Certainly! Let's break down the problem step-by-step to find the slope and the y-intercept, and then use this information to complete the slope-intercept form of the line.
### Step 1: Identify the given data points
We have a set of data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 17 \\
\hline
-1 & 13 \\
\hline
1 & 5 \\
\hline
2 & 1 \\
\hline
3 & -3 \\
\hline
\end{array}
\][/tex]
### Step 2: Determine the rate of change (slope)
The rate of change (slope) is given as [tex]\(-4\)[/tex]. This tells us how much [tex]\(y\)[/tex] changes for a change in [tex]\(x\)[/tex].
### Step 3: Use the slope to find the y-intercept
We need to determine the y-intercept [tex]\(b\)[/tex] in the slope-intercept form [tex]\(y = mx + b\)[/tex]. We can use any of the given data points and the slope. Let's use the point [tex]\((-2, 17)\)[/tex]:
Recall the equation of the line in slope-intercept form:
[tex]\[ y = mx + b \][/tex]
Substitute [tex]\(m = -4\)[/tex], [tex]\(x = -2\)[/tex], and [tex]\(y = 17\)[/tex]:
[tex]\[ 17 = -4(-2) + b \][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[ 17 = 8 + b \][/tex]
[tex]\[ b = 17 - 8 \][/tex]
[tex]\[ b = 9 \][/tex]
### Step 4: Write the slope-intercept form
Now we have both the slope ([tex]\(m = -4\)[/tex]) and the y-intercept ([tex]\(b = 9\)[/tex]). So the slope-intercept form of the line is:
[tex]\[
y = -4x + 9
\][/tex]
### Summary
- The rate of change (slope) is:
[tex]\[ \boxed{-4} \][/tex]
- The complete slope-intercept form of the line is:
[tex]\[ y = -4x + \boxed{9} \][/tex]
