To determine which function has the steeper slope, we first need to find the slopes of both functions.
Step-by-Step Solution:
### Function 1:
1. Identify the Points:
- [tex]\(x\)[/tex]-intercept: [tex]\((3, 0)\)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\((0, 4)\)[/tex]
2. Calculate the Slope: The slope [tex]\(m\)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Substitute the Values:
- [tex]\( (x_1, y_1) = (3, 0) \)[/tex]
- [tex]\( (x_2, y_2) = (0, 4) \)[/tex]
Therefore,
[tex]\[
m_1 = \frac{4 - 0}{0 - 3} = \frac{4}{-3} = -\frac{4}{3}
\][/tex]
### Function 2:
1. Select Two Points from the Table:
- Let's use [tex]\((-12, -4)\)[/tex] and [tex]\( (4, 8) \)[/tex].
2. Calculate the Slope: Using the formula for slope:
[tex]\[
m_2 = \frac{y_4 - y_3}{x_4 - x_3}
\][/tex]
3. Substitute the Values:
- [tex]\( (x_3, y_3) = (-12, -4) \)[/tex]
- [tex]\( (x_4, y_4) = (4, 8) \)[/tex]
Therefore,
[tex]\[
m_2 = \frac{8 - (-4)}{4 - (-12)} = \frac{8 + 4}{4 + 12} = \frac{12}{16} = \frac{3}{4}
\][/tex]
### Comparison of Slopes:
To determine which function has the steeper slope, compare the absolute values of the slopes.
- Slope of Function 1: [tex]\( -\frac{4}{3} \)[/tex] has an absolute value of [tex]\( \left|-\frac{4}{3}\right| = \frac{4}{3} \)[/tex].
- Slope of Function 2: [tex]\( \frac{3}{4} \)[/tex] has an absolute value of [tex]\( \left|\frac{3}{4}\right| = \frac{3}{4} \)[/tex].
Since [tex]\( \frac{4}{3} > \frac{3}{4} \)[/tex], Function 1 has the steeper slope.
### Conclusion:
- The slope of Function 1 is [tex]\( -\frac{4}{3} \)[/tex].
- The slope of Function 2 is [tex]\( \frac{3}{4} \)[/tex].
- Therefore, Function 1 has the steeper slope.
The correct answer is:
D. Function 1 has a steeper slope of [tex]\(-\frac{4}{3}\)[/tex].
