Quadratic graphs GCSE Questions and Worked Answers
A quadratic graph plots the output of a rule containing x² as its highest power. Build coordinate pairs, draw a smooth curve, then use its height, intercepts and turning point to answer questions.
Foundation & Higher6 worked examples10 original questions
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Start with the meaning
What you need to know about quadratic graphs
A square with side length 1 has area 1, side 2 gives area 4, and side 3 gives area 9. The output changes by more each time: it does not follow a straight-line rule. Plotting input and output as pairs makes that changing relationship visible.
See the idea first
From a rule to a smooth curve
Write x for the input and y for the output. In y = x² − 4, square x and then subtract 4. Here x is any real number, including negatives; unlike the square-length example, it is not restricted to lengths. A point (x, y) tells you how far across and how high to plot. The resulting U-shaped curve is a parabola.
Each dot satisfies y = x² − 4. The smooth curve includes the values between the dots. It meets the x-axis at −2 and 2.
Substitution gives the plotted points
x
−3
−2
−1
0
1
2
3
y
5
0
−3
−4
−3
0
5
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Plot a quadratic
What the problem asks: Draw y = x² − 4 for −3 ≤ x ≤ 3.
How to solve it: Use the table to plot each pair, then join them with a smooth curve. The curve turns at (0, −4); do not join the points with straight segments.
Read solutions
What the problem asks: Use the graph to solve x² − 4 = 0.
How to solve it: The required output is zero, so read where the curve meets the horizontal x-axis: x = −2 or 2. These solutions are called roots.
Find a turning point
What the problem asks: Find the lowest point of y = (x − 2)² − 3.
How to solve it: A square cannot be negative. It is smallest, zero, when x = 2, giving y = −3. The turning point is (2, −3). Algebraic turning-point work is Higher.
A reliable routine
Plot from a table of values
Use a table when asked to draw a quadratic over an interval. Substitution produces points that make the equation true; a smooth curve represents the intermediate inputs too.
Choose x-values across the requested interval, including values either side of the turn.
Substitute carefully, putting negative inputs in brackets.
Plot each (x, y) on labelled axes with a consistent scale.
Draw a smooth curve and check its symmetry. Add extra table values near the turn if needed.
Check: An x-intercept has y = 0; the y-intercept has x = 0. A quadratic may have two, one or no real roots.
Fully worked
Quadratic graphs GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Complete a table
3 marks
Question
Find y for x = −2, −1, 0, 1, 2 in y=x2+1.
At −2, y=(−2)2+1=5. At −1, y=1+1=2. At 0, y=1. At 1 and 2 the squares equal those at −1 and −2, so y = 2 and 5 again. In order: 5, 2, 1, 2, 5. Tip: matching values either side of zero check symmetry.
Example 2
Read the key points
3 marks
Question
Use the graph of y=x2−4 to state its roots, y-intercept and minimum point.
Each dot satisfies y = x² − 4. The smooth curve includes the values between the dots. It meets the x-axis at −2 and 2.
The graph meets y = 0 at x = −2 and 2: these are the roots. At x = 0, y = −4, so the y-intercept is (0, −4). This is also the minimum, where the curve stops falling and starts rising. They coincide here because the symmetry line is x = 0; the y-intercept is not generally the minimum.
Example 3
Find roots from factors
3 marks
Question
Find the x-intercepts of y=x2−x−6.
At an x-intercept y = 0.
x2−x−6=0(x−3)(x+2)=0
A product is zero if a factor is zero: x = 3 or −2. The intercepts are (3, 0) and (−2, 0).
Example 4
A horizontal line
3 marks
Question
Use y=x2−4 to solve x2−4=2, giving x to one decimal place.
The line y = 2 meets the curve near x = −2.4 and 2.4. To check,
x2=6x=±6=±2.4494…
Therefore x ≈ −2.4 or 2.4 to one decimal place. A graph gives approximate readings.
Example 5
A turning point by completing the square
Higher only4 marks
Question
Find the minimum point of y=x2−6x+7.
y=(x−3)2−9+7y=(x−3)2−2
The square has least value zero at x = 3. Then y = −2, so the minimum is (3, −2) and the symmetry line is x = 3.
Example 6
A downward curve
3 marks
Question
Describe the turning point and roots of y=9−x2.
Since x2≥0, subtracting it makes y at most 9. At x = 0 the maximum is (0, 9). For roots:
9−x2=0x2=9x=−3 or 3
The curve opens downward.
10 original questions · total 20 marks
Quadratic graphs GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1
One coordinate
1 mark
For y=x2+3, find y when x = −4.
Show worked answer
y=(−4)2+3=16+3=19
The point is (−4, 19).
2
A table
3 marks
Find y for x = −2, −1, 0, 1, 2 in y=x2−1.
Show worked answer
Square each input, then subtract 1. At −2: (−2)2−1=3; at −1: (−1)2−1=0; at 0: 02−1=−1. Inputs 1 and 2 give the same outputs as −1 and −2. Outputs are 3, 0, −1, 0, 3 in that order.
3
Y-intercept
1 mark
Find the y-intercept of y=2x2+5x−7.
Show worked answer
Set x = 0: y=0+0−7=−7. The point is (0, −7).
4
Two roots
2 marks
Find the roots of y=x2−25.
Show worked answer
x2−25=0x2=25x=−5 or 5
Both signs square to 25.
5
Factorised roots
2 marks
Find the x-intercepts of y=(x+4)(x−1).
Show worked answer
Set each factor to zero: x = −4 or 1. The points are (−4, 0) and (1, 0).
6
Symmetry
2 marks
A parabola has roots −3 and 7. Find its vertical line of symmetry.
Show worked answer
The line is halfway between the roots:
x=2−3+7=2
Give a line equation, not just 2.
7
A minimum
Higher only2 marks
Find the minimum point of y=(x+1)2+4.
Show worked answer
The square is zero at x = −1 and never negative. Then y = 4. Minimum: (−1, 4).
8
No real roots
2 marks
Explain why y=x2+6 never meets the x-axis.
Show worked answer
Since x2≥0, y is always at least 6. It cannot be zero, so there are no real roots.
9
A repeated root
2 marks
How many distinct roots has y=(x−5)2?
Show worked answer
The square is zero only at x = 5. There is one distinct root; the curve touches the axis at (5, 0).
10
A line intersects a curve
Higher only3 marks
Find where y=x2 and y=x+6 intersect.
Show worked answer
The outputs match at an intersection:
x2=x+6x2−x−6=0(x−3)(x+2)=0
Thus x = 3 or −2. Using y = x² gives (3, 9) and (−2, 4).
Examiner-style feedback
Common quadratic graphs mistakes
A polygon instead of a parabola
A quadratic is smooth, not a chain of straight segments between the table points.
Only one square root
Both positive and negative inputs may produce the required square.
Confusing the axes
Roots are x-values at y = 0. The y-intercept is found at x = 0.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
Generate pairs by substitution.
Plot and draw smoothly.
Read axes before naming intercepts.
Use symmetry to check the curve.
Quick answers
Quadratic graphs FAQ
Does every quadratic cross the axis twice?
No: it can cross twice, touch once or stay entirely on one side.
What is a turning point?
The minimum or maximum of the parabola: its direction changes from falling to rising, or vice versa.
A11–A12 plotting and graphical interpretation across tiers; completing-square turning points and linear–quadratic intersections labelled Higher. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
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