Students practice identifying linear patterns in function tables and graphing the equation y = 2x + 3.

Let’s solve each function table step by step.

We’re looking for a rule in the form y = mx + b, where:
- m is the slope (how much y changes when x increases by 1)
- b is the y-intercept (value of y when x = 0)

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Problem 1:

x | y
0 | 1
1 | 9
2 | 17
3 | 25
10 | ?

From x=0 to x=1, y goes from 1 to 9 → change of +8
From x=1 to x=2, y goes from 9 to 17 → +8
So slope m = 8

When x=0, y=1 → so b = 1

Rule: y = 8x + 1

Check:
x=3 → 8*3 +1 = 25 ✔️
x=10 → 8*10 +1 = 81

✔ Fill in 81

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Problem 2:

x | y
1 | 2
2 | 0
3 | -2
4 | -4
12 | ?

From x=1 to x=2, y goes from 2 to 0 → change of -2
Slope m = -2

Use point (1,2):
y = mx + b → 2 = -2(1) + b → b = 4

Rule: y = -2x + 4

Check:
x=3 → -6 + 4 = -2 ✔️
x=4 → -8 + 4 = -4 ✔️
x=12 → -24 + 4 = -20

✔ Fill in -20

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Problem 3:

x | y
3 | 21
6 | 42
9 | 63
12 | 84
16 | ?

From x=3 to x=6 (Δx=3), y from 21 to 42 (Δy=21) → slope = 21/3 = 7
Check next: x=6 to 9 → y=42 to 63 → Δy=21, Δx=3 → slope=7 ✔️

So m=7

Use point (3,21):
21 = 7*3 + b → 21 = 21 + b → b=0

Rule: y = 7x

Check:
x=12 → 7*12=84 ✔️
x=16 → 7*16=112

✔ Fill in 112

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Problem 4:

x | y
4 | -2
5 | -4
6 | -6
7 | -8
15 | ?

From x=4 to 5 → y from -2 to -4 → Δy=-2 → slope m=-2

Use (4,-2):
-2 = -2*4 + b → -2 = -8 + b → b=6

Rule: y = -2x + 6

Check:
x=5 → -10+6=-4 ✔️
x=7 → -14+6=-8 ✔️
x=15 → -30+6 = -24

✔ Fill in -24

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Problem 5:

x | y
-3 | -21
-1 | -11
1 | -1
3 | 9
18 | ?

From x=-3 to -1 (Δx=2), y from -21 to -11 → Δy=10 → slope = 10/2 = 5

Use (-1,-11):
-11 = 5*(-1) + b → -11 = -5 + b → b = -6

Rule: y = 5x - 6

Check:
x=1 → 5-6=-1 ✔️
x=3 → 15-6=9 ✔️
x=18 → 5*18 -6 = 90-6=84

✔ Fill in 84

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Problem 6:

x | y
1 | 8
2 | 10
3 | 12
4 | 14
12 | ?

From x=1 to 2 → y from 8 to 10 → Δy=2 → slope m=2

Use (1,8):
8 = 2*1 + b → b=6

Rule: y = 2x + 6

Check:
x=3 → 6+6=12 ✔️
x=4 → 8+6=14 ✔️
x=12 → 24+6=30

✔ Fill in 30

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Problem 7:

x | y
-3 | 6
0 | 4
3 | 2
6 | 0
9 | ?

From x=-3 to 0 (Δx=3), y from 6 to 4 → Δy=-2 → slope = -2/3? Wait — let’s check again.

Actually, from x=0 to x=3 → y from 4 to 2 → Δy=-2, Δx=3 → slope = -2/3? But that doesn’t fit integer steps.

Wait — look at x=0 → y=4 → so b=4

From x=0 to x=3 → y drops by 2 → so for every 3 units right, y drops 2 → slope = -2/3? But let’s test with another pair.

x=3 → y=2 → if rule is y = mx + 4 → 2 = 3m + 4 → 3m = -2 → m = -2/3

But then x=6 → y = -2/3 *6 +4 = -4 +4 = 0 ✔️
x=9 → y = -2/3 *9 +4 = -6 +4 = -2

But wait — maybe I made a mistake. Let’s recheck differences:

x: -3 → 0 → 3 → 6 → 9 (each step +3)
y: 6 → 4 → 2 → 0 → ? (each step -2)

So yes, slope = -2/3

Rule: y = (-2/3)x + 4

But the problem says “put it in y = mx + b form” — fractions are okay.

x=9 → y = (-2/3)*9 + 4 = -6 + 4 = -2

✔ Fill in -2

Alternatively, maybe they expect integer slope? Let me double-check.

Wait — perhaps I misread. From x=-3 to x=0: y from 6 to 4 → drop of 2 over 3 → slope -2/3. Yes.

But let’s see if there’s an error in assumption. Maybe it’s not linear? No, it is linear.

So rule is y = (-2/3)x + 4

But let’s write it as fraction: y = -\frac{2}{3}x + 4

For x=9: y = -6 + 4 = -2

✔ Fill in -2

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Problem 8:

x | y
2 | 10
4 | 18
6 | 26
8 | 34
19 | ?

From x=2 to 4 → y from 10 to 18 → Δy=8, Δx=2 → slope = 4

Use (2,10):
10 = 4*2 + b → 10 = 8 + b → b=2

Rule: y = 4x + 2

Check:
x=4 → 16+2=18 ✔️
x=6 → 24+2=26 ✔️
x=8 → 32+2=34 ✔️
x=19 → 4*19 +2 = 76+2=78

✔ Fill in 78

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Problem 9a:

Equation: y = 2x + 3

Table given:

x | y
-4 | -5 → 2*(-4)+3 = -8+3 = -5 ✔️
-2 | -1 → 2*(-2)+3 = -4+3 = -1 ✔️
0 | 3 → 0+3=3 ✔️
1 | 5 → 2+3=5 ✔️
2 | 7 → 4+3=7 ✔️

All filled correctly.

Now graph on coordinate plane.

Points to plot:
(-4, -5), (-2, -1), (0, 3), (1, 5), (2, 7)

Draw a straight line through them.

Since this is text-based, I can’t draw, but you would:

- Mark each point on the grid.
- Connect them with a straight line.
- Extend the line in both directions.

The line should have slope 2 and cross y-axis at (0,3).

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Final Answers:

1. Rule: y = 8x + 1; missing value: 81
2. Rule: y = -2x + 4; missing value: -20
3. Rule: y = 7x; missing value: 112
4. Rule: y = -2x + 6; missing value: -24
5. Rule: y = 5x - 6; missing value: 84
6. Rule: y = 2x + 6; missing value: 30
7. Rule: y = -\frac{2}{3}x + 4; missing value: -2
8. Rule: y = 4x + 2; missing value: 78
9a. Table already complete; graph the points and draw line.

Final Answer:
1. y = 8x + 1, 81
2. y = -2x + 4, -20
3. y = 7x, 112
4. y = -2x + 6, -24
5. y = 5x - 6, 84
6. y = 2x + 6, 30
7. y = -\frac{2}{3}x + 4, -2
8. y = 4x + 2, 78
9a. Table complete; graph points (-4,-5), (-2,-1), (0,3), (1,5), (2,7) and draw line.