This worksheet helps students practice finding the equation of a line by analyzing 10 different graphs to determine the gradient and y-intercept.

Let's solve each of the 10 problems on this worksheet step by step.

We are asked to find:
- Gradient (m): The slope of the line, calculated as $ m = \frac{\text{rise}}{\text{run}} $
- Y-intercept (c): The point where the line crosses the y-axis (i.e., when $ x = 0 $)
- Equation of the line: In the form $ y = mx + c $

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1)

Graph: Line passes through (-2, 0), (0, 2), and (2, 4)

- Y-intercept: At $ x = 0 $, $ y = 2 $ → $ c = 2 $
- Gradient: From (-2, 0) to (0, 2):
Rise = $ 2 - 0 = 2 $, Run = $ 0 - (-2) = 2 $ → $ m = \frac{2}{2} = 1 $
- Equation: $ y = 1x + 2 $ → $ \boxed{y = x + 2} $

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2)

Graph: Line passes through (0, -1), (1, 1), (2, 3)

- Y-intercept: At $ x = 0 $, $ y = -1 $ → $ c = -1 $
- Gradient: From (0, -1) to (1, 1):
Rise = $ 1 - (-1) = 2 $, Run = $ 1 - 0 = 1 $ → $ m = \frac{2}{1} = 2 $
- Equation: $ y = 2x - 1 $

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3)

Graph: Line passes through (0, 2), (1, 5), (-1, -1)

- Y-intercept: At $ x = 0 $, $ y = 2 $ → $ c = 2 $
- Gradient: From (0, 2) to (1, 5):
Rise = $ 5 - 2 = 3 $, Run = $ 1 - 0 = 1 $ → $ m = 3 $
- Equation: $ y = 3x + 2 $

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4)

Graph: Line passes through (0, -2), (1, -4), (-1, 0)

- Y-intercept: At $ x = 0 $, $ y = -2 $ → $ c = -2 $
- Gradient: From (0, -2) to (1, -4):
Rise = $ -4 - (-2) = -2 $, Run = $ 1 - 0 = 1 $ → $ m = -2 $
- Equation: $ y = -2x - 2 $

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5)

Graph: Line passes through (0, 4), (1, 1), (2, -2)

- Y-intercept: At $ x = 0 $, $ y = 4 $ → $ c = 4 $
- Gradient: From (0, 4) to (1, 1):
Rise = $ 1 - 4 = -3 $, Run = $ 1 - 0 = 1 $ → $ m = -3 $
- Equation: $ y = -3x + 4 $

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6)

Graph: Line passes through (-2, -2), (0, 1), (2, 4)

- Y-intercept: At $ x = 0 $, $ y = 1 $ → $ c = 1 $
- Gradient: From (-2, -2) to (0, 1):
Rise = $ 1 - (-2) = 3 $, Run = $ 0 - (-2) = 2 $ → $ m = \frac{3}{2} $
- Equation: $ y = \frac{3}{2}x + 1 $

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7)

Graph: Line passes through (0, -5), (1, -2), (2, 1)

- Y-intercept: At $ x = 0 $, $ y = -5 $ → $ c = -5 $
- Gradient: From (0, -5) to (1, -2):
Rise = $ -2 - (-5) = 3 $, Run = $ 1 - 0 = 1 $ → $ m = 3 $
- Equation: $ y = 3x - 5 $

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8)

Graph: Line passes through (0, 0), (1, -2), (-1, 2)

- Y-intercept: At $ x = 0 $, $ y = 0 $ → $ c = 0 $
- Gradient: From (0, 0) to (1, -2):
Rise = $ -2 - 0 = -2 $, Run = $ 1 - 0 = 1 $ → $ m = -2 $
- Equation: $ y = -2x $

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9)

Graph: Line passes through (-2, 0), (0, 1), (2, 2)

- Y-intercept: At $ x = 0 $, $ y = 1 $ → $ c = 1 $
- Gradient: From (-2, 0) to (0, 1):
Rise = $ 1 - 0 = 1 $, Run = $ 0 - (-2) = 2 $ → $ m = \frac{1}{2} $
- Equation: $ y = \frac{1}{2}x + 1 $

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10)

Graph: Line passes through (0, 1), (2, 0), (-2, 2)

- Y-intercept: At $ x = 0 $, $ y = 1 $ → $ c = 1 $
- Gradient: From (0, 1) to (2, 0):
Rise = $ 0 - 1 = -1 $, Run = $ 2 - 0 = 2 $ → $ m = -\frac{1}{2} $
- Equation: $ y = -\frac{1}{2}x + 1 $

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✔ Final Answers:

| # | Gradient | Y-intercept | Equation |
|---|----------|-------------|----------|
| 1 | 1 | 2 | $ y = x + 2 $ |
| 2 | 2 | -1 | $ y = 2x - 1 $ |
| 3 | 3 | 2 | $ y = 3x + 2 $ |
| 4 | -2 | -2 | $ y = -2x - 2 $ |
| 5 | -3 | 4 | $ y = -3x + 4 $ |
| 6 | $ \frac{3}{2} $ | 1 | $ y = \frac{3}{2}x + 1 $ |
| 7 | 3 | -5 | $ y = 3x - 5 $ |
| 8 | -2 | 0 | $ y = -2x $ |
| 9 | $ \frac{1}{2} $ | 1 | $ y = \frac{1}{2}x + 1 $ |
|10 | $ -\frac{1}{2} $ | 1 | $ y = -\frac{1}{2}x + 1 $ |

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🔍 Summary Tips:

- Always look for the point where the line crosses the y-axis → that’s the y-intercept.
- To find the gradient, pick two points on the line and use $ m = \frac{y_2 - y_1}{x_2 - x_1} $.
- Then write the equation in the form $ y = mx + c $.

Let me know if you'd like a printable version or help with any specific graph!