The Lighthouse Beam: Finding Distance
How the slope and intercept of a line relate to the distance of points from the origin.
Equation
y = x
Graph
Table
| x | y |
|---|---|
| -5 | -5 |
| -4 | -4 |
| -3 | -3 |
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
What this lesson covers
What you do
You shape the function y = m*x + c and watch the graph answer.
Challenges to clear
- Make the beam pass through (3, 4). This point is exactly 5 units from the origin.
- Make the beam pass through (5, 12). This point is exactly 13 units from the origin.
- Swing the beam somewhere new: through (0, −5), which is 5 units straight down from the lighthouse.
- Now through (4, 3) as well — also exactly 5 units out, because 4² + 3² = 5². Both ends of the beam sit on the same circle.
Check yourself
You found a line passing through (3, 4). If you slide 'c' up by 1 unit, what happens to the distance of the point (3, y) from the origin?
- The distance stays 5 units because x didn't change.
- The distance becomes 6 units because y increased by 1.
- The distance changes, but not necessarily by 1 unit. — correct
- The distance becomes 0 because the line moved away from the origin.
Think about it
- If you increase m, does the point (3,4) stay on the line?
- If you increase c, does the point (3,4) stay on the line?