Question

humanize this: Part I — Graph and description

How to graph y = (1/2)x + 5 (step-by-step):
1. Start with the y-intercept: when x = 0, y = 5. Plot the point (0, 5).
2. Use the slope 1/2. That means “rise 1, run 2.” From (0,5) go right 2 and up 1 to get the point (2, 6). You can also go left 2 and down 1 to get (-2, 4).
3. Plot a few points (for example: (0,5), (2,6), (-2,4), and the x-intercept found by setting y = 0: 0 = (1/2)x + 5 ⇒ x = −10, so (−10,0)).
4. Draw a straight line through those points — that is the graph.

Two features of the graph:
- Slope: The line has slope 1/2, so it rises slowly to the right (for every 2 units you move right, it goes up 1 unit).
- Y-intercept (and x-intercept): The y-intercept is 5 (the line crosses the y-axis at (0,5)). The x-intercept is −10 (the line crosses the x-axis at (−10,0)). Also, it’s a straight line (domain and range are all real numbers).

Part II — A perpendicular equation

A line perpendicular to another has a slope that is the negative reciprocal. The original slope is 1/2, so the negative reciprocal is −2.

Any perpendicular line has slope −2, so its equation looks like y = −2x + b for some number b. One specific perpendicular line that also crosses the y-axis at 5 is:

y = −2x + 5

Check: (1/2) × (−2) = −1, so the slopes are negative reciprocals and the lines are perpendicular.

Answer 1

Here’s a more conversational, easy-to-follow version.

Part I — Graphing y = (1/2)x + 5

- Start with the y-intercept. When x = 0, y = 5, so plot (0, 5).
- Use the slope 1/2. That means “rise 1, run 2.” From (0,5) go right 2 and up 1 to get (2,6). You can also go left 2 and down 1 to get (−2,4).
- Add a few more points for accuracy. For example: (0,5), (2,6), (−2,4). To find the x-intercept set y = 0: 0 = (1/2)x + 5 → x = −10, so the x-intercept is (−10, 0).
- Draw a straight line through those points — that’s the graph.

Quick notes about the line
- Slope: 1/2 — the line rises slowly to the right (every 2 units right, it goes up 1).
- Y-intercept: 5 (crosses the y-axis at (0,5)).
- X-intercept: −10 (crosses the x-axis at (−10,0)).
- It’s a straight line, so domain and range are all real numbers.

Part II — A perpendicular line

Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of 1/2 is −2. So any line perpendicular to the given one has slope −2 and looks like y = −2x + b for some b.

If we want a perpendicular line that also crosses the y-axis at 5, just use b = 5:
y = −2x + 5

Check: (1/2) × (−2) = −1, so the slopes are negative reciprocals and the lines are perpendicular.