Plotting Straight-Line Graphs Revision Notes | GCSE AQA Higher Maths

1. Why Straight-Line Graphs Matter

Every time a taxi charges a fixed rate per mile, or a phone plan adds a set cost per minute, the relationship between cost and usage is a straight line. Plotting straight-line graphs is one of the most tested skills in GCSE Maths — and once you see the pattern, every question becomes predictable.

The key equation you need is $y = mx + c$, where $m$ controls the steepness and $c$ tells you where the line crosses the $y$-axis. Your job in this lesson is to go from an equation like $y = 2x + 1$ to a correctly plotted line on a coordinate grid — every time, without error.

2. The Equation of a Straight Line

Reading $m$ and $c$ from an equation is a quick win. In $y = 3x - 2$, the gradient is $3$ and the line crosses the $y$-axis at $-2$. In $y = -x + 5$, the gradient is $-1$ (the line slopes downward) and the y-intercept is $5$. Spotting these values instantly saves time in the exam.

3. Building a Table of Values

Knowing $m$ and $c$ tells you a lot about a line, but to plot it accurately you need coordinates. That is where a table of values comes in — you choose a set of $x$-values, substitute each one into the equation, and record the resulting $y$-values as coordinate pairs.

Choose $x$-values that are spread across the grid you are given, typically three to five values including at least one negative. Wider spacing makes it easier to draw an accurate line and spot any arithmetic errors.

Worked example: Plotting $y = 2x + 1$ from a Table of Values

Complete a table of values for $y = 2x + 1$ using $x = -2, -1, 0, 1, 2$, then describe the coordinates you would plot.

1. $x$: $-2,\ -1,\ 0,\ 1,\ 2$ — Choose a spread of $x$-values that covers the grid range. Including negatives and zero gives a reliable picture of the line. [M1 for method]
2. $y = 2 \times (-2) + 1 = -4 + 1 = -3$ — Multiply first, then add. The coordinate is $(-2, -3)$.
3. $y = 2 \times (-1) + 1 = -2 + 1 = -1$ — The coordinate is $(-1, -1)$.
4. $y = 2 \times 0 + 1 = 0 + 1 = 1$ — When $x = 0$, $y$ equals the y-intercept $c = 1$. The coordinate is $(0, 1)$. This is a useful check.
5. $y = 2 \times 1 + 1 = 2 + 1 = 3$ — The coordinate is $(1, 3)$.
6. $y = 2 \times 2 + 1 = 4 + 1 = 5$ — The coordinate is $(2, 5)$. Notice each $y$-value increases by $2$ as $x$ increases by $1$ — that is the gradient $m = 2$ in action. [A1 for all correct coordinates]

Answer: Coordinates to plot: $(-2, -3)$, $(-1, -1)$, $(0, 1)$, $(1, 3)$, $(2, 5)$. Join with a straight line and extend to the edges of the grid.

Notice how the five plotted points all lie perfectly on the same straight line. If one point looks out of place when you draw the line, go back and recheck that substitution — a single arithmetic slip is almost always the cause.

4. Plotting the Graph Accurately

Once you have your table of values, plotting is a three-step process: mark each coordinate as a small cross or dot, check they appear to lie in a straight line, then draw a single ruled line through all of them extending to the edges of the grid.

• Use a sharp pencil and a ruler — a wobbly freehand line will lose marks.
• Mark points with a small, precise cross ($\times$) so the exact location is clear.
• Extend the line to the edges of the grid unless the question specifies a limited domain.
• Label the line with its equation, especially if more than one line is plotted.

Worked example: Plotting $y = -x + 3$ (Negative Gradient)

Complete a table of values for $y = -x + 3$ using $x = -1, 0, 1, 2, 3$ and describe the graph.

1. $y = -(-1) + 3 = 1 + 3 = 4$ — Negative times negative gives positive. Coordinate: $(-1, 4)$. [M1 for correct handling of negative gradient]
2. $y = -(0) + 3 = 0 + 3 = 3$ — The y-intercept is $3$. Coordinate: $(0, 3)$.
3. $y = -(1) + 3 = -1 + 3 = 2$ — Coordinate: $(1, 2)$. Notice $y$ decreases by $1$ each time $x$ increases by $1$ — that is the gradient $m = -1$.
4. $y = -(2) + 3 = -2 + 3 = 1$ — Coordinate: $(2, 1)$.
5. $y = -(3) + 3 = -3 + 3 = 0$ — Coordinate: $(3, 0)$. The line crosses the $x$-axis here. [A1 for all correct coordinates]
6. Points: $(-1, 4)$, $(0, 3)$, $(1, 2)$, $(2, 1)$, $(3, 0)$ — The line slopes downward from left to right (negative gradient). It crosses the $y$-axis at $3$ and the $x$-axis at $3$. Draw a ruled line through all five points.

Answer: A straight line with gradient $-1$, crossing the $y$-axis at $(0, 3)$ and the $x$-axis at $(3, 0)$, sloping downward from left to right.

5. Special Cases: Lines Through the Origin and Horizontal Lines

Two types of straight line appear regularly in exam questions and are worth recognising immediately. The first is any line of the form $y = mx$, where $c = 0$. These lines pass directly through the origin $(0, 0)$ because when $x = 0$, $y = 0$ as well.

The second type is a horizontal line such as $y = 4$. Here, $y$ is always $4$ regardless of the value of $x$, so the gradient is zero. Vertical lines like $x = 3$ are also straight lines — every point on them has $x = 3$ regardless of $y$.

Worked example: Plotting $y = 3x$ (Through the Origin)

Plot the graph of $y = 3x$ using $x = -2, -1, 0, 1, 2$.

1. $y = 3 \times (-2) = -6$ — Coordinate: $(-2, -6)$.
2. $y = 3 \times (-1) = -3$ — Coordinate: $(-1, -3)$.
3. $y = 3 \times 0 = 0$ — Coordinate: $(0, 0)$. The line passes through the origin — this confirms $c = 0$. [M1]
4. $y = 3 \times 1 = 3$ — Coordinate: $(1, 3)$.
5. $y = 3 \times 2 = 6$ — Coordinate: $(2, 6)$. Plot all five points and draw a ruled line through them. [A1]

Answer: A straight line through the origin with gradient $3$, passing through $(-2, -6)$, $(-1, -3)$, $(0, 0)$, $(1, 3)$, $(2, 6)$.

6. Deliberate Practice

Now that you have seen the method across three different equation types, it is time to apply it independently. Work through each problem below before checking the solution — the act of attempting it first is what builds exam confidence.

7. Exam Technique and Common Mistakes

Most marks lost on straight-line graph questions come from a small number of predictable errors. Knowing what they are — and actively checking for them — is the difference between full marks and dropping two or three on a question you understood.

Plotting straight-line graphs accurately — by building a table of values, substituting carefully, and drawing a ruled line — is the core skill assessed in higher-tier specification point 3.2.2. Every other straight-line topic (finding equations, solving simultaneous equations graphically) builds directly on this foundation.