To determine which graph represents the given linear relationship, we need to identify the equation of the line that passes through the points given in the table.
First, let's identify two points from the table and use them to find the slope of the line. Consider the points (-1, 2) and (0, 0):
1. Calculate the slope (m):
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((-1, 2)\)[/tex] and [tex]\((0, 0)\)[/tex], we have:
[tex]\[ m = \frac{0 - 2}{0 + 1} = \frac{-2}{1} = -2 \][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex].
2. Calculate the y-intercept (b):
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept [tex]\( b \)[/tex], we use one of the points along with the slope. Let's use the point [tex]\((0, 0)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 0 = -2 \cdot 0 + b \][/tex]
[tex]\[ b = 0 \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
3. Write the equation of the line:
Using the slope [tex]\(-2\)[/tex] and y-intercept [tex]\(0\)[/tex], the equation of the line is:
[tex]\[ y = -2x \][/tex]
4. Validate the equation with the remaining points:
We should check the equation [tex]\( y = -2x \)[/tex] against the given points to ensure all points lie on this line.
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -2(-1) = 2 \][/tex] (Matches the table)
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -2(0) = 0 \][/tex] (Matches the table)
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -2(1) = -2 \][/tex] (Matches the table)
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -2(2) = -4 \][/tex] (Matches the table)
Since all points satisfy the equation [tex]\( y = -2x \)[/tex], we conclude that the equation correctly represents the linear relationship given in the table.
Now, among your graph options, select the graph that shows a linear relationship with a slope of [tex]\(-2\)[/tex] and y-intercept of [tex]\(0\)[/tex]. It should be a straight line starting at the origin [tex]\((0,0)\)[/tex] and passing through the points [tex]\((-1, 2)\)[/tex], [tex]\((1, -2)\)[/tex], and [tex]\((2, -4)\)[/tex].
