Curve Sketching with Maxima and Minima Revision Notes | GCSE AQA Higher Further Maths

1. What a Sketch Actually Tells You

Imagine you are driving along a hilly road. The road rises to a peak, then dips into a valley, then rises again. A curve on a graph behaves in exactly the same way — and a sketch captures those hills and valleys without needing to plot every single point.

In GCSE maths, when an exam question says sketch, it does not mean draw a precise plot. It means draw the approximate shape and label the key features: turning points, intercepts, and the general direction of the curve. Those labels are where the marks live.

The two most important features on any sketch are the maximum point and the minimum point — collectively called turning points. Knowing how to find them, label them, and interpret them is the core skill this lesson builds.

2. Maxima and Minima — The Key Definitions

Think of a maximum point as the top of a hill: the curve was climbing, it reaches the peak, then it starts to fall. A minimum point is the bottom of a valley: the curve was falling, it reaches the lowest point, then it starts to climb again.

• At a maximum point: the curve goes up, reaches the peak, then comes back down. The gradient changes from positive to zero to negative.
• At a minimum point: the curve goes down, reaches the trough, then rises again. The gradient changes from negative to zero to positive.
• Both are called turning points because the curve literally turns direction at these coordinates.
• On a sketch, turning points are labelled with their coordinates, e.g. $(2, 5)$ for a maximum or $(1, -3)$ for a minimum.

3. Reading the Key Features Before You Sketch

Before putting pencil to paper, gather all the information the question gives you. A well-organised list of features makes the sketch almost draw itself.

At GCSE, the question will usually give you the coordinates of the turning points. Your job is to use those coordinates correctly on the sketch and interpret what they mean — not to derive them from scratch using calculus.

Sketching $y = x^2 - 4x + 3$

4. Worked Example — Sketching a Curve with Given Turning Points

Here is the type of question you will meet in an exam. The turning points are given to you — your task is to use them to produce a correct, labelled sketch.

Worked example: Sketching a Cubic Curve with Known Turning Points

A curve has equation $y = x^3 - 3x^2 - 9x + 2$. The curve has a maximum point at $(-1, 7)$ and a minimum point at $(3, -25)$. The curve crosses the y-axis at $(0, 2)$. Sketch the curve, labelling the turning points and the y-intercept.

1. Leading term is $x^3$ with a positive coefficient. — A positive cubic starts bottom-left and ends top-right. The curve will come from below on the left and rise to the right. This tells you the overall direction before you plot anything. [M1 — correct shape identified]
2. Plot the point $(0, 2)$ on the y-axis. — The y-intercept is given directly. Mark it clearly — it anchors the left side of your sketch.
3. Plot $(-1, 7)$ and label it 'maximum $(-1, 7)$'. — This is the local high point. The curve rises to this peak from the left, then falls away to the right. [A1 — correct coordinates labelled]
4. Plot $(3, -25)$ and label it 'minimum $(3, -25)$'. — This is the local low point. The curve falls to this trough from the left, then rises again to the right. [A1 — correct coordinates labelled]
5. Connect the points with a smooth continuous curve: rising from bottom-left, peaking at $(-1, 7)$, falling through $(0, 2)$, continuing down to $(3, -25)$, then rising to the top-right. — The curve must pass through all labelled points without sharp corners. A smooth, flowing line shows the gradient is changing continuously. [M1 — smooth curve through all labelled features]
6. Confirm: maximum is higher than minimum. Curve comes from bottom-left and exits top-right. All labels are clear. — A quick sanity check catches errors before you move on. The maximum at $y = 7$ is indeed above the minimum at $y = -25$, which is consistent.

Answer: A smooth cubic curve rising from bottom-left, reaching a maximum at $(-1, 7)$, passing through $(0, 2)$, falling to a minimum at $(3, -25)$, then rising to the top-right. All three points are labelled with coordinates.

5. Interpreting Turning Points — What Do They Mean?

Sketching is only half the skill. Exam questions also ask you to interpret what the turning points mean — either in a pure maths context or within a real-world scenario.

When you interpret a maximum point, you are saying: at this $x$-value, the quantity described by $y$ is at its greatest local value. When you interpret a minimum point, you are saying: at this $x$-value, the quantity described by $y$ is at its smallest local value.

Worked example: Interpreting a Maximum Point in Context

A ball is thrown upward and its height $h$ metres above the ground after $t$ seconds is modelled by a curve. The curve has a maximum point at $(3, 44)$ and crosses the horizontal axis at $t = 0$ and $t = 6$. Interpret the maximum point in context.

1. Maximum point: $(3, 44)$, so $t = 3$ and $h = 44$. — Always read both coordinates — the $x$-value tells you when, the $y$-value tells you how much.
2. $t = 3$ seconds after the ball is thrown. — The horizontal axis represents time, so $t = 3$ means 3 seconds have elapsed.
3. $h = 44$ metres above the ground. — The vertical axis represents height, so $h = 44$ is the height at that moment.
4. The ball reaches its greatest height of 44 metres at 3 seconds after being thrown. — A complete interpretation links both coordinates to their real-world meaning. This is what earns the mark in an exam. [A1 — both values interpreted in context]

Answer: The maximum point $(3, 44)$ means the ball reaches its greatest height of 44 metres, 3 seconds after being thrown.

6. Quadratic Curves — A Special Case Worth Knowing

Quadratic curves (parabolas) have exactly one turning point. If the coefficient of $x^2$ is positive, the parabola opens upward and the single turning point is a minimum. If the coefficient of $x^2$ is negative, the parabola opens downward and the single turning point is a maximum.

7. Practice — Try These Yourself

The best way to consolidate this skill is to work through questions where you apply the full process: identify features, mark them, and draw the curve. Here are two practice problems to test your method.

8. Exam Technique — Marks, Traps, and Winning Moves

Sketch questions are generous with method marks — but only if you show the examiner what you know. A sketch with no labels earns very few marks, even if the shape is correct. Every coordinate you label is a potential mark.

Interpretation questions require you to link both the $x$-value and the $y$-value back to the context. Writing only one of the two values will cost you a mark. Always write a full sentence: "At $x =...$, the value of $y$ is..., which means..."