To find the equation in slope-intercept form of the linear function represented by the given table, we will follow these steps:
1. Calculate the slope (m):
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Let's use the first two points from the table: [tex]\((-6, -18)\)[/tex] and [tex]\((-1, -8)\)[/tex].
[tex]\[
m = \frac{-8 - (-18)}{-1 - (-6)} = \frac{-8 + 18}{-1 + 6} = \frac{10}{5} = 2
\][/tex]
2. Verify the linearity:
To ensure the table represents a linear function, the slope between subsequent points should be the same. Let's use the points [tex]\((-1, -8)\)[/tex] and [tex]\( (4, 2) \)[/tex]:
[tex]\[
m = \frac{2 - (-8)}{4 - (-1)} = \frac{2 + 8}{4 + 1} = \frac{10}{5} = 2
\][/tex]
Now let's use the points [tex]\( (4, 2) \)[/tex] and [tex]\( (9, 12) \)[/tex]:
[tex]\[
m = \frac{12 - 2}{9 - 4} = \frac{12 - 2}{9 - 4} = \frac{10}{5} = 2
\][/tex]
Since the slope is consistent between all points, the function is linear with slope [tex]\( m = 2 \)[/tex].
3. Find the y-intercept (b):
Using the slope-intercept form of the line equation [tex]\( y = mx + b \)[/tex], we can find the y-intercept [tex]\( b \)[/tex] by using one of the points. Let's use the point [tex]\((-6, -18)\)[/tex].
Substitute the values into the equation:
[tex]\[
-18 = 2(-6) + b
\][/tex]
Simplify to solve for [tex]\( b \)[/tex]:
[tex]\[
-18 = -12 + b \implies b = -18 + 12 = -6
\][/tex]
4. Form the equation:
Now that we have the slope [tex]\( m = 2 \)[/tex] and the y-intercept [tex]\( b = -6 \)[/tex], we can write the equation in slope-intercept form:
[tex]\[
y = 2x - 6
\][/tex]
Given the choices provided:
- [tex]\( y = -2x - 6 \)[/tex]
- [tex]\( y = -2x + 6 \)[/tex]
- [tex]\( y = 2x - 6 \)[/tex]
- [tex]\( y = 2x + 6 \)[/tex]
The correct equation matches the option [tex]\( y = 2x - 6 \)[/tex].
Therefore, the correct equation in slope-intercept form of the linear function represented by the table is:
[tex]\[ \boxed{y = 2x - 6} \][/tex]
