Which type of graph is made by \(y = x^2\)?
Quadratic graphs usually curve.
\(y = x^2\) is a quadratic graph. Its shape is a U-shaped curve called a parabola.
Graphs of Quadratic Functions
Quadratic graphs form curved shapes known as parabolas. Understanding their shape and key features helps link algebra with graphical interpretation.
Overview
A quadratic graph is a curved graph called a parabola.
It comes from an equation where the highest power of x is 2.
\( y = ax^2 + bx + c \)
Unlike linear graphs, quadratic graphs are not straight lines. They curve upward or downward and often have a highest or lowest point.
What you should understand after this topic
- Recognise what a quadratic graph looks like
- Understand how the value of a affects the shape
- Understand what the turning point represents
- Identify symmetry in a quadratic graph
- Understand how roots relate to where the graph crosses the x-axis
Key Definitions
Quadratic Function
A function with highest power \(x^2\).
Parabola
The curved shape made by a quadratic graph.
Turning Point
The highest or lowest point on the graph.
Axis of Symmetry
The vertical line that cuts the parabola into two matching halves.
Root / Solution
A point where the graph crosses the x-axis.
y-intercept
The point where the graph crosses the y-axis.
Minimum Point
The lowest point on a parabola that opens upward.
Maximum Point
The highest point on a parabola that opens downward.
Key Rules
If \(a > 0\)
The parabola opens upward.
If \(a < 0\)
The parabola opens downward.
Bigger \(|a|\)
The parabola becomes narrower or steeper.
Smaller \(|a|\)
The parabola becomes wider.
How to Solve
Step 1: Recognise a quadratic graph
A quadratic graph is a curved graph called a parabola. It is always symmetrical and has one turning point. These graphs come from quadratic equations.
\( y = ax^2 + bx + c \)
Exam tip: Quadratic graphs are always smooth curves, not straight lines.
Step 2: Understand the shape (value of \(a\))
Positive \(a\)
Graph opens upwards (minimum point).
Negative \(a\)
Graph opens downwards (maximum point).
Large \(|a|\)
Graph is narrow and steep.
Small \(|a|\)
Graph is wide.
Step 3: Find the intercepts
Intercepts help you sketch the graph quickly.
Exam thinking: Always find intercepts first when sketching.
y-intercept
Set \(x = 0\). The value of \(c\) gives the point.
x-intercepts (roots)
Solve \( ax^2 + bx + c = 0 \). These are where the graph crosses the x-axis. See solving quadratics.
Step 4: Use symmetry
Quadratic graphs are symmetrical about a vertical line called the axis of symmetry.
The turning point lies on the axis of symmetry.
Points on one side mirror the other side.
Exam tip: You only need to calculate half the points.
Step 5: Identify the turning point
The turning point is the highest or lowest point of the graph.
Exam thinking: The turning point is usually halfway between the roots.
Key idea
Upward graph → minimum point.
Downward graph → maximum point.
Step 6: Sketching efficiently
Use key features instead of a full table. This builds on factorising expressions.
Sketch \( y = x^2 - 4 \)
Exam tip: This method is faster than using a full table.
- Find y-intercept: \( (0,-4) \).
- Solve \( x^2 - 4 = 0 \) → roots \( x = -2, 2 \).
- Plot the intercepts.
- Use symmetry to guide the curve.
- Draw a smooth parabola.
Step 7: Number of roots
Two roots
Graph crosses the x-axis twice.
One root
Graph touches the x-axis once.
No roots
Graph does not meet the x-axis.
Step 8: Link to solving quadratics
Solving a quadratic gives the x-intercepts of the graph.
\( x^2 - 5x + 6 = 0 \)
The solutions are where the graph crosses the x-axis.
See solving quadratics for methods.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on plotting, interpreting, and analysing quadratic graphs.
Edexcel
Complete the table of values for \( y = x^2 \) when \( x = -2, -1, 0, 1, 2 \).
Edexcel
On a set of axes, plot the graph of \( y = x^2 - 4 \).
Edexcel
Write down the coordinates of the turning point of \( y = x^2 - 6x + 5 \).
Edexcel
Find the values of \( x \) where the graph of \( y = x^2 - 5x + 6 \) crosses the x-axis.
Edexcel
Write down the equation of the line of symmetry of \( y = x^2 - 8x + 7 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on interpreting quadratic graphs and solving equations graphically.
AQA
Sketch the graph of \( y = x^2 + 2x - 3 \).
AQA
Use the graph of \( y = x^2 - 3x - 4 \) to solve \( x^2 - 3x - 4 = 0 \).
AQA
State the coordinates of the turning point of \( y = (x - 2)^2 + 1 \).
AQA
The graph of \( y = x^2 \) is transformed to \( y = x^2 + 5 \). Describe the transformation.
AQA
A student says that the graph of \( y = x^2 + 4 \) crosses the x-axis.
Tick one box. True ☐ False ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, transformations, and interpreting quadratic graphs.
OCR
The graph of \( y = x^2 \) is translated to \( y = (x - 3)^2 \). Describe this transformation.
