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Consider the linear functions below.

Function 1:
- [tex]$x$[/tex]-intercept: [tex]$(3,0)$[/tex]
- [tex]$y$[/tex]-intercept: [tex]$(0,4)$[/tex]

Function 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-12 & -4 \\
\hline
-8 & -1 \\
\hline
-4 & 2 \\
\hline
0 & 5 \\
\hline
4 & 8 \\
\hline
\end{array}
\][/tex]

Find the slope of each function and determine which has the steeper slope.

A. Function 2 has a steeper slope of [tex]$\frac{3}{4}$[/tex].
B. Function 1 has a steeper slope of [tex]$\frac{4}{3}$[/tex].
C. Function 1 has a steeper slope of [tex]$-\frac{4}{3}$[/tex].
D. Function 2 has a steeper slope of [tex]$-\frac{3}{4}$[/tex].


Sagot :

Let's determine the slope of each function and see which one is steeper.

### Function 1:
Given points:
- [tex]\(x\)[/tex]-intercept: [tex]\((3, 0)\)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\((0, 4)\)[/tex]

The formula for 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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Applying the given points to the slope formula:
[tex]\[ m_1 = \frac{4 - 0}{0 - 3} = \frac{4}{-3} = -\frac{4}{3} \][/tex]

So, the slope of Function 1 is [tex]\(-\frac{4}{3}\)[/tex].

### Function 2:
Given the points:
[tex]\((-12, -4)\)[/tex] and [tex]\((-8, -1)\)[/tex]

Using the slope formula for these points:
[tex]\[ m_2 = \frac{-1 - (-4)}{-8 - (-12)} = \frac{-1 + 4}{-8 + 12} = \frac{3}{4} \][/tex]

So, the slope of Function 2 is [tex]\(\frac{3}{4}\)[/tex].

### Comparison:
To determine which slope is steeper, we compare the absolute values of the slopes:
[tex]\[ |\text{slope of Function 1}| = \left|-\frac{4}{3}\right| = \frac{4}{3} \][/tex]
[tex]\[ |\text{slope of Function 2}| = \left|\frac{3}{4}\right| = \frac{3}{4} \][/tex]

Since [tex]\(\frac{4}{3}\)[/tex] is greater than [tex]\(\frac{3}{4}\)[/tex], the slope of Function 1 is steeper.

Thus, the correct answer is:

C. Function 1 has a steeper slope of [tex]\(-\frac{4}{3}\)[/tex].