Question

which graph best represents the relationship between the x - values and the y - values in the table?
the table has two columns, x and y. the rows are: when x = 0, y = - 2; when x = 2, y = 0; when x = 4, y = 2; when x = 6, y = 4.
there are four graphs labeled a, b, c, d. each graph is a coordinate plane with a line. graph a: the line has a certain slope and y - intercept. graph b: the line has a different slope and y - intercept. graph c: the line has another slope and y - intercept. graph d: the line has yet another slope and y - intercept.

Explanation:

Step1: Find the slope

First, we calculate the slope \( m \) using the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Taking two points from the table, say \( (0, - 2) \) and \( (2,0) \), we have \( m=\frac{0 - (-2)}{2 - 0}=\frac{2}{2} = 1 \).

Step2: Find the equation of the line

Using the slope - intercept form \( y=mx + b \), where \( m = 1 \) and using the point \( (0,-2) \) (when \( x = 0 \), \( y=-2 \)), we get \( b=-2 \). So the equation of the line is \( y=x - 2 \).

Step3: Analyze the graphs

• For graph A: Let's check the slope. The line in graph A has a negative slope (since it goes from top - left to bottom - right), so it can't be A.
• For graph B: Let's check the slope. The slope of the line in graph B is positive. Let's check the y - intercept. When \( x = 0 \), from the equation \( y=x - 2 \), \( y=-2 \). Let's also check a point. When \( x = 2 \), \( y=2 - 2=0 \), which matches the table. When \( x = 4 \), \( y=4 - 2 = 2 \), and when \( x = 6 \), \( y=6 - 2=4 \), which also matches the table.
• For graph C: The line in graph C has a negative slope, so it can't be C.
• For graph D: The line in graph D has a negative slope, so it can't be D.

Answer:

The graph labeled B