Question

Sketch the graphs of all three equations on the same coordinate plane.
y= x,y=x+1, andy= x-2

Answer 1

To sketch the graphs of the three equations \(y = x\), \(y = x + 1\), and \(y = x - 2\) on the same coordinate plane, we can follow these steps:

1. **Graph of \(y = x\)**:
- This is a straight line that passes through the origin (0,0).
- It has a slope of 1, meaning for every unit increase in \(x\), \(y\) also increases by 1.
- Points on this line include (1,1), (2,2), (-1,-1), etc.

2. **Graph of \(y = x + 1\)**:
- This line also has a slope of 1, but it intercepts the y-axis at (0,1).
- It is parallel to the line \(y = x\), and it is shifted one unit up.
- Points on this line include (0,1), (1,2), (2,3), (-1,0), etc.

3. **Graph of \(y = x - 2\)**:
- This line has a slope of 1 as well but intercepts the y-axis at (0,-2).
- It is parallel to the other two lines, but it is shifted two units down.
- Points on this line include (0,-2), (1,-1), (2,0), (-1,-3), etc.

### Sketching Steps:
1. Draw a coordinate plane with x and y axes.
2. For \(y = x\), draw a line starting from the origin and going upwards to the right at 45 degrees.
3. For \(y = x + 1\), draw another line parallel to the first line, starting from the point (0,1) and going upwards.
4. For \(y = x - 2\), draw a third line parallel to the other two, starting from (0,-2) and going upwards.

### Intersections:
These three lines do not intersect each other, as they are all parallel lines. Each line has a different y-intercept.

### Summary of Graphs:
- **\(y = x\)**: Passes through (0,0)
- **\(y = x + 1\)**: Passes through (0,1)
- **\(y = x - 2\)**: Passes through (0,-2)

When you plot these lines on the same coordinate plane, you'll see three parallel lines, spaced evenly apart vertically.