Question

x | -1 | 0 | 1 | 2 | 3
--- | --- | --- | --- | --- | ---
h(x) | -7 | -4 | -1 | 2 | 5

order the above functions, from least to greatest, by the rate of change of functions over the interval 0, 2.

a. g, f, h
b. f, g, h

Explanation:

Step1: Analyze function \( f \) (the horizontal line)

The function \( f \) is a horizontal line, so its rate of change (slope) over any interval is \( 0 \). Because for a horizontal line \( y = c \) (constant), the change in \( y \) (\( \Delta y \)) is \( 0 \) for any \( \Delta x \), so rate of change \(=\frac{\Delta y}{\Delta x}=0\).

Step2: Analyze function \( g \) (the curve)

First, find two points on \( g \) over \([0, 2]\). At \( x = 0 \), from the graph, \( y = 4 \) (since it crosses the y - axis at \( (0, 4) \)). At \( x = 2 \), let's assume the grid is 1 unit per square. Looking at the graph, when \( x = 2 \), the \( y \) - value of \( g \) is, say, let's check the curve. Wait, actually, maybe we can estimate. Wait, no, maybe the curve is an exponential or something, but let's get two points. At \( x = 0 \), \( y = 4 \); at \( x = 2 \), let's see, the curve at \( x = 2 \) is above \( y = 6 \) (from the graph, the red curve at \( x = 2 \) is higher than \( y = 6 \)? Wait, no, maybe I misread. Wait, the y - axis: the grid lines, so at \( x = 0 \), \( y = 4 \); at \( x = 2 \), let's say the point is \( (2, y_2) \). Wait, maybe the curve is \( g(x) \), let's take \( x = 0 \), \( y = 4 \); \( x = 2 \), let's see, the graph shows that at \( x = 2 \), the \( y \) - value is, for example, if we look at the grid, each square is 1 unit. So at \( x = 0 \), \( y = 4 \); at \( x = 2 \), let's say \( y \) is, like, 8? Wait, no, maybe better to calculate the rate of change. Rate of change for \( g \) over \([0, 2]\) is \( \frac{g(2)-g(0)}{2 - 0} \). From the graph, \( g(0)=4 \), and at \( x = 2 \), the curve is above \( y = 6 \), let's say \( g(2) \) is, for example, 8 (maybe it's an exponential function like \( g(x)=4 + \text{something} \)). Wait, maybe I made a mistake. Wait, actually, the horizontal line \( f \) has rate of change 0. Now for \( h(x) \):

Step3: Analyze function \( h(x) \) (the table)

For \( h(x) \), we have the table:
| \( x \) | - 1 | 0 | 1 | 2 | 3 |
| --- | --- | --- | --- | --- | --- |
| \( h(x) \) | - 7 | - 4 | - 1 | 2 | 5 |

Over the interval \([0, 2]\), \( x_1 = 0 \), \( h(x_1)=-4 \); \( x_2 = 2 \), \( h(x_2)=2 \). The rate of change is \( \frac{h(2)-h(0)}{2 - 0}=\frac{2-(-4)}{2}=\frac{6}{2}=3 \).

Now, for \( g \): at \( x = 0 \), \( y = 4 \); let's find \( g(2) \). Looking at the graph, when \( x = 2 \), the \( y \) - coordinate of \( g \) is, let's see, the curve is increasing, and at \( x = 2 \), it's above \( y = 6 \), maybe \( y = 8 \) (since the curve is steepening). So rate of change for \( g \) is \( \frac{g(2)-g(0)}{2-0}=\frac{8 - 4}{2}=\frac{4}{2}=2 \)? Wait, no, maybe my estimation is wrong. Wait, maybe the curve at \( x = 0 \) is \( (0, 4) \) and at \( x = 2 \) is \( (2, 8) \)? Or maybe another point. Wait, actually, the key is: \( f \) has rate 0, \( g \) has a positive rate (since it's increasing), and \( h \) has rate 3. Wait, but wait, maybe I messed up \( g \)'s points. Wait, let's re - examine the graph. The red curve: at \( x = 0 \), \( y = 4 \); as \( x \) increases from 0 to 2, the curve goes up. Let's count the grid. Each square is 1 unit. At \( x = 0 \), \( y = 4 \); at \( x = 2 \), let's see, the curve is at \( y = 8 \)? No, maybe \( y = 6 \)? Wait, no, the point labeled \( g \) is near \( x = 1 \), \( y = 6 \)? Wait, maybe the correct way is:

Wait, the function \( f \) is horizontal, so rate of change \( m_f = 0 \).

Function \( h \): rate of change \( m_h=\frac{h(2)-h(0)}{2 - 0}=\frac{2-(-4)}{2}=3 \).

Function \( g \): let's take two points on \( g \) in \([0, 2]\). At \( x = 0 \), \( y = 4 \); at \( x = 2 \), let's say from the graph, when \( x = 2 \), the \( y \) - value is, for example, 8 (since the curve is exponential - like, increasing rapidly). So \( m_g=\frac{8 - 4}{2}=2 \). Wait, but then the rates are: \( f:0 \), \( g:2 \), \( h:3 \). So the order from least to greatest is \( f \) (rate 0), \( g \) (rate 2), \( h \) (rate 3)? No, wait, the options are A. \( g,f,h \) and B. \( f,g,h \).

Wait, I think I made a mistake with \( f \). The function \( f \) is the horizontal line, so its rate of change is 0. \( g \) is increasing, so its rate of change is positive. \( h \) has a rate of change of 3. Now, what's the rate of change of \( g \)? Let's take more accurate points. At \( x = 0 \), \( y = 4 \); at \( x = 2 \), looking at the graph, the \( y \) - coordinate: let's see, the grid has x - axis from - 8 to 8, y - axis from - 8 to 8. The curve at \( x = 0 \) is (0, 4), at \( x = 2 \), let's count the squares. From \( x = 0 \) to \( x = 2 \) (2 units right), the \( y \) - value goes from 4 to, say, 8 (since the curve is steep). So rate of change is \( (8 - 4)/2 = 2 \). \( h \) has rate 3. \( f \) has rate 0. So the order from least to greatest is \( f \) (0), \( g \) (2), \( h \) (3), which is \( f,g,h \), so option B.

Answer:

B. \( f,g,h \)