Step-by-step worked examples and graded practice questions on quadratic graphs — plotting from a table of values, identifying key features, and using graphs to solve equations.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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A quadratic graph is the graph of an equation containing an x² term, such as y = x² − 4x + 3. Every quadratic graph is a smooth curve called a parabola, with four key features to identify:
The roots (x-intercepts) — where the curve crosses the x-axis
The y-intercept — where the curve crosses the y-axis
The turning point — the single minimum (or maximum) point of the curve
The line of symmetry — a vertical line through the turning point, which the curve mirrors either side of
If the coefficient of x² is positive, the graph is U-shaped with a minimum turning point. If it's negative, the graph is ∩-shaped with a maximum turning point.
The graph of y = x² − 4x + 3, showing the roots, y-intercept, turning point, and line of symmetry.
Plotting from a table of values
To plot a quadratic graph, substitute each x-value into the equation to build a table of values, then plot the points and join them with a smooth curve — never straight lines between the points.
Worked Example 1 — Completing a table of values
Complete a table of values for y = x² − 4x + 3, for x from −1 to 5, and use it to plot the graph.
1
Substitute each x-value: x = −1 gives y = 1 + 4 + 3 = 8. x = 0 gives y = 3. x = 1 gives y = 0.
2
Continuing: x = 2 gives y = −1. x = 3 gives y = 0. x = 4 gives y = 3. x = 5 gives y = 8.
3
Plot each point and join them with a smooth curve — notice the symmetry either side of x = 2
The table of values plotted and joined with a smooth curve.
Finding roots, intercepts and the turning point
Rather than plotting a full table every time, you can find each key feature directly from the equation:
Roots: factorise the expression and set it equal to zero
y-intercept: substitute x = 0
Turning point: its x-coordinate is exactly halfway between the two roots (by symmetry), or use completing the square
Worked Example 2 — Key features from the equation
Find the roots, y-intercept and turning point of y = x² − 4x + 3.
1
Roots: factorise x² − 4x + 3 = (x − 1)(x − 3) = 0, so x = 1 and x = 3
2
y-intercept: substitute x = 0, giving y = 3, so the y-intercept is (0, 3)
3
Turning point: the x-coordinate is halfway between the roots, (1 + 3) ÷ 2 = 2. Substitute x = 2: y = 4 − 8 + 3 = −1
AnswerRoots: x = 1, x = 3. y-intercept: (0, 3). Turning point (minimum): (2, −1)
Using graphs to solve equations
A quadratic graph can be used to solve equations, not just plot the curve. To solve x² − 4x + 3 = 0, read the x-values where the curve crosses the x-axis. To solve x² − 4x + 3 = k for some other number k, draw the horizontal line y = k and read the x-values where it crosses the curve.
Worked Example 3 — Solving using the roots
Use the graph of y = x² − 4x + 3 to solve x² − 4x + 3 = 0.
1
The solutions are the x-values where the curve crosses the x-axis (where y = 0)
2
Reading the graph: the curve crosses at x = 1 and x = 3
Answerx = 1 or x = 3
Worked Example 4 — Solving using a horizontal line
Use the graph of y = x² − 4x + 3 to estimate the solutions to x² − 4x + 3 = 2, giving your answers to 1 decimal place.
1
Draw the horizontal line y = 2 on the same axes
2
Read the x-values where the line crosses the curve
Answerx ≈ 0.3 or x ≈ 3.7
Reading the solutions to x² − 4x + 3 = 2 from where y = 2 crosses the curve.
