GCSE Maths Quadratic Graph Turning Points Explained
The turning point of a quadratic graph is its highest or lowest point. GCSE Higher questions may ask you to read it from a diagram, estimate it from a table or find it exactly using algebra. A completed-square form often makes the answer visible immediately. These examples show how the equation, axis of symmetry and roots fit together.
Recognise the shape of a quadratic graph
A function such as y = x² produces a U-shaped parabola opening upwards. Its turning point is (0, 0), where the graph reaches its minimum. A negative coefficient of x², as in y = −x², produces a downward-opening parabola with a maximum instead. Vertical and horizontal shifts move the turning point. The x-coordinate of the turning point also gives the equation of the graph’s vertical line of symmetry. Sketch the shape first so you know whether the turning point should be a maximum or a minimum.
Find the turning point by completing the square
Take y = x² − 6x + 5. Half of −6 is −3, so write y = (x − 3)² − 9 + 5 = (x − 3)² − 4. Because (x − 3)² cannot be negative for real x, the smallest y-value is −4. That happens when x = 3, so the turning point is (3, −4). The axis of symmetry is x = 3. You can verify the result by substituting x = 3 into the original expression: 9 − 18 + 5 = −4.
Connect the roots to the axis of symmetry
Factorising the same expression gives y = (x − 1)(x − 5). The graph crosses the x-axis where y = 0, at x = 1 and x = 5. The axis of symmetry lies halfway between the roots: (1 + 5)/2 = 3. That matches the turning point’s x-coordinate found by completing the square. The y-intercept is found by putting x = 0, which gives y = 5. These three features help you make an accurate sketch without constructing a long table of values.
A downward-opening example with a maximum
Now consider y = −x² + 4x + 1. Factor out the negative from the squared and linear terms: y = −(x² − 4x) + 1. Complete the square inside: x² − 4x = (x − 2)² − 4. Therefore y = −(x − 2)² + 5. This graph reaches a maximum of 5 at x = 2, giving turning point (2, 5). The coefficient of the squared expression is negative, so any move away from x = 2 decreases y.
Avoid confusing the coordinates of the turning point
For y = (x + 4)² − 9, the square is smallest when x = −4, not +4. The turning point is therefore (−4, −9). Always solve the bracket equal to zero rather than copying the sign from inside it. Remember that the number outside the square is the y-value of the turning point. Students also sometimes give only x = −4, which is the axis of symmetry rather than the full coordinate pair. State clearly what the question asks for.
Use graph skills in exam-style problems
A GCSE question may provide a graph and ask for its roots, intercept, turning point or values for a specified x. Read the axes and scale carefully. For an exact algebraic question, choose factorisation for roots or completed-square form for the turning point. An approximate sketch should show the correct opening direction and symmetry. Practising both visual and algebraic representations helps students understand that these features describe one function, not separate unrelated techniques.
Frequently asked questions
Is a quadratic turning point always on the x-axis?
No. It lies on the x-axis only when its y-coordinate is zero.
Does completing the square always help find a turning point?
Yes for a quadratic, though coefficients other than 1 may require additional algebra.
More GCSE learning resources
Explore these existing guides when a prerequisite skill or another aspect of your GCSE preparation needs attention. Use the links selectively to find the explanation that matches your current problem.
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