The graph of linear function k passes through the points (−7, 0)
 and (1, 8)
.


Which statement must be true?
A.The zero of k is 7
.     
B.The graph of k passes through the point (−1, −8)
.       
C.The x-intercept of the graph of k is −7
. 
D.The slope of the graph of k is −43
.

We are given that the graph of the linear function \( k \) passes through the points \((-7, 0)\) and \((1, 8)\).

### Step 1: Check the x-intercept
The x-intercept is the point where the function crosses the x-axis, meaning the y-coordinate is 0. Since the point \((-7, 0)\) is given, this is the x-intercept. Thus:

- **C: The x-intercept of the graph of k is \(-7\)** must be true.

### Step 2: Find the slope
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((-7, 0)\) and \((1, 8)\):

\[
m = \frac{8 - 0}{1 - (-7)} = \frac{8}{1 + 7} = \frac{8}{8} = 1
\]

So, the slope is 1, not \(-43\), which means **D is false**.

### Step 3: Evaluate other options
- **A: The zero of \( k \) is 7** is false because the zero (x-intercept) is already determined to be \(-7\), not 7.
- **B: The graph of \( k \) passes through the point \((-1, -8)\)** is false. With a slope of 1 and passing through the x-intercept at \((-7, 0)\), we can calculate the equation of the line to verify this.

### Step 4: Write the equation of the line
Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), where \( m = 1 \) and a point on the line is \((-7, 0)\):

\[
y - 0 = 1(x - (-7)) \implies y = x + 7
\]

Substitute \( x = -1 \) to check if the line passes through \((-1, -8)\):

\[
y = (-1) + 7 = 6
\]

Since \( y = 6 \), not \(-8\), **B is false**.

### Conclusion:
The correct statement is **C: The x-intercept of the graph of \( k \) is \(-7\)**.

Would you like further details or have any questions?

Here are five related questions:
1. How is the equation of a line derived from two points?
2. What is the significance of the slope in a linear function?
3. How can you determine the y-intercept from two points?
4. Why is the x-intercept important in understanding the graph of a line?
5. How would the equation change if the slope was negative?

**Tip:** To find the slope of a line between two points, always remember the formula \(\frac{y_2 - y_1}{x_2 - x_1}\).

The graph of linear function k passes through the points (-7, 0) and (1, 8). Which statement must be true? A. The zero of k is 7. B. The graph of k passes through the point (-1, -8). C. The x-intercept of the graph of k is -7. D. The slope of the graph of k is -43.

Explore how to find the x-intercept and slope of a linear function passing through the points (-7, 0) and (1, 8). This problem uses key concepts such as the slope formula and the point-slope form to derive the correct answer. Suitable for students in Grades 8-10.

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