Question

1. the functions ( f(x) ), ( g(x) ), and ( h(x) ) are shown below. sort the options that represent the functions according to their average rates of change on the interval ( 1 leq x leq 4 ) from least to greatest. graph of ( f(x) ), table for ( g(x) ): ( \begin{array}{c|c} x & g(x) \\ hline 1 & 6 \\ 2 & 6 \\ 3 & 8 \\ 4 & 12 \\ 5 & 18 end{array} ), ( h(x) = x^2 - 3x + 4 ). options: a. ( f(x), h(x), g(x) ); b. ( h(x), f(x), g(x) ); c. ( g(x), f(x), h(x) ); d. ( g(x), h(x), f(x) ); e. ( h(x), g(x), f(x) ); f. ( f(x), g(x), h(x) ). 4. the functions ( f(x) ), ( g(x) ), and ( h(x) ) are shown below. sort the options that represent the functions according to their average rates of change on the interval ( -2 leq x leq 3 ) from least to greatest. graph of ( f(x) ), table for ( g(x) ): ( \begin{array}{c|c} x & g(x) \\ hline -1 & 6 \\ 0 & 3 \\ 1 & 2 \\ 2 & 3 \\ 3 & 6 end{array} ), ( h(x) = -x^2 + 7x - 4 ). options: a. ( f(x), g(x), h(x) ); b. ( g(x), f(x), h(x) ); c. ( f(x), h(x), g(x) ); d. ( h(x), f(x), g(x) ); e. ( g(x), h(x), f(x) ); f. ( h(x), g(x), f(x) ).

Answer:

To solve these problems, we need to calculate the average rate of change for each function over the given interval and then order them from least to greatest (or fastest to slowest, depending on the context). The average rate of change of a function \( y = f(x) \) over the interval \([a, b]\) is given by:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Left Problem (Interval \( 1 \leq x \leq 4 \))

Step 1: Analyze \( f(x) \) (Graph)

• From the graph, we can observe the behavior of \( f(x) \). Let's assume the key points (though the graph is a bit unclear, we can infer from the shape). However, since we have \( g(x) \) (table) and \( h(x) \) (formula), we'll calculate their average rates first.

Step 2: Calculate Average Rate for \( g(x) \) (Table)

• \( g(x) \) values: \( x = 1 \) (6), \( x = 2 \) (6), \( x = 3 \) (8), \( x = 4 \) (12)
• Interval: \( [1, 4] \)
• Average Rate of Change:

\[
\frac{g(4) - g(1)}{4 - 1} = \frac{12 - 6}{3} = \frac{6}{3} = 2
\]

Step 3: Calculate Average Rate for \( h(x) \) (Formula: \( h(x) = x^2 - 3x + 4 \))

• \( h(1) = 1^2 - 3(1) + 4 = 1 - 3 + 4 = 2 \)
• \( h(4) = 4^2 - 3(4) + 4 = 16 - 12 + 4 = 8 \)
• Average Rate of Change:

\[
\frac{h(4) - h(1)}{4 - 1} = \frac{8 - 2}{3} = \frac{6}{3} = 2
\]

Wait, this can't be right. Maybe I misread the formula. Let me check again. The formula for \( h(x) \) in the left problem: the user wrote \( h(x) = x^2 - 3x + 4 \)? Wait, the table for \( g(x) \) is:

\( x \): 1, 2, 3, 4, 5

\( g(x) \): 6, 6, 8, 12, 18

Wait, maybe \( h(x) \) is a different formula. Wait, the left problem's options are about ordering \( f(x) \), \( g(x) \), \( h(x) \) by average rate of change from least to greatest.

Wait, maybe the graph of \( f(x) \) has a peak and a valley. Let's re-express:

• For \( g(x) \): from \( x=1 \) to \( x=4 \), the change is \( 12 - 6 = 6 \) over 3 units, so rate 2.
• For \( h(x) \): Let's recalculate \( h(1) \) and \( h(4) \) correctly. If \( h(x) = x^2 - 3x + 4 \):

\( h(1) = 1 - 3 + 4 = 2 \)

\( h(4) = 16 - 12 + 4 = 8 \)

So rate is \( (8 - 2)/3 = 2 \).

But the graph of \( f(x) \): if it's a function with a peak and a valley, maybe its average rate from 1 to 4 is less than 2? Wait, maybe I made a mistake. Alternatively, maybe the formula for \( h(x) \) is different. Wait, the right problem has \( h(x) = -x^2 + 7x - 4 \). Maybe the left problem's \( h(x) \) is different. Alternatively, maybe the table is for \( g(x) \), and the formula is for \( h(x) \), and the graph is \( f(x) \).

Alternatively, let's look at the options. The left problem's options are:

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

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

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

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

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

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

Wait, maybe the average rate of \( f(x) \) is negative? If the graph has a peak at \( x=2 \) or something, then from \( x=1 \) to \( x=4 \), maybe it decreases then increases. Let's assume:

• \( f(x) \): from \( x=1 \) to \( x=4 \), if the graph goes up then down then up, maybe the average rate is lower.

• \( g(x) \): average rate 2 (as calculated)

• \( h(x) \): average rate 2 (as calculated)

But this is confusing. Maybe the left problem's \( h(x) \) is \( x^2 - 3x + 4 \), and \( g(x) \) is the table, and \( f(x) \) is the graph. Let's check the right problem.

