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On a coordinate plane, a line goes through points (negative 2, 4) and (0, negative 4).

Linear functions are expressed by the graph and equation. Select all that apply.
The slope is positive for both functions.
The equation has a steeper slope than the line in the graph.
The y-intercept is the same for both.
The graph and the equation express an equivalent function.

1 answer

To analyze the situation, let's determine the slope of the line passing through the points \((-2, 4)\) and \((0, -4)\).

  1. Calculating the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 4}{0 - (-2)} = \frac{-8}{2} = -4 \] The slope of the line is \(-4\), which is negative.

  2. Finding the y-intercept (b): The y-intercept can be found using the point \((0, -4)\). Hence, \(b = -4\).

  3. Equation of the line: The equation of the line in slope-intercept form \(y = mx + b\) is: \[ y = -4x - 4 \]

Now, let's analyze the provided statements:

  1. The slope is positive for both functions: This is false because the slope we calculated is \(-4\), which is negative.

  2. The equation has a steeper slope than the line in the graph: This makes no sense in this context since the equation represents the same line. Therefore, we cannot compare slopes of functions representing the same line. This is false.

  3. The y-intercept is the same for both: The y-intercept of the line is \(-4\), and if we're referring to the same line (the equation we derived) then yes, the y-intercept is the same. This is true.

  4. The graph and the equation express an equivalent function: Since both represent the same line, this is true.

Summary

  • The slope is positive for both functions: False
  • The equation has a steeper slope than the line in the graph: False
  • The y-intercept is the same for both: True
  • The graph and the equation express an equivalent function: True

Thus, the correct selections are:

  • The y-intercept is the same for both.
  • The graph and the equation express an equivalent function.