Fill a table of values, plot the points, join with a smooth curve, then read roots, the turning point and the line of symmetry. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.
The graph of y=ax2+bx+c is a U-shaped parabola if a>0, and an n-shape if a<0. It cuts the y-axis at (0,c). The x-intercepts are the roots. The turning point is on the vertical line of symmetry x=−2ab, and you find y by substituting that x.
To draw one, make a table from x=−3 to 3 or as asked, plot, and join with a smooth curve — not a ruler through the bottom. Read solutions of x2−4x=5 as intersections of the curve with y=5, or rearrange to =0 and read the axis crossings.
A table of values, plot, join with a smooth U (or n). Roots are the x-intercepts. The line of symmetry is halfway between the roots. The turning point sits on that line.
Objectives
Find a y-value on y = x^2 + c.
Read the roots of y = x^2 − a.
Read the y-intercept, and the crossings of a factorised curve.
Key points
Substitute negative and positive values carefully into y=ax2+bx+c to complete a table.
Plot the points and draw a smooth U-shaped (or n-shaped) curve — never straight lines or a flat bottom.
Read the roots (where the curve crosses the x-axis) as the solutions of y=0.
Read the turning point and the equation of the line of symmetry.
Worked examples
Example 1
On the curve y=x^2-4, work out the value of y when x=3.
2 marks
Substitute x = 3 into the formula: y = 3^2 - 4.
Calculate: 3^2 = 9, then 9 - 4 = 5.
The resulting y-value is 5.
Answer: 5
Common slips
Slip: Square 3 and stop without subtracting 4. The constant -4 must be subtracted from x^2.
Slip: Daisy stopped at 9. Subtract 4 from 9 to get 5.
Slip: Conclude y = 9. Evaluating 9 - 4 gives 5.
Example 2
Find the roots of the curve y=x^2-4.
2 marks
Set y = 0 to find x-intercepts: x^2 - 4 = 0.
Solve x^2 = 4 by taking square roots: x = -2 or 2.
The roots of the curve are -2 or 2.
Answer: -2 or 2
Common slips
Slip: Read -4 directly as the root. The roots occur where y = 0, so x^2 = 4.
Slip: Eliot wrote -4. Taking square root of 4 gives ±2.
Slip: Conclude -4. The curve crosses the x-axis at -2 and 2.
Example 3
Find the x-values where the curve y=(x-2)(x-3) crosses the x-axis.
3 marks
Set y = 0 to find axis crossings: (x-2)(x-3) = 0.
Solve each bracket: x - 2 = 0 gives 2, and x - 3 = 0 gives 3.
The curve crosses at 2 or 3.
Answer: 2 or 3
Common slips
Slip: Add 2 and 3 together. The curve crosses the x-axis where each factor is zero.
Slip: Flora added 2 + 3 = 5. The crossings are the two separate values 2 or 3.
Slip: Conclude 5. The crossing points are x = 2 and x = 3.
Questions and answers
GCSE Maths drawing quadratic graphs questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
On the curve y=x^2-4, work out the value of y when x=2.
Show answer
Answer: 0
Substitute x = 2: 2^2 - 4.
Calculate: 4 - 4 = 0.
M1 — Substitute x = 2: 2^2 - 4.
A1 — Calculate: 4 - 4 = 0.
Question 2 Easy2 marks
On the curve y=x^2+1, work out the value of y when x=3.
Show answer
Answer: 10
Substitute x = 3: 3^2 + 1.
Calculate: 9 + 1 = 10.
M1 — Substitute x = 3: 3^2 + 1.
A1 — Calculate: 9 + 1 = 10.
Question 3 Medium2 marks
Find the roots of the curve y=x^2-9.
Show answer
Answer: -3 or 3
Set y = 0: x^2 = 9.
Take square roots: x = -3 or 3.
M1 — Set y = 0: x^2 = 9.
A1 — Take square roots: x = -3 or 3.
Question 4 Medium2 marks
Find the roots of the curve y=x^2-16.
Show answer
Answer: -4 or 4
Set y = 0: x^2 = 16.
Find x-intercepts: x = -4 or 4.
M1 — Set y = 0: x^2 = 16.
A1 — Find x-intercepts: x = -4 or 4.
Question 5 Hard2 marks
Find the y-intercept of the curve y=x^2-4.
Show answer
Answer: -4
Substitute x = 0: y = 0^2 - 4.
Intercept is -4.
M1 — Substitute x = 0: y = 0^2 - 4.
A1 — Intercept is -4.
Question 6 Hard2 marks
Find the y-intercept of the curve y=x^2+3.
Show answer
Answer: 3
Substitute x = 0: y = 0^2 + 3.
Intercept is 3.
M1 — Substitute x = 0: y = 0^2 + 3.
A1 — Intercept is 3.
Question 7 Exam-style2 marks
On the curve y=x^2-9, work out the value of y when x=5.
Show answer
Answer: 16
Square the x-value: 5^2 = 25.
Subtract 9: 25 - 9 = 16.
M1 — 5^2 - 9
A1 — 16
Common mistakes
Most lost marks on this topic are one slip.
Stopping at x squared and ignoring the constant
Add the constant, including a negative one
Giving the constant as a root
The roots are the square roots of that constant
Adding the two bracket numbers instead of listing them
The curve crosses at each of those numbers
Spot the slip
On the curve y=x^2-4, work out y when x=3. Ada squares 3 and stops at 9. What is y?
5
9
-4
Show the slip
5 3 squared is 9, and 9 - 4 = 5.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Check the symmetry of your y-values before plotting — if the pattern does not mirror, recheck the odd one out.
Use a sharp pencil and draw the curve in one movement; a wobbly or straight-edged curve can lose the accuracy mark.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.