To determine which graph correctly represents the given linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to identify the equation of the line that passes through the points provided. This equation takes the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's go through the steps to find the slope ([tex]\( m \)[/tex]) and y-intercept ([tex]\( b \)[/tex]).
### Step 1: Calculate the Slope ([tex]\( m \)[/tex])
The slope of a line [tex]\( m \)[/tex] is calculated as the change in [tex]\( y \)[/tex] divided by the change in [tex]\( x \)[/tex]. Given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the formula for the slope is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((-1, 2)\)[/tex] and [tex]\( (0, 0) \)[/tex]:
[tex]\[
m = \frac{0 - 2}{0 - (-1)} = \frac{-2}{1} = -2
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex].
### Step 2: Determine the Y-intercept ([tex]\( b \)[/tex])
The y-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. From the table, we see that when [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. Hence, the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
### Step 3: Write the Equation of the Line
With the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] known, the equation of the line is:
[tex]\[ y = -2x + 0 \][/tex]
or simply,
[tex]\[ y = -2x \][/tex]
### Step 4: Plot the Line Using the Equation
We can now use the equation [tex]\( y = -2x \)[/tex] to plot the graph and verify it against the given points.
1. When [tex]\( x = -1 \)[/tex], [tex]\( y = -2(-1) = 2 \)[/tex].
2. When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) = 0 \)[/tex].
3. When [tex]\( x = 1 \)[/tex], [tex]\( y = -2(1) = -2 \)[/tex].
4. When [tex]\( x = 2 \)[/tex], [tex]\( y = -2(2) = -4 \)[/tex].
The points [tex]\((-1, 2)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((1, -2)\)[/tex], and [tex]\((2, -4)\)[/tex] all lie on the line described by [tex]\( y = -2x \)[/tex].
### Conclusion
The graph of the linear relationship will be a straight line passing through the points [tex]\((-1, 2)\)[/tex], [tex]\((0, 0)\)[/tex], [tex]\((1, -2)\)[/tex], and [tex]\((2, -4)\)[/tex] with a slope of [tex]\(-2\)[/tex].
So, the correct graph is the one that shows a straight line descending with a slope of [tex]\(-2\)[/tex] and intercepting the origin (0,0).
