To determine if the points \((-2, -4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), \((2, 4)\), and \((3, 9)\) represent a linear relationship, you need to check if they all lie on a straight line. One way to do this is to calculate the slope between consecutive points and see if the slopes are the same.
The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's calculate the slopes between consecutive points:
-
Between \((-2, -4)\) and \((-1, 1)\): \[ m = \frac{1 - (-4)}{-1 - (-2)} = \frac{5}{1} = 5 \]
-
Between \((-1, 1)\) and \((0, 0)\): \[ m = \frac{0 - 1}{0 - (-1)} = \frac{-1}{1} = -1 \]
-
Between \((0, 0)\) and \((1, 1)\): \[ m = \frac{1 - 0}{1 - 0} = \frac{1}{1} = 1 \]
-
Between \((1, 1)\) and \((2, 4)\): \[ m = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3 \]
-
Between \((2, 4)\) and \((3, 9)\): \[ m = \frac{9 - 4}{3 - 2} = \frac{5}{1} = 5 \]
The slopes obtained from the pairs of points are \(5\), \(-1\), \(1\), \(3\), and \(5\), which are not all equal. Because the slopes between consecutive points are different, the points do not lie on a straight line.
Therefore, the set of points does not represent a linear relationship.