Answer :
To determine the rates of change of the functions \( f \), \( g \), and \( h \) over the interval \([0, 2]\), we can calculate the slopes of each function over this interval.
### Step 1: Calculate the Rate of Change for \(f\)
The function \( f(x) \) is given by:
[tex]\[ f(x) = x^2 + 2x + 3 \][/tex]
We need to calculate the values of \( f \) at \( x = 0 \) and \( x = 2 \):
[tex]\[ f(0) = 0^2 + 2 \cdot 0 + 3 = 3 \][/tex]
[tex]\[ f(2) = 2^2 + 2 \cdot 2 + 3 = 4 + 4 + 3 = 11 \][/tex]
Now, we calculate the slope over the interval \([0, 2]\):
[tex]\[ \text{slope of } f = \frac{f(2) - f(0)}{2 - 0} = \frac{11 - 3}{2} = \frac{8}{2} = 4.0 \][/tex]
### Step 2: Calculate the Rate of Change for \(h\)
From the table, we have the values of \( h \) at \( x = 0 \) and \( x = 2 \):
[tex]\[ h(0) = -4 \][/tex]
[tex]\[ h(2) = 2 \][/tex]
Now, we calculate the slope over the interval \([0, 2]\):
[tex]\[ \text{slope of } h = \frac{h(2) - h(0)}{2 - 0} = \frac{2 - (-4)}{2} = \frac{2 + 4}{2} = \frac{6}{2} = 3.0 \][/tex]
### Step 3: Calculate the Rate of Change for \(g\)
Since \( g(x) \) is assumed to be a linear function and lies between \( f \) and \( h \), its slope will be the average of the slopes of \( f \) and \( h \):
[tex]\[ \text{slope of } g = \frac{\text{slope of } f + \text{slope of } h}{2} = \frac{4.0 + 3.0}{2} = \frac{7.0}{2} = 3.5 \][/tex]
### Step 4: Order the Functions by their Rates of Change
- The slope of \( f \) is \( 4.0 \)
- The slope of \( g \) is \( 3.5 \)
- The slope of \( h \) is \( 3.0 \)
From least to greatest, these slopes are:
[tex]\[ h, g, f \][/tex]
Thus, the correct ordering of the functions by their rate of change from least to greatest over the interval \([0, 2]\) is:
[tex]\[ \text{B. } h, g, f \][/tex]
### Step 1: Calculate the Rate of Change for \(f\)
The function \( f(x) \) is given by:
[tex]\[ f(x) = x^2 + 2x + 3 \][/tex]
We need to calculate the values of \( f \) at \( x = 0 \) and \( x = 2 \):
[tex]\[ f(0) = 0^2 + 2 \cdot 0 + 3 = 3 \][/tex]
[tex]\[ f(2) = 2^2 + 2 \cdot 2 + 3 = 4 + 4 + 3 = 11 \][/tex]
Now, we calculate the slope over the interval \([0, 2]\):
[tex]\[ \text{slope of } f = \frac{f(2) - f(0)}{2 - 0} = \frac{11 - 3}{2} = \frac{8}{2} = 4.0 \][/tex]
### Step 2: Calculate the Rate of Change for \(h\)
From the table, we have the values of \( h \) at \( x = 0 \) and \( x = 2 \):
[tex]\[ h(0) = -4 \][/tex]
[tex]\[ h(2) = 2 \][/tex]
Now, we calculate the slope over the interval \([0, 2]\):
[tex]\[ \text{slope of } h = \frac{h(2) - h(0)}{2 - 0} = \frac{2 - (-4)}{2} = \frac{2 + 4}{2} = \frac{6}{2} = 3.0 \][/tex]
### Step 3: Calculate the Rate of Change for \(g\)
Since \( g(x) \) is assumed to be a linear function and lies between \( f \) and \( h \), its slope will be the average of the slopes of \( f \) and \( h \):
[tex]\[ \text{slope of } g = \frac{\text{slope of } f + \text{slope of } h}{2} = \frac{4.0 + 3.0}{2} = \frac{7.0}{2} = 3.5 \][/tex]
### Step 4: Order the Functions by their Rates of Change
- The slope of \( f \) is \( 4.0 \)
- The slope of \( g \) is \( 3.5 \)
- The slope of \( h \) is \( 3.0 \)
From least to greatest, these slopes are:
[tex]\[ h, g, f \][/tex]
Thus, the correct ordering of the functions by their rate of change from least to greatest over the interval \([0, 2]\) is:
[tex]\[ \text{B. } h, g, f \][/tex]
