Gradient and Coordinates Worksheet

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📚 10C Standard — Foundation Skills

Instructions: Show all working. Use the gradient formula: m = (y₂ - y₁)/(x₂ - x₁). Leave answers in simplest form.

1. Calculate the gradient of the line through the points (1, 2) and (4, 8).

2. Find the gradient of the line joining A(0, 3) and B(6, 9).

3. For each line below, choose the correct classification:

Horizontal (gradient = 0)

Vertical (undefined gradient)

Positive gradient

Negative gradient

a) The line y = 3 is ____________.

b) The line x = 5 is ____________.

c) The line through (0, 0) and (2, 4) has ____________.

4. State the gradient and y-intercept for each linear equation:

a) y = 2x + 5

b) y = -3x + 1

✏️ 10B Standard — Application and Problem Solving

5. Find the gradient of the line joining P(-2, 4) and Q(3, -6). Give your answer in simplest fractional form.

6. A line passes through points R(1, 3) and S(5, k) and has gradient 2. Find the value of k.

7. Determine if lines AB and CD are parallel by comparing their gradients:

Line AB: A(-1, 2), B(3, 10)

Line CD: C(0, -1), D(2, 3)

8. Rearrange the equation 3x + 2y = 12 into the form y = mx + c and identify the gradient.

9. Find the value of p if the gradient of the line through (p, 7) and (4, 1) is -2.

🧠 10A Standard — Advanced Reasoning and Extension

10. Error Analysis: A student calculated the gradient between M(-4, 6) and N(2, -3) as (-4-6)/(2-(-3)) = -2. Identify and correct the error, then find the actual gradient.

11. Three points A(1, 2), B(4, 8), and C(7, k) are collinear (lie on the same straight line). Use the fact that gradients AB and BC must be equal to find the value of k.

12. Investigation: Find the gradients of all four sides of quadrilateral PQRS with vertices P(0, 0), Q(6, 2), R(8, 8), and S(2, 6). Use your results to classify the quadrilateral and justify your answer.

13. Challenge: Two lines are perpendicular if the product of their gradients equals -1. Line 1 passes through (1, 3) and (4, 9). Line 2 passes through (0, 5) and (6, t). Find the value of t that makes these lines perpendicular.

Teacher Note: Worked solutions provided separately for differentiated feedback and explicit teaching examples.