Linear Graphs | MathsHKO - Free Printable

I'll work through each table systematically, calculating the y-values for each x-value, then identify the patterns.

Table 1: y = 3x - 2, y = 3x - 1, y = 3x, y = 3x + 1, y = 3x + 2

For x = -2, -1, 0, 1, 2:

y = 3x - 2:
- x = -2: y = 3(-2) - 2 = -6 - 2 = -8
- x = -1: y = 3(-1) - 2 = -3 - 2 = -5
- x = 0: y = 3(0) - 2 = 0 - 2 = -2
- x = 1: y = 3(1) - 2 = 3 - 2 = 1
- x = 2: y = 3(2) - 2 = 6 - 2 = 4

y = 3x - 1:
- x = -2: y = 3(-2) - 1 = -6 - 1 = -7
- x = -1: y = 3(-1) - 1 = -3 - 1 = -4
- x = 0: y = 3(0) - 1 = 0 - 1 = -1
- x = 1: y = 3(1) - 1 = 3 - 1 = 2
- x = 2: y = 3(2) - 1 = 6 - 1 = 5

y = 3x:
- x = -2: y = 3(-2) = -6
- x = -1: y = 3(-1) = -3
- x = 0: y = 3(0) = 0
- x = 1: y = 3(1) = 3
- x = 2: y = 3(2) = 6

y = 3x + 1:
- x = -2: y = 3(-2) + 1 = -6 + 1 = -5
- x = -1: y = 3(-1) + 1 = -3 + 1 = -2
- x = 0: y = 3(0) + 1 = 0 + 1 = 1
- x = 1: y = 3(1) + 1 = 3 + 1 = 4
- x = 2: y = 3(2) + 1 = 6 + 1 = 7

y = 3x + 2:
- x = -2: y = 3(-2) + 2 = -6 + 2 = -4
- x = -1: y = 3(-1) + 2 = -3 + 2 = -1
- x = 0: y = 3(0) + 2 = 0 + 2 = 2
- x = 1: y = 3(1) + 2 = 3 + 2 = 5
- x = 2: y = 3(2) + 2 = 6 + 2 = 8

Pattern: All have slope 3 (same steepness). Each equation is shifted vertically by different amounts. The lines are parallel.

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Table 2: y = -2x + 1, y = 1 - 2x, y + 2x = 1

Let me check if these are the same equation:
- y = -2x + 1
- y = 1 - 2x (same as above, just rearranged)
- y + 2x = 1 → y = 1 - 2x (same as above)

All three equations are identical! Let me calculate:

y = -2x + 1:
- x = -2: y = -2(-2) + 1 = 4 + 1 = 5
- x = -1: y = -2(-1) + 1 = 2 + 1 = 3
- x = 0: y = -2(0) + 1 = 0 + 1 = 1
- x = 1: y = -2(1) + 1 = -2 + 1 = -1
- x = 2: y = -2(2) + 1 = -4 + 1 = -3

Pattern: All three equations represent the SAME line. They're just written differently.

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Table 3: y = 2x + 1, y = -½x + 1

y = 2x + 1:
- x = -2: y = 2(-2) + 1 = -4 + 1 = -3
- x = -1: y = 2(-1) + 1 = -2 + 1 = -1
- x = 0: y = 2(0) + 1 = 0 + 1 = 1
- x = 1: y = 2(1) + 1 = 2 + 1 = 3
- x = 2: y = 2(2) + 1 = 4 + 1 = 5

y = -½x + 1:
- x = -2: y = -½(-2) + 1 = 1 + 1 = 2
- x = -1: y = -½(-1) + 1 = 0.5 + 1 = 1.5
- x = 0: y = -½(0) + 1 = 0 + 1 = 1
- x = 1: y = -½(1) + 1 = -0.5 + 1 = 0.5
- x = 2: y = -½(2) + 1 = -1 + 1 = 0

Pattern: Both lines pass through (0, 1) - they have the same y-intercept but different slopes. One goes up steeply (slope 2), one goes down gently (slope -½).

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Table 4: y = ½x, y = x, y = 2x, y = 3x, y = 4x

y = ½x:
- x = -2: y = ½(-2) = -1
- x = -1: y = ½(-1) = -0.5
- x = 0: y = ½(0) = 0
- x = 1: y = ½(1) = 0.5
- x = 2: y = ½(2) = 1

y = x:
- x = -2: y = -2
- x = -1: y = -1
- x = 0: y = 0
- x = 1: y = 1
- x = 2: y = 2

y = 2x:
- x = -2: y = 2(-2) = -4
- x = -1: y = 2(-1) = -2
- x = 0: y = 2(0) = 0
- x = 1: y = 2(1) = 2
- x = 2: y = 2(2) = 4

y = 3x:
- x = -2: y = 3(-2) = -6
- x = -1: y = 3(-1) = -3
- x = 0: y = 3(0) = 0
- x = 1: y = 3(1) = 3
- x = 2: y = 3(2) = 6

y = 4x:
- x = -2: y = 4(-2) = -8
- x = -1: y = 4(-1) = -4
- x = 0: y = 4(0) = 0
- x = 1: y = 4(1) = 4
- x = 2: y = 4(2) = 8

Pattern: All pass through origin (0, 0). As the coefficient increases (½, 1, 2, 3, 4), the lines get steeper.

Final Answer:

Table 1 (y = 3x series):
| x | -2 | -1 | 0 | 1 | 2 |
|---|----|----|---|---|---|
| y = 3x - 2 | -8 | -5 | -2 | 1 | 4 |
| y = 3x - 1 | -7 | -4 | -1 | 2 | 5 |
| y = 3x | -6 | -3 | 0 | 3 | 6 |
| y = 3x + 1 | -5 | -2 | 1 | 4 | 7 |
| y = 3x + 2 | -4 | -1 | 2 | 5 | 8 |

*Notice: All lines have the same slope (3) and are parallel. Each is shifted up by 1 unit from the previous.*

Table 2 (y = -2x + 1 series):
| x | -2 | -1 | 0 | 1 | 2 |
|---|----|----|---|---|---|
| y = -2x + 1 | 5 | 3 | 1 | -1 | -3 |
| y = 1 - 2x | 5 | 3 | 1 | -1 | -3 |
| y + 2x = 1 | 5 | 3 | 1 | -1 | -3 |

*Notice: All three equations are the SAME line written in different forms.*

Table 3:
| x | -2 | -1 | 0 | 1 | 2 |
|---|----|----|---|---|---|
| y = 2x + 1 | -3 | -1 | 1 | 3 | 5 |
| y = -½x + 1 | 2 | 1.5 | 1 | 0.5 | 0 |

*Notice: Both lines share the same y-intercept (0, 1) but have different slopes.*

Table 4 (y = kx series):
| x | -2 | -1 | 0 | 1 | 2 |
|---|----|----|---|---|---|
| y = ½x | -1 | -0.5 | 0 | 0.5 | 1 |
| y = x | -2 | -1 | 0 | 1 | 2 |
| y = 2x | -4 | -2 | 0 | 2 | 4 |
| y = 3x | -6 | -3 | 0 | 3 | 6 |
| y = 4x | -8 | -4 | 0 | 4 | 8 |

*Notice: All lines pass through the origin (0, 0). Larger coefficients create steeper lines.*