Question

A coordinate plane with 2 lines drawn. The first line is labeled f(x) and passes through the points (0, negative 2) and (1, 1). The second line is labeled g(x) and passes through the points (negative 4, 0) and (0, 2). The lines intersect at about (2.5, 3.2)
How does the slope of g(x) compare to the slope of f(x)?

The slope of g(x) is the opposite of the slope of f(x).
The slope of g(x) is less than the slope of f(x).
The slope of g(x) is greater than the slope of f(x).
The slope of g(x) is equal to the slope of f(x).

Answer 1

To compare the slopes of the lines \( f(x) \) and \( g(x) \), we first need to find their slopes based on the given points.

1. **Determine the slope of \( f(x) \)**:
The line \( f(x) \) passes through the points \( (0, -2) \) and \( (1, 1) \).
The slope \( m_f \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the points, we have:
\[
m_f = \frac{1 - (-2)}{1 - 0} = \frac{1 + 2}{1} = \frac{3}{1} = 3
\]

2. **Determine the slope of \( g(x) \)**:
The line \( g(x) \) passes through the points \( (-4, 0) \) and \( (0, 2) \).
The slope \( m_g \) is calculated similarly:
\[
m_g = \frac{2 - 0}{0 - (-4)} = \frac{2 - 0}{0 + 4} = \frac{2}{4} = \frac{1}{2}
\]

3. **Comparison of slopes**:
Now we have:
- The slope of \( f(x) \): \( m_f = 3 \)
- The slope of \( g(x) \): \( m_g = \frac{1}{2} \)

To compare the two slopes:
Since \( 3 > \frac{1}{2} \), we can conclude:
- The slope of \( g(x) \) is less than the slope of \( f(x) \).

Thus, the correct statement is:
**The slope of g(x) is less than the slope of f(x).**