To analyze the linear function \( f \) that passes through the points \( (-3, 0) \) and \( (1, 8) \), we can determine the following:
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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 \( 2 \).
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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 \( f(x) = 2x + 6 \).
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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.
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