Which is the graph of the linear function that is represented by the equation y = one-half x minus 2?

Question

On a coordinate plane, a line goes through points (negative 1, 0) and (0, 2).
On a coordinate plane, a line goes through points (negative 2, 0) and (0, 1).
On a coordinate plane, a line goes through points (0, negative 2) and (4, 0).
On a coordinate plane, a line goes through points (0, negative 2) and (1, 0).

Answer 1

To determine which graph represents the linear function given by the equation \( y = \frac{1}{2}x - 2 \), we can identify key properties of the function:

1. Y-Intercept: The y-intercept occurs when \( x = 0 \).

   • Plugging \( x = 0 \) into the equation:
     \[ y = \frac{1}{2}(0) - 2 = -2 \]
     So, the line crosses the y-axis at the point (0, -2).

2. Slope: The slope of the line is \( \frac{1}{2} \), which means for every 2 units the line moves horizontally to the right, it rises 1 unit vertically.

3. Finding another point: To find another point on the line, we can choose a value for \( x \):

   • If we let \( x = 2 \):
     \[ y = \frac{1}{2}(2) - 2 = 1 - 2 = -1 \]
     This gives us the point (2, -1).

Now, we check which of the given options matches with these coordinates:

• Option 1: (−1, 0) and (0, 2) does not cross (0, -2).
• Option 2: (−2, 0) and (0, 1) does not cross (0, -2).
• Option 3: (0, −2) and (4, 0) crosses (0, -2), let's check the line:
  • The point (4, 0) gives us:
    \[ y = \frac{1}{2}(4) - 2 = 2 - 2 = 0 \] which confirms the slope.
• Option 4: (0, −2) and (1, 0) also crosses (0, -2), check:
  • The point (1, 0) gives us:
    \[ y = \frac{1}{2}(1) - 2 = \frac{1}{2} - 2 = -\frac{3}{2} \] does not match.

Thus, the correct option that accurately represents the function \( y = \frac{1}{2}x - 2 \) is the one that goes through the points (0, -2) and (4, 0).