Line Graph Practice Test Worksheets
Line graph questions ask you to compare line segments on a coordinate grid. The key is to calculate gradients carefully, then use the rules for parallel lines, perpendicular lines, translations, and missing coordinates.
For students who can read plotted points, but lose marks using gradients, parallel lines, perpendicular lines, and translations.
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What Do Line Graph Questions Look Like?
Line graph questions ask you to compare line segments plotted on a coordinate grid. You must calculate gradients, test whether lines are parallel or perpendicular, and determine translations or missing endpoint coordinates. Here is a worked example.
Check Whether Two Lines Are Parallel
Question: Line A passes through (3, 4) and (6, 2). Line B passes through (1, 7) and (4, 5). Are lines A and B parallel?
Worked Method
Step 1: Gradient of line A = 2 - 46 - 3 = -23.
Step 2: A parallel line must also have gradient -23.
Step 3: Line B, through (1, 7) and (4, 5), has gradient 5 - 74 - 1 = −23, matching line A.
Step 4: At x = 4, line A has y = 4 − 23 = 103, whereas line B has y = 5. They are distinct lines with equal gradients, so yes, they are parallel.
Tip: Use the same point order in the numerator and denominator. Reversing only one subtraction gives the wrong sign.
Rules to Help You Solve Line Graph Questions
The papers focus on coordinate line segments: finding gradients, comparing parallel lines, checking perpendicular lines, translating a line, and using a known gradient to find a missing coordinate:
📈 Rule 1: Gradient is rise over run
Subtract the y-values, then divide by the change in x-values, keeping the points in the same order in both subtractions. A horizontal line has gradient 0. A vertical line has an undefined gradient because its change in x is 0; do not divide by zero.
Example: From (3, 4) to (6, 2), gradient = 2 - 46 - 3 = -23.
∥ Rule 2: Parallel lines have the same gradient
Distinct non-vertical parallel lines have equal gradients and different y-intercepts (where they cross the y-axis). Equal gradients and equal y-intercepts mean the segments lie on the same straight line. Distinct vertical lines are parallel to each other.
Example: A line with gradient -12 is parallel to any other line with gradient -12 and a different y-intercept.
⊥ Rule 3: Perpendicular gradients multiply to -1
For a line with a non-zero, defined gradient, flip the fraction and change the sign to find the perpendicular gradient. These two gradients multiply to -1. The special case is a horizontal line and a vertical line: they are perpendicular, but the vertical gradient is undefined.
Example: A line with gradient -1 has perpendicular gradient 1.
↔ Rule 4: Translations move both endpoints
Apply the same translation vector to each endpoint: use the horizontal change for each x-coordinate and the vertical change for each y-coordinate.
Example: Moving (5, 1) and (7, 2) two left and one up gives (3, 2) and (5, 3).
Two More Worked Examples
Translate Both Endpoints
Question: A line segment joins (5, 1) to (7, 2). Find its new endpoints after a translation 2 units left and 1 unit up.
Worked Method
Step 1: A move 2 left and 1 up means x - 2 and y + 1.
Step 2: (5, 1) becomes (3, 2).
Step 3: (7, 2) becomes (5, 3).
Tip: Translate both endpoints. Moving only one endpoint does not correctly translate the whole segment.
Use the Parallel Gradient to Find a Missing Coordinate
Question: A second line joins (2, 5) to (6, y). If it is parallel to a line from (3, 1) to (7, 2), what is y?
Worked Method
Step 1: Gradient of the first line = 2 - 17 - 3 = 14.
Step 2: The second line must also have gradient 14.
Step 3: Its run is 6 - 2 = 4, so the rise must be 1.
Step 4: y = 5 + 1 = 6.
Tip: The rise is the change in y, not the final y-coordinate. A rise of 1 means add 1 to the starting y-coordinate of 5, giving y = 6.
