To determine the equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] from the given points on the table, we need to follow these steps:
### Step 1: Calculate the Slope [tex]\( m \)[/tex]
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
We will use the first two points [tex]\((-3, -8)\)[/tex] and [tex]\((-1, -2)\)[/tex]:
[tex]\[
x_1 = -3, \quad y_1 = -8
\][/tex]
[tex]\[
x_2 = -1, \quad y_2 = -2
\][/tex]
Plug these values into the slope formula:
[tex]\[
m = \frac{-2 - (-8)}{-1 - (-3)} = \frac{-2 + 8}{-1 + 3} = \frac{6}{2} = 3
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( 3.0 \)[/tex].
### Step 2: Determine the Y-Intercept [tex]\( b \)[/tex]
The slope-intercept form of the equation is [tex]\( y = mx + b \)[/tex]. We have [tex]\( m = 3.0 \)[/tex], and we need to determine [tex]\( b \)[/tex]. We can use any point from the table. Let's use the point [tex]\((-3, -8)\)[/tex]:
Plug in the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( m \)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[
-8 = 3(-3) + b
\][/tex]
Simplify this equation to find [tex]\( b \)[/tex]:
[tex]\[
-8 = -9 + b
\][/tex]
Add 9 to both sides to solve for [tex]\( b \)[/tex]:
[tex]\[
b = 1
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( 1.0 \)[/tex].
### Final Equation
By combining the slope [tex]\( m = 3.0 \)[/tex] and the y-intercept [tex]\( b = 1.0 \)[/tex], the equation of the line in slope-intercept form is:
[tex]\[
y = 3.0x + 1.0
\][/tex]
Therefore, the equation representing the function shown in the table is:
[tex]\[
y = \boxed{3.0}x + \boxed{1.0}
\][/tex]
