To analyze the statements regarding the linear function \( k \) that passes through the points \( (-7, 0) \) and \( (1, 8) \), we'll follow these steps:
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Calculate the slope of the line using the formula \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\) where \((x_1, y_1) = (-7, 0)\) and \((x_2, y_2) = (1, 8)\).
\[ \text{slope} = \frac{8 - 0}{1 - (-7)} = \frac{8}{1 + 7} = \frac{8}{8} = 1 \]
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Find the y-intercept using the point-slope form or slope-intercept form \( y = mx + b \). Here, we can use one of the points, say \((-7, 0)\):
\[ 0 = 1 \cdot (-7) + b \implies 0 = -7 + b \implies b = 7 \]
Hence, the equation of the line is: \[ y = 1x + 7 \quad \text{or} \quad y = x + 7 \]
Now let's analyze the statements:
A. The slope of the graph of k is −4 over 3.
- This statement is false since we calculated the slope to be 1.
B. The graph of k passes through the point (−1, −8).
- To check this, substitute \( x = -1 \) into the equation \( y = x + 7 \): \[ y = -1 + 7 = 6 \quad \text{(not -8)} \]
- So this statement is false.
C. The zero of k is 7.
- The zero of the function is the x-value where \( y = 0 \): \[ 0 = x + 7 \implies x = -7 \]
- Thus, this statement is false.
D. The x-intercept of the graph of k is −7.
- The x-intercept (where \( y = 0 \)) is already found to be \( x = -7 \) as shown above.
- So this statement is true.
Therefore, the correct answer is:
D. The x-intercept of the graph of k is −7.
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