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Which of the following graphs represents a linear function?

coordinate plane with a graph drawn that passes through the points 2 comma negative 2 and 2 comma negative 1 and 2 comma 0
coordinate plane with a graph drawn that passes through the points negative 1 comma 0 and 0 comma negative 2 and 1 comma negative 4
coordinate plane with a graph that passes through the points negative 2 comma 0 and 0 comma negative 4 and 2 comma 0
coordinate plane with a graph that passes through the points negative 3 comma 0 and 0 comma negative 3 and 3 comma 0 and 0 comma 3

1 answer

To determine which of the given graphs represents a linear function, we need to understand the definition of a linear function. A linear function has the form \( y = mx + b \), which means it can be represented as a straight line on a coordinate plane.

Let's evaluate each option based on the points provided:

  1. First graph: Points (2, -2), (2, -1), and (2, 0).

    • All these points have the same x-coordinate (2) but different y-coordinates. This means the graph is a vertical line, which is not a function. Therefore, this is not a linear function.

  2. Second graph: Points (-1, 0), (0, -2), and (1, -4).

    • We can find the slope between any two points:
      • Slope between (-1, 0) and (0, -2): \( \frac{-2 - 0}{0 - (-1)} = \frac{-2}{1} = -2 \)
      • Slope between (0, -2) and (1, -4): \( \frac{-4 - (-2)}{1 - 0} = \frac{-2}{1} = -2 \)
      • Slope between (-1, 0) and (1, -4): \( \frac{-4 - 0}{1 - (-1)} = \frac{-4}{2} = -2 \)
    • The slopes are the same, and since all points lie on the same line, this is a linear function.

  3. Third graph: Points (-2, 0), (0, -4), and (2, 0).

    • We calculate the slope:
      • Slope between (-2, 0) and (0, -4): \( \frac{-4 - 0}{0 - (-2)} = \frac{-4}{2} = -2 \)
      • Slope between (0, -4) and (2, 0): \( \frac{0 - (-4)}{2 - 0} = \frac{4}{2} = 2 \)
    • The slopes are different, indicating that these points do not lie on the same straight line. Therefore, this is not a linear function.

  4. Fourth graph: Points (-3, 0), (0, -3), (3, 0), and (0, 3).

    • Here, we can see that three points: (-3, 0), (3, 0) show a horizontal line, while (0, -3) and (0, 3) show a vertical line. Since there are distinct vertical and horizontal components, this suggests the graph consists of disjoint segments, which cannot be represented as a single line and does not satisfy the definition of a linear function. Therefore, this is not a linear function.

In conclusion, the second graph represents a linear function.