Straight-Line Graphs
The Skill
The Equation of a Straight Line
$$y = mx + c$$
- $m$ = gradient (steepness of the line)
- $c$ = y-intercept (where the line crosses the y-axis)
Finding the Gradient
$$m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
- Positive gradient: line goes up from left to right ↗
- Negative gradient: line goes down from left to right ↘
- Zero gradient: horizontal line
- Undefined gradient: vertical line
Finding the Equation of a Line
Method 1: From a graph
- Find where the line crosses the y-axis → that's $c$
- Pick two points and calculate the gradient → that's $m$
- Write $y = mx + c$
Method 2: From two points
- Calculate gradient: $m = \frac{y_2 - y_1}{x_2 - x_1}$
- Substitute one point into $y = mx + c$ to find $c$
Parallel and Perpendicular Lines
Parallel lines have the same gradient. If $y = 3x + 1$ and $y = 3x - 5$ are parallel (both have $m = 3$).
Perpendicular lines have gradients that multiply to give $-1$. If one line has gradient $m$, the perpendicular has gradient $-\frac{1}{m}$.
Example: $y = 2x + 3$ and $y = -\frac{1}{2}x + 1$ are perpendicular.
Plotting Straight Lines
- Make a table of values (at least 3 points)
- Plot the points
- Draw a straight line through them
The Traps
Common misconceptions and how to avoid them.
Calculating gradient with x over y instead of y over x "The Gradient Flip"
The Mistake in Action
Find the gradient of the line through (1, 2) and (4, 8).
Wrong: $m = \frac{4 - 1}{8 - 2} = \frac{3}{6} = 0.5$
Why It Happens
Students mix up which values go on top and bottom. They may remember "rise over run" but forget which is which.
The Fix
Gradient = change in y over change in x
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Memory aids:
- "y over x" — alphabetically, y comes after x, so y is on top
- "Rise over run" — rise (vertical/y) comes before run (horizontal/x)
$$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$
Spot the Mistake
Find the gradient through (1, 2) and (4, 8)
$m = \frac{4-1}{8-2} = \frac{3}{6}$
$= 0.5$
Click on the line that contains the error.
Confusing gradient and y-intercept "The m-c Mixup"
The Mistake in Action
For the line $y = 5x - 3$, state the gradient and y-intercept.
Wrong: Gradient = $-3$ y-intercept = $5$
Why It Happens
Students remember that the equation has two important numbers but mix up which is which. The order in the equation ($mx$ before $c$) can be confusing.
The Fix
In $y = mx + c$:
- $m$ is the coefficient of $x$ (the number multiplied by $x$) = gradient
- $c$ is the constant (the number on its own) = y-intercept
For $y = 5x - 3$:
- Gradient = 5 (the coefficient of $x$)
- y-intercept = $-3$ (the constant)
Memory aid: "$m$ for multiplied by $x$; $c$ for constant"
Spot the Mistake
For $y = 5x - 3$:
Gradient = $-3$, y-intercept = $5$
Click on the line that contains the error.
Thinking perpendicular lines have the same gradient "The Perpendicular Puzzle"
The Mistake in Action
Line A has equation $y = 2x + 1$. Line B is perpendicular to Line A. Find the gradient of Line B.
Wrong: Gradient of Line B = $2$ (same as Line A)
Why It Happens
Students confuse perpendicular with parallel, or don't know the relationship between perpendicular gradients.
The Fix
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to give $-1$.
If Line A has gradient $m$, then a perpendicular line has gradient $-\frac{1}{m}$.
Line A: gradient = $2$ Line B: gradient = $-\frac{1}{2}$
Check: $2 \times (-\frac{1}{2}) = -1$ ✓
Memory aid: "Flip and negate" — turn the gradient upside down and change the sign.
Spot the Mistake
Line A: $y = 2x + 1$. Line B is perpendicular.
Gradient of Line B = 2
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Find the gradient of the line passing through the points (2, 3) and (6, 11).
Solution
Use the gradient formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Label the points: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (6, 11)$
$$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$$
Answer: Gradient = 2
Check: The line goes up from left to right, so the gradient should be positive ✓
Question
A line has gradient 3 and y-intercept $-2$. Write its equation.
Solution
The general form is $y = mx + c$ where:
- $m$ = gradient = 3
- $c$ = y-intercept = $-2$
Substitute: $$y = 3x + (-2)$$ $$y = 3x - 2$$
Answer: $y = 3x - 2$
Question
Find the gradient and y-intercept of the line $2y - 6x = 10$.
Solution
Step 1: Rearrange into $y = mx + c$ form. $$2y - 6x = 10$$ $$2y = 6x + 10$$ $$y = 3x + 5$$
Step 2: Read off the values. Comparing with $y = mx + c$:
- Gradient ($m$) = 3
- y-intercept ($c$) = 5
Answer: Gradient = 3, y-intercept = 5
Level 2: Scaffolded
Fill in the key steps.
Question
Find the equation of the line through (1, 4) and (3, 10).
Level 3: Solo
Try it yourself!
Question
Find the equation of the line parallel to $y = 4x - 3$ that passes through the point (2, 5).
Show Solution
Step 1: Find the gradient. Parallel lines have the same gradient. The given line has gradient 4, so our line also has gradient 4.
Step 2: Use $y = mx + c$ with the point. $$5 = 4(2) + c$$ $$5 = 8 + c$$ $$c = -3$$
Answer: $y = 4x - 3$
Wait — this is the same as the original line! Let me check...
Check: Does (2, 5) lie on $y = 4x - 3$? $4(2) - 3 = 8 - 3 = 5$ ✓
The point (2, 5) actually lies on the original line, so the "parallel" line through it is the same line.
Question
Find the equation of the line perpendicular to $y = 2x + 1$ that passes through the point (4, 3).
Show Solution
Step 1: Find the gradient. The original line has gradient 2. For perpendicular lines, gradients multiply to give $-1$. $$2 \times m = -1$$ $$m = -\frac{1}{2}$$
Step 2: Use $y = mx + c$ with the point. $$3 = -\frac{1}{2}(4) + c$$ $$3 = -2 + c$$ $$c = 5$$
Answer: $y = -\frac{1}{2}x + 5$
Check: Gradients: $2 \times (-\frac{1}{2}) = -1$ ✓ Point: $-\frac{1}{2}(4) + 5 = -2 + 5 = 3$ ✓
Examiner's View
Mark allocation: Finding gradient is 1-2 marks. Finding equation is 2-3 marks. Parallel/perpendicular is 2-3 marks.
Common errors examiners see:
- Gradient upside down (change in x over change in y)
- Wrong sign on gradient
- Confusing m and c
- Not recognising rearranged equations
What gains marks:
- Showing the gradient calculation clearly
- Drawing accurate lines using a ruler
- Checking your equation with a point
AQA Notes
AQA often gives equations not in $y = mx + c$ form — rearrange first!