Let's solve each problem step by step.
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Part 1: Find the slope of the line through each pair of points
The formula for slope $ m $ between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$
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1. (1, 5), (-1, -3)
$$
m = \frac{-3 - 5}{-1 - 1} = \frac{-8}{-2} = 4 \\
\boxed{m = 4}
$$
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2. (0, 2), (3, 0)
$$
m = \frac{0 - 2}{3 - 0} = \frac{-2}{3} \\
\boxed{m = -\frac{2}{3}}
$$
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3. (1, 9), (0, 6)
$$
m = \frac{6 - 9}{0 - 1} = \frac{-3}{-1} = 3 \\
\boxed{m = 3}
$$
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4. (8, -5), (4, -2)
$$
m = \frac{-2 - (-5)}{4 - 8} = \frac{3}{-4} = -\frac{3}{4} \\
\boxed{m = -\frac{3}{4}}
$$
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5. (-3, 5), (-3, -1)
$$
m = \frac{-1 - 5}{-3 - (-3)} = \frac{-6}{0} \quad \text{(undefined)} \\
\boxed{\text{undefined}}
$$
> This is a vertical line — slope is undefined.
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6. (-2, -2), (10, -2)
$$
m = \frac{-2 - (-2)}{10 - (-2)} = \frac{0}{12} = 0 \\
\boxed{m = 0}
$$
> This is a horizontal line — slope is 0.
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Part 2: Determine the rate of change of each graph
We'll estimate the slope from the graphs using rise over run.
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7. Graph 7:
Line goes from (0, 0) to (4, 2).
Rise = 2, Run = 4 → $ m = \frac{2}{4} = \frac{1}{2} $
$$
\boxed{\frac{1}{2}}
$$
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8. Graph 8:
Line goes from (0, 4) to (4, 0).
Rise = -4, Run = 4 → $ m = \frac{-4}{4} = -1 $
$$
\boxed{-1}
$$
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9. Graph 9:
Line goes from (-4, 0) to (4, 2).
Rise = 2, Run = 8 → $ m = \frac{2}{8} = \frac{1}{4} $
$$
\boxed{\frac{1}{4}}
$$
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10. Graph 10:
Line goes from (-2, -2) to (1, 4).
Rise = 6, Run = 3 → $ m = \frac{6}{3} = 2 $
$$
\boxed{2}
$$
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11. Graph 11:
Line goes from (-1, -4) to (1, 4).
Rise = 8, Run = 2 → $ m = \frac{8}{2} = 4 $
$$
\boxed{4}
$$
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12. Graph 12:
Line goes from (-4, 4) to (2, -2).
Rise = -6, Run = 6 → $ m = \frac{-6}{6} = -1 $
$$
\boxed{-1}
$$
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Problem 16: Hiking Rate of Change
Naomi:
- Starts at 7400 feet at 7:00 A.M.
- Ends at 9800 feet at 11:00 A.M.
Time elapsed: $ 11:00 - 7:00 = 4 $ hours
Change in elevation: $ 9800 - 7400 = 2400 $ feet
Rate of change = $ \frac{\text{change in elevation}}{\text{change in time}} = \frac{2400}{4} = 600 $ feet per hour
$$
\boxed{600}
$$
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✔ Final Answers:
1. $ m = 4 $
2. $ m = -\frac{2}{3} $
3. $ m = 3 $
4. $ m = -\frac{3}{4} $
5. undefined
6. $ m = 0 $
7. $ \frac{1}{2} $
8. $ -1 $
9. $ \frac{1}{4} $
10. $ 2 $
11. $ 4 $
12. $ -1 $
16. $ 600 $
Let me know if you'd like these formatted for printing or need explanations on how to read the graphs!
Parent Tip: Review the logic above to help your child master the concept of slope rate of change worksheet.
