Question

which graph matches this table?
input (x) | -2 | 0 | 2
output (y) | -2 | -1 | 0

Explanation:

Step1: Find the equation of the line

First, calculate the slope \( m \) using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using the points \((-2,-2)\) and \((0,-1)\):
\( m=\frac{-1-(-2)}{0 - (-2)}=\frac{-1 + 2}{2}=\frac{1}{2} \)
The y - intercept \( b \) is the value of \( y \) when \( x = 0 \). From the point \((0,-1)\), \( b=-1 \).
So the equation of the line is \( y=\frac{1}{2}x-1 \).

Step2: Check the points on the line

• For \( x=-2 \): \( y=\frac{1}{2}(-2)-1=-1 - 1=-2 \) (matches the table).
• For \( x = 0 \): \( y=\frac{1}{2}(0)-1=-1 \) (matches the table).
• For \( x = 2 \): \( y=\frac{1}{2}(2)-1=1 - 1=0 \) (matches the table).

Now, analyze the graphs:

• The first graph: The line does not have a slope of \( \frac{1}{2} \) (checking the rise over run, it has a slope of 1).
• The second graph: Let's check the slope. Taking two points, say \((-4,-3)\) and \((0,-1)\), the slope is \( \frac{-1-(-3)}{0-(-4)}=\frac{2}{4}=\frac{1}{2} \), and the y - intercept is - 1. The points from the table \((-2,-2)\), \((0,-1)\), \((2,0)\) lie on this line.
• The third graph: It has a negative slope, while our line has a positive slope (\( \frac{1}{2} \)), so it can't be this one.

Answer:

The second graph (the middle one)