Question

A graph is shown with the points (-6, -6), (-2, -4), (2, -2) and (6, 0)

Compare the function ƒ(x) = 3x − 1/2 to the function shown in the graph. Which statements are correct?
Responses
A The function ƒ(x) = 3x −
























and the graphed function have the same slope.The function ƒ(x) = 3x − 1 2 and the graphed function have the same slope.
B The graphed function has a lesser rate of change than the function ƒ(x) = 3x −
























.The graphed function has a lesser rate of change than the function ƒ(x) = 3x − 1 2 .
C The function ƒ(x) = 3x −
























has a lesser rate of change than the graphed function.The function ƒ(x) = 3x − 1 2 has a lesser rate of change than the graphed function.
D The graphed function has a y-intercept with a lesser value than the function ƒ(x) = 3x −
























.The graphed function has a y-intercept with a lesser value than the function ƒ(x) = 3x − 1 2 .
E The function ƒ(x) = 3x −
























has a x-intercept with a lesser value than the graphed function.

Answer 1

To compare the function \( f(x) = 3x - \frac{1}{2} \) with the points provided in the graph (-6, -6), (-2, -4), (2, -2), and (6, 0), let’s analyze the characteristics of both functions.

1. **Slope Comparison**:
The slope of the function \( f(x) = 3x - \frac{1}{2} \) is 3.

For the graphed function, we can find the slope between two points. Let's calculate the slope between the points (-2, -4) and (2, -2):
\[
\text{slope} = \frac{-2 - (-4)}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}
\]

Thus, the slope of the graphed function is \( \frac{1}{2} \), which is less than 3.

2. **Rate of Change**:
Since the slope of the graphed function (\( \frac{1}{2} \)) is less than the slope of \( f(x) \) (3), we can confirm that:
- **B is correct**: "The graphed function has a lesser rate of change than the function \( f(x) = 3x - \frac{1}{2} \)."
- **C is incorrect**: The function \( f(x) = 3x - \frac{1}{2} \) does not have a lesser rate of change than the graphed function; it has a greater slope.

3. **Y-Intercept Comparison**:
The y-intercept of \( f(x) = 3x - \frac{1}{2} \) is -1/2 (or -0.5). To find the y-intercept of the graphed function, we can see that at \( x = 0 \) (interpolating or checking the points), the graph does not cross above y = -1. It likely stays below -0.5, so this suggests:
- **D is correct**: "The graphed function has a y-intercept with a lesser value than the function \( f(x) = 3x - \frac{1}{2} \)."

4. **X-Intercept Comparison**:
The x-intercept of the function \( f(x) = 3x - \frac{1}{2} \) can be found by setting \( f(x) = 0 \):
\[
3x - \frac{1}{2} = 0 \implies 3x = \frac{1}{2} \implies x = \frac{1}{6}
\]

For the graphed function, we can find the x-intercept by locating where y = 0 in the given points; that is at x = 6. Since \( \frac{1}{6} < 6 \):
- **E is correct**: "The function \( f(x) = 3x - \frac{1}{2} \) has a x-intercept with a lesser value than the graphed function."

So the correct statements are:
- **A**: (Incorrect). They have different slopes.
- **B**: (Correct). The graphed function has a lesser rate of change.
- **C**: (Incorrect). The function \( f(x) \) has a greater rate of change.
- **D**: (Correct). The graphed function has a lesser y-intercept value.
- **E**: (Correct). The function \( f(x) \) has a lesser x-intercept.

The overall correct responses are **B**, **D**, and **E**.