Student's completed homework sheet demonstrating how to calculate slope from graphs and coordinate points.

You’ve uploaded a completed worksheet on Slopes of Lines, Parallel & Perpendicular Lines. The student (Trent H.) has filled in most answers in green ink. Let’s go through each section, verify the answers, and explain the solutions clearly.

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## 📌 SECTION 1: Find the slope of the lines graphed below

The formula for slope from a graph is:

> Slope = rise / run = (change in y) / (change in x)

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1. Line going up to the right

- Rise = +4, Run = +2 → Slope = 4/2 = 2
- ✔ Correct — labeled “positive”

2. Line going down to the right

- Rise = -3, Run = +5 → Slope = -3/5
- ✘ Student wrote 3/5 — incorrect sign!
- Should be -3/5

3. Line going up to the right

- Rise = +2, Run = +1 → Slope = 2/1 = 2
- ✔ Correct — labeled “positive”

4. Vertical line

- Change in x = 0 → Slope = undefined
- ✔ Correct — labeled “undefined”

5. Line going down to the right

- Rise = -5, Run = +1 → Slope = -5/1 = -5
- ✔ Correct — labeled “negative”

6. Horizontal line

- Rise = 0 → Slope = 0
- ✔ Correct — labeled “zero”

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## 📌 SECTION 2: Find the slope between two points

Formula:

> m = (y₂ - y₁) / (x₂ - x₁)

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7. (-1, -11) and (-6, -7)

- m = (-7 - (-11)) / (-6 - (-1)) = (4) / (-5) = -4/5
- ✘ Student got 4/-7 — wrong calculation
- Correct answer: -4/5

8. (-7, -13) and (1, -5)

- m = (-5 - (-13)) / (1 - (-7)) = (8) / (8) = 1
- ✔ Correct!

9. (8, 3) and (-5, 3)

- m = (3 - 3) / (-5 - 8) = 0 / (-13) = 0
- ✘ Student wrote “–” — incomplete
- Correct answer: 0 (horizontal line)

10. (15, 7) and (3, -2)

- m = (-2 - 7) / (3 - 15) = (-9) / (-12) = 3/4
- ✘ Student got 1/3 — wrong simplification
- Correct: 3/4

11. (-5, -1) and (-5, -10)

- m = (-10 - (-1)) / (-5 - (-5)) = (-9) / (0) → undefined
- ✔ Correct — vertical line

12. (-12, 16) and (-4, -2)

- m = (-2 - 16) / (-4 - (-12)) = (-18) / (8) = -9/4
- ✔ Correct!

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## 📌 SECTION 3: Determine if lines PQ and RS are parallel, perpendicular, or neither

Given:
- P(-9, -4), Q(-7, -1) → find m(PQ)
- R(-2, 5), S(-6, -1) → find m(RS)

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Step 1: Find m(PQ)

- m = (-1 - (-4)) / (-7 - (-9)) = (3) / (2) = 3/2

Step 2: Find m(RS)

- m = (-1 - 5) / (-6 - (-2)) = (-6) / (-4) = 3/2

Wait — student calculated m(RS) as -3/2, which is incorrect.

Let’s recalculate m(RS):
- R(-2, 5), S(-6, -1)
- Δy = -1 - 5 = -6
- Δx = -6 - (-2) = -4
- m = -6 / -4 = 3/2

✔ So both slopes are 3/2

→ Therefore, lines are parallel

Student wrote:
- m(PQ) = 3/2 ✔
- m(RS) = -3/2 ✘ (should be 3/2)
- Types of lines: wrote “1” — unclear, but probably meant “parallel”

✔️ Final Answer: Parallel

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## ✍️ Summary of Corrections Needed:

| Problem | Student’s Answer | Correct Answer | Notes |
|--------|------------------|----------------|-------|
| 2 | 3/5 | -3/5 | Sign error — line goes down |
| 7 | 4/-7 | -4/5 | Arithmetic error |
| 9 | – | 0 | Horizontal line |
| 10 | 1/3 | 3/4 | Simplified incorrectly |
| 13 (RS)| -3/2 | 3/2 | Sign error — both points go down and left → positive slope |
| 13 (Type) | 1 | Parallel | Since slopes equal |

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## 🎯 Final Tips for Student:

- Always double-check signs when calculating slope.
- For horizontal lines → slope = 0
- For vertical lines → slope = undefined
- Two lines are parallel if slopes are equal.
- Two lines are perpendicular if slopes are negative reciprocals (e.g., 2 and -1/2).
- Use the formula carefully: m = (y₂ - y₁)/(x₂ - x₁) — order matters!

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✔ Overall, good effort! Just watch out for sign errors and arithmetic slips. Keep practicing!

Let me know if you want a clean version of this worksheet with all correct answers filled in!