Question

The graph of linear function f passes through the points (-3, 0) and (1, 8)

Which of the following statements must be true about the function?
Responses
A The x-intercept of the graph of f is -3.The x -intercept of the graph of f is -3.
B The function has a slope of -4.The function has a slope of -4.
C The graph of f passes through the point (-1, -8).The graph of f passes through the point (-1, -8).
D The zero of f is 3.

Answer 1

To analyze the linear function \( f \) that passes through the points \( (-3, 0) \) and \( (1, 8) \), we can determine the following:

1. **Finding the Slope**:
The slope \( m \) of a linear function can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points \( (-3, 0) \) (which we'll label as \( (x_1, y_1) \)) and \( (1, 8) \) (which we'll label as \( (x_2, y_2) \)):
\[
m = \frac{8 - 0}{1 - (-3)} = \frac{8}{1 + 3} = \frac{8}{4} = 2
\]
So the slope of the function is \( 2 \).

2. **The y-intercept**:
We can use the slope-intercept form of a linear equation \( f(x) = mx + b \) to find the y-intercept \( b \). We know the slope \( m \) and one of the points. Using point \( (-3, 0) \):
\[
0 = 2(-3) + b
\]
\[
0 = -6 + b
\]
\[
b = 6
\]
Thus, the equation of the linear function is \( f(x) = 2x + 6 \).

3. **Finding the x-intercept**:
The x-intercept occurs where \( f(x) = 0 \):
\[
0 = 2x + 6
\]
\[
2x = -6
\]
\[
x = -3
\]

Now we can analyze each of the given statements:

A. **The x-intercept of the graph of f is -3.**
This is true, as we calculated that the x-intercept is -3.

B. **The function has a slope of -4.**
This is false; the slope is actually \( 2 \), not \( -4 \).

C. **The graph of f passes through the point (-1, -8).**
We can check this point by substituting \( x = -1 \) into the function:
\[
f(-1) = 2(-1) + 6 = -2 + 6 = 4
\]
So the point \( (-1, -8) \) is not on the graph of the function.

D. **The zero of f is 3.**
The zero of \( f \) refers to the x-value for which \( f(x) = 0 \). Since we found that the x-intercept (zero) is -3, this statement is false.

The only statement that must be true is:
**A. The x-intercept of the graph of f is -3.**