Right Problem (Interval \( -2 \leq x \leq 3 \))

Step 1: Analyze \( f(x) \) (Graph)

• The graph of \( f(x) \) has a peak, so it's a downward-opening parabola? Wait, no, the graph shows a peak, so it's a quadratic with a maximum.

Step 2: Calculate Average Rate for \( g(x) \) (Table)

• \( g(x) \) values: \( x = -2 \) (11), \( x = -1 \) (6), \( x = 0 \) (3), \( x = 1 \) (2), \( x = 2 \) (3), \( x = 3 \) (6)
• Interval: \( [-2, 3] \)
• \( g(-2) = 11 \), \( g(3) = 6 \)
• Average Rate of Change:

\[
\frac{g(3) - g(-2)}{3 - (-2)} = \frac{6 - 11}{5} = \frac{-5}{5} = -1
\]

Step 3: Calculate Average Rate for \( h(x) \) (Formula: \( h(x) = -x^2 + 7x - 4 \))

• \( h(-2) = -(-2)^2 + 7(-2) - 4 = -4 - 14 - 4 = -22 \)
• \( h(3) = -(3)^2 + 7(3) - 4 = -9 + 21 - 4 = 8 \)
• Average Rate of Change:

\[
\frac{h(3) - h(-2)}{3 - (-2)} = \frac{8 - (-22)}{5} = \frac{30}{5} = 6
\]

Step 4: Analyze \( f(x) \) (Graph)

• From the graph, \( f(x) \) passes through \( (-2, \text{low}) \) and \( (3, \text{peak}) \). Let's assume the key points. If the graph at \( x=-2 \) is, say, 0, and at \( x=3 \) is, say, 8? Wait, no, the graph of \( f(x) \) intersects \( g(x) \) and \( h(x) \). Alternatively, let's check the average rate.

• \( f(x) \) is a parabola with a maximum. Let's assume \( f(-2) \) is, say, 0, and \( f(3) \) is 8 (since \( h(3) = 8 \)). Then the average rate would be \( (8 - 0)/5 = 1.6 \), but this is guesswork. Alternatively, let's use the options.

The right problem's options are:

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

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

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

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

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

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

We calculated:

• \( g(x) \) average rate: \( -1 \)

• \( h(x) \) average rate: \( 6 \)

• \( f(x) \): Let's assume the graph of \( f(x) \) has a positive average rate (since it goes from a low to a peak). Let's say \( f(-2) = 0 \) and \( f(3) = 5 \), so average rate \( (5 - 0)/5 = 1 \). Then the order from fastest to slowest (or least to greatest) would be \( g(x) \) (rate -1), \( f(x) \) (rate 1), \( h(x) \) (rate 6). So the order from least to greatest (slowest to fastest) is \( g(x), f(x), h(x) \), which is option B.

Left Problem (Revisited)

Let's re-express:

• \( g(x) \) table: \( x=1 \) (6), \( x=4 \) (12). So average rate \( (12 - 6)/(4 - 1) = 2 \).

• \( h(x) \) formula: \( h(x) = x^2 - 3x + 4 \). \( h(1) = 2 \), \( h(4) = 8 \). Average rate \( (8 - 2)/3 = 2 \).

• \( f(x) \) graph: If the graph has a peak at \( x=2 \), then from \( x=1 \) to \( x=4 \), the function might decrease then increase. Let's say \( f(1) = 10 \), \( f(4) = 5 \). Then average rate \( (5 - 10)/3 = -5/3 \approx -1.67 \).

So the average rates would be:

• \( f(x) \): ~ -1.67

• \( g(x) \): 2

• \( h(x) \): 2

Wait, but the options include \( h(x), f(x), g(x) \) (option B) or \( f(x), h(x), g(x) \) (option A). If \( f(x) \) has a negative rate, \( h(x) \) and \( g(x) \) have positive rates (2), then the order from least to greatest (slowest to fastest) would be \( f(x) \) (negative), \( h(x) \) (2), \( g(x) \) (2)? No, that doesn't make sense. Alternatively, maybe the formula for \( h(x) \) is different. Wait, the left problem's \( h(x) \) is written as \( h(x) = x^2 - 3x + 4 \)? Maybe it's \( h(x) = x^2 - 3x + 4 \), and \( g(x) \) is the table, and \( f(x) \) is the graph.

Alternatively, maybe the left problem's correct answer is B: \( h(x), f(x), g(x) \), but this is uncertain without clearer graph details. However, based on the right problem's calculation, the answer for the right problem is B: \( g(x), f(x), h(x) \).

Final Answers

Left Problem (Assuming Correct Calculation)

After re-evaluating, if \( f(x) \) has a lower average rate (e.g., negative), \( h(x) \) and \( g(x) \) have positive rates, the order from least to greatest (slowest to fastest) would be \( f(x), h(x), g(x) \) (option A) or \( h(x), f(x), g(x) \) (option B). However, due to graph ambiguity, we'll proceed with the right problem.

Right Problem

The average rates are:

• \( g(x) \): \( -1 \)

• \( f(x) \): Positive (e.g., 1)

• \( h(x) \): \( 6 \)

Thus, the order from least to greatest (slowest to fastest) is \( g(x), f(x), h(x) \), which is option B.

Final Answers

Left Problem (Best Guess): \(\boldsymbol{B}\) ( \( h(x), f(x), g(x) \) )

Right Problem: \(\boldsymbol{B}\) ( \( g(x), f(x), h(x) \) )