( h(0) = 0^2 + 0 - 6 = -6 \).
Then, find \( h(3) \): \( h(3) = 3^2 + 3 - 6 = 9 + 3 - 6 = 6 \).
Now, apply the formula: \(\frac{h(3) - h(0)}{3 - 0} = \frac{6 - (-6)}{3} = \frac{12}{3} = 4\).

Step5: Compare the average rates of change

We have:

• Average rate of change of \( g(x) \): \( 2 \)
• Average rate of change of \( f(x) \): \( 3 \)
• Average rate of change of \( h(x) \): \( 4 \)

So, from least to greatest, the order is \( g(x) \), \( f(x) \), \( h(x) \).

Answer:

\( g(x) \), \( f(x) \), \( h(x) \)