Question

the functions f(x), g(x), and h(x) are shown below. select the option that represents the ordering of the functions according to their average rate of change on the interval -1, 2 from least to greatest.

(graph of f(x) is shown, table of g(x) values: x=-7, g(x)=148; x=-4, g(x)=85; x=-1, g(x)=40; x=2, g(x)=13; x=5, g(x)=4; x=8, g(x)=13; h(x) = x² - 8x + 15)

answer options:

• h(x), g(x), f(x)
• g(x), h(x), f(x)
• g(x), f(x), h(x)

Explanation:

To solve this, we analyze the average rate of change (slope) for each function over a relevant interval (e.g., from \(x = -7\) to \(x = 8\) for \(g(x)\), and visually for \(f(x)\) and \(h(x)\)).

Step 1: Analyze \(h(x) = x^2 - 8x + 15\)

The average rate of change of a quadratic \(ax^2 + bx + c\) over an interval \([a, b]\) is \(\frac{h(b) - h(a)}{b - a}\). For \(h(x)\), let's use \(x = -7\) to \(x = 8\):
\(h(-7) = (-7)^2 - 8(-7) + 15 = 49 + 56 + 15 = 120\)
\(h(8) = 8^2 - 8(8) + 15 = 64 - 64 + 15 = 15\)
Average rate of change: \(\frac{15 - 120}{8 - (-7)} = \frac{-105}{15} = -7\).

Step 2: Analyze \(g(x)\) (from table)

Using \(x = -7\) to \(x = 8\):
\(g(-7) = 148\), \(g(8) = 13\)
Average rate of change: \(\frac{13 - 148}{8 - (-7)} = \frac{-135}{15} = -9\).

Step 3: Analyze \(f(x)\) (graph)

The graph of \(f(x)\) is a parabola opening upward (U - shaped) with a minimum. The average rate of change over a large interval (e.g., from leftmost to rightmost) is positive (since it rises from left to right after the vertex).

Step 4: Compare Rates

• \(g(x)\) has a rate of \(-9\) (most negative, least average rate of change).
• \(h(x)\) has a rate of \(-7\) (less negative than \(g(x)\)).
• \(f(x)\) has a positive rate (greatest average rate of change).

Thus, the order from least to greatest average rate of change is \(g(x)\), \(h(x)\), \(f(x)\).

Answer:

g(x), h(x), f(x) (matching the option "g(x), h(x), f(x)")