Q. Which Graph Represents the Function \( y = x – 2 \)?

Solution — step-by-step explanation

1. Recognize the form of the equation. The given function is

   \( y = x – 2 \).

   This is in slope-intercept form \( y = mx + b \), so the slope is

   \( m = 1 \)

   and the y-intercept is

   \( b = -2 \).

2. Find the y-intercept point. Set \( x = 0 \) and compute \( y \):

   \( y = 0 – 2 = -2 \).

   The y-intercept point is \( (0, -2) \). This is where the graph crosses the y-axis.

3. Find the x-intercept. Set \( y = 0 \) and solve for \( x \):

   \( 0 = x – 2 \)

   \( x = 2 \).

   The x-intercept point is \( (2, 0) \). This is where the graph crosses the x-axis.

4. Use the slope to get another point. The slope \( m = 1 \) means rise over run is \( 1/1 \): for each 1 unit to the right, go up 1 unit. Starting from \( (0, -2) \):

   Move right 1 to \( x = 1 \) and up 1 to \( y = -1 \). That gives the point \( (1, -1) \).

   Optionally list several points by plugging in x values:

   • \( x = -1 \Rightarrow y = -1 – 2 = -3 \), point \( (-1, -3) \)
   • \( x = 0 \Rightarrow y = -2 \), point \( (0, -2) \)
   • \( x = 1 \Rightarrow y = -1 \), point \( (1, -1) \)
   • \( x = 2 \Rightarrow y = 0 \), point \( (2, 0) \)
   • \( x = 3 \Rightarrow y = 1 \), point \( (3, 1) \)

5. Describe the graph. The graph is a straight line through the points found above. It:

   • crosses the y-axis at \( (0, -2) \),
   • crosses the x-axis at \( (2, 0) \),
   • rises from left to right with slope \( 1 \) (a 45-degree incline on a square grid).

6. Conclusion: The correct graph is the straight line that passes through \( (0, -2) \) and \( (2, 0) \) and has slope \( 1 \). To identify it among given choices, select the graph that matches those characteristics.