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1For y = x² − 2x, find the value of y when x = 3.Foundation
Q2For y = x² − 2x, find the value of y when x = −1.Foundation
Q3The graph of y = x² − 4x + 3 crosses the y-axis. State the y-intercept.Foundation
Q4A quadratic graph has roots at x = 2 and x = 6. State the x-coordinate of the line of symmetry.Foundation
Q5State whether the graph of y = −x² + 3x + 1 is U-shaped or ∩-shaped, and explain why.Foundation
Higher (Grade 5–7)
Q6Factorise y = x² − 5x + 6, and find its roots.Higher
Q7Find the turning point of y = x² − 6x + 8 using the symmetry of its roots.Higher
Q8Find the roots of y = x² − 9.Higher
Q9A quadratic graph has turning point (2, −5) and passes through (0, −1). State whether the graph is U-shaped or ∩-shaped, giving a reason.Higher
Q10Use the graph of y = x² − 4x + 3 (shown above) to solve x² − 4x + 3 = 0.Higher
Graph for Q7, showing the turning point found by symmetry:
The graph of y = x² − 6x + 8 for Question 7.
Higher — Hard (Grade 8–9)
Q11Find the coordinates of the turning point of y = x² − 4x + 3 by completing the square.Grade 8–9
Q12Using the graph of y = x² − 4x + 3, estimate the solutions to x² − 4x + 3 = 2, to 1 decimal place.Grade 8–9
Q13A quadratic graph y = x² + bx + c has roots at x = −2 and x = 5. Find the values of b and c.Grade 8–9
Q14A quadratic graph has a minimum point at (3, −2) and passes through (5, 6). By symmetry, find another point on the graph with the same y-value as (5, 6).Grade 8–9
Q15The graph of y = x² − 4x + 3 is translated so that its new turning point is at (2, 3). Write down the equation of the new graph, and state its new y-intercept.Grade 8–9
Answers
Foundation (Q1–Q5)
Q13(9 − 6)
Q23(1 + 2)
Q3(0, 3)
Q4x = 4((2+6) ÷ 2)
Q5∩-shaped, because the coefficient of x² is negative (−1)
Higher (Q6–Q10)
Q6(x − 2)(x − 3); roots x = 2, x = 3
Q7(3, −1)(roots x=2,4; midpoint x=3; y = 9−18+8 = −1)
Q8x = 3, x = −3(difference of two squares: (x−3)(x+3) = 0)
Q9U-shaped — the y-intercept (−1) is higher than the turning point (−5), so the turning point must be a minimum
Q13b = −3, c = −10(sum of roots = 3 = −b; product of roots = −10 = c)
Q14(1, 6)((5,6) is 2 units right of the line of symmetry x=3, so the matching point is 2 units left, at x=1)
Q15y = (x − 2)² + 3; new y-intercept (0, 7)(shift the turning point from −1 to 3, i.e. +4; expanding gives x² − 4x + 7)
Common mistakes
These are the errors Alamin sees most frequently with quadratic graphs at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Joining plotted points with straight lines
A quadratic graph is a smooth curve, not a series of straight segments. Ruler-drawn "V" shapes between points lose marks even if every point is correctly plotted.
Common Mistake 2
Sign errors when substituting negative x-values
For y = x² − 4x + 3 at x = −1: (−1)² = +1 (not −1), and −4 × (−1) = +4. Losing track of these signs is the most common table-of-values error.
Common Mistake 3
Confusing the turning point with the y-intercept
The y-intercept is where the curve crosses the y-axis (x = 0); the turning point is the minimum or maximum of the whole curve, and is rarely at x = 0. They're only the same point by coincidence.
Common Mistake 4
Forgetting the turning point's x-coordinate is the midpoint of the roots
This shortcut only works when the graph actually has two real roots. If the discriminant is negative, use completing the square instead.
Common Mistake 5
Reading estimated solutions as exact values
When a question says "use the graph to estimate", your answer should be read from the graph to an appropriate degree of accuracy (often 1 decimal place) — not solved exactly using the quadratic formula, even if you could.
Common Mistake 6
Assuming every quadratic graph is U-shaped
If the coefficient of x² is negative, the graph is ∩-shaped with a maximum turning point, not a minimum. Always check the sign of the x² term first.
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