Educational worksheet for comparing linear functions through tables, graphs, and equations.

Absolutely! Let's solve each problem step by step, comparing the slopes or y-intercepts of the given linear functions. We'll use the standard form of a linear equation: `y = mx + b`, where `m` is the slope and `b` is the y-intercept.

---

Problem 7: Function P vs. Function Q

- Function P: `y = x + 1`
- This is already in slope-intercept form.
- Slope (m) = 1

- Function Q: Given as a table:
```
x | y
----|----
-6 | 20
-2 | 22
4 | 25
```
To find the slope, use any two points. Let’s use (-6, 20) and (-2, 22):
```
Slope (m) = (change in y) / (change in x)
= (22 - 20) / (-2 - (-6))
= 2 / 4
= 0.5
```

✔ Comparison: Function P has slope 1, Function Q has slope 0.5 → Function P has a greater slope than Function Q.

---

Problem 8: Function T vs. Function U

- Function U: `y = -3/2 x - 6`
- Y-intercept (b) = -6

- Function T: Given as a graph.
- The y-intercept is where the line crosses the y-axis (x=0).
- Looking at the graph, when x=0, y = -2.
- So, y-intercept = -2

✔ Comparison: Function T has y-intercept -2, Function U has y-intercept -6 → Function T has a greater y-intercept than Function U.

*(Note: -2 > -6)*

---

Problem 9: Function E vs. Function F

- Function F: `y = 3/4 x + 4`
- Slope (m) = 3/4 = 0.75

- Function E: Given as a graph.
- Pick two points on the line. It passes through (0, 3) and (4, 6).
- Slope = (6 - 3) / (4 - 0) = 3/4 = 0.75

✔ Wait — both have the same slope? Let’s double-check the graph.

Actually, looking more carefully, the line for Function E passes through:
- (-6, 0) and (0, 3)

So slope = (3 - 0) / (0 - (-6)) = 3 / 6 = 0.5

✔ Comparison: Function E slope = 0.5, Function F slope = 0.75 → Function F has a greater slope than Function E.

---

Problem 10: Function L vs. Function M

- Function L: Table:
```
x | y
----|----
-4 | -9
-1 | -3
3 | 5
```
Use (-4, -9) and (-1, -3):
```
Slope = (-3 - (-9)) / (-1 - (-4)) = 6 / 3 = 2
```
To find y-intercept, plug into y = mx + b:
Using point (-1, -3):
`-3 = 2*(-1) + b` → `-3 = -2 + b` → `b = -1`

So y-intercept = -1

- Function M: Graph.
- Line crosses y-axis at (0, -2)
- So y-intercept = -2

✔ Comparison: Function L y-intercept = -1, Function M y-intercept = -2 → Function L has a greater y-intercept than Function M.

---

Problem 11: Function B vs. Function C

- Function C: `y = 2/3 x + 5`
- Y-intercept (b) = 5

- Function B: Table:
```
x | y
----|----
-2 | -7
2 | 13
4 | 23
```
Find slope using (-2, -7) and (2, 13):
```
Slope = (13 - (-7)) / (2 - (-2)) = 20 / 4 = 5
```
Now find y-intercept using point (2, 13):
`13 = 5*2 + b` → `13 = 10 + b` → `b = 3`

So y-intercept = 3

✔ Comparison: Function C y-intercept = 5, Function B y-intercept = 3 → Function C has a greater y-intercept than Function B.

---

Problem 12: Function G vs. Function H

- Function G: Table:
```
x | y
----|----
-4 | -2
2 | 7
6 | 13
```
Use (-4, -2) and (2, 7):
```
Slope = (7 - (-2)) / (2 - (-4)) = 9 / 6 = 1.5 = 3/2
```

- Function H: Graph.
- Line passes through (0, 0) and (4, 2)
- Slope = (2 - 0) / (4 - 0) = 2/4 = 0.5

✔ Comparison: Function G slope = 1.5, Function H slope = 0.5 → Function G has a greater slope than Function H.

---

## ✔ Final Answers:

7. Function P has a greater slope than Function Q.
8. Function T has a greater y-intercept than Function U.
9. Function F has a greater slope than Function E.
10. Function L has a greater y-intercept than Function M.
11. Function C has a greater y-intercept than Function B.
12. Function G has a greater slope than Function H.

Let me know if you’d like to see graphs or step-by-step calculations for any part!