OCR
Find the coordinates of the turning point of \( y = -x^2 + 4x - 1 \).
OCR
Determine the y-intercept of \( y = 2x^2 - 3x + 1 \).
OCR
Find the roots of \( 2x^2 - 7x + 3 = 0 \) from its graph.
OCR
The graphs of \( y = x^2 \) and \( y = x^2 - 4 \) are drawn on the same axes. State the coordinates of their points of intersection.
Exam Checklist
Step 1
Check whether the graph should open upward or downward.
Step 2
Find key points such as intercepts and the turning point if possible.
Step 3
Use symmetry to help plot matching points.
Step 4
Draw a smooth parabola, not straight line segments.
Most common exam mistakes
Shape mistake
Forgetting that a quadratic graph is curved, not straight.
Symmetry mistake
Not using the matching shape on both sides of the turning point.
Root mistake
Confusing roots with the y-intercept.
Direction mistake
Using the wrong sign of \( a \) and drawing the parabola the wrong way.
Common Mistakes
These are common mistakes students make when working with graphs of quadratic functions in GCSE Maths.
Drawing straight lines instead of curves
Incorrect
A student joins points with straight lines.
Correct
Quadratic graphs are smooth curves (parabolas), not straight lines. Always draw a smooth curve through the points.
Forgetting symmetry
Incorrect
A student draws one side of the graph differently from the other.
Correct
Quadratic graphs are symmetrical about a vertical line through the turning point. Both sides should mirror each other.
Mixing up x-intercepts and y-intercepts
Incorrect
A student labels intercepts incorrectly.
Correct
The y-intercept is where the graph crosses the y-axis (x = 0). The x-intercepts are where the graph crosses the x-axis (y = 0).
Assuming every quadratic has two roots
Incorrect
A student expects the graph to cross the x-axis twice.
Correct
A quadratic can have two roots, one root (touching the axis), or no real roots depending on the equation.
Missing the turning point
Incorrect
A student sketches the graph without identifying its highest or lowest point.
Correct
The turning point (vertex) is a key feature of the graph. Always identify and plot it when sketching.
Try It Yourself
Practise plotting and analysing quadratic graphs.
Foundation
Foundation Practice
Recognise quadratic graphs, complete tables and identify simple features.
Complete the value: when \(x = 3\), find \(y\) for \(y = x^2\).
Square the value of x.
Substitute \(x = 3\): \(y = 3^2 = 9\).
For \(y = x^2\), what is \(y\) when \(x = -4\)?
A negative number squared becomes positive.
\((-4)^2 = 16\), so \(y = 16\).
For \(y = x^2 + 1\), find \(y\) when \(x = 2\).
Square first, then add 1.
\(y = 2^2 + 1 = 4 + 1 = 5\).
Which coordinate lies on the graph \(y = x^2\)?
Substitute x = 3.
When \(x = 3\), \(y = 3^2 = 9\), so \((3, 9)\) is on the graph.
For \(y = x^2 - 2\), find \(y\) when \(x = -3\).
Square \(-3\) first.
\((-3)^2 = 9\). Then \(9 - 2 = 7\).
The graph of \(y = x^2\) has its lowest point at which coordinate?
Think about the smallest value of \(x^2\).
The smallest value of \(x^2\) is 0, when \(x = 0\). So the lowest point is \((0, 0)\).
For \(y = x^2 + 3\), what is the y-intercept?
The y-intercept is when \(x = 0\).
When \(x = 0\), \(y = 0^2 + 3 = 3\). So the y-intercept is 3.
A student says \((-2)^2 = -4\). What mistake did they make?
\((-2)^2\) means \((-2) \times (-2)\).
\((-2)^2 = (-2) \times (-2) = 4\), not \(-4\).
For \(y = x^2 - x\), find \(y\) when \(x = 3\).
Substitute 3 into both parts of the expression.
\(y = 3^2 - 3 = 9 - 3 = 6\).
Higher
Higher Practice
Analyse turning points, roots, intercepts and quadratic graph features.
For \(y = x^2 - 4\), where does the graph cross the y-axis?
Set \(x = 0\) to find the y-intercept.
When \(x = 0\), \(y = 0^2 - 4 = -4\). So the y-intercept is \((0, -4)\).
Find the roots of \(y = x^2 - 9\). Give your answer as two x-values separated by a comma.
Roots happen when \(y = 0\).
Set \(x^2 - 9 = 0\). Then \(x^2 = 9\), so \(x = -3\) or \(x = 3\).
Which graph has a turning point at \((0, 5)\)?
\(y = x^2 + c\) moves the graph up by c.
\(y = x^2 + 5\) is \(y = x^2\) shifted up 5 units, so the turning point is \((0, 5)\).
For \(y = (x - 2)^2\), what is the turning point?
In \(y = (x - a)^2\), the turning point is \((a, 0)\).
\(y = (x - 2)^2\) has turning point \((2, 0)\).
For \(y = -x^2 + 4\), what is the shape of the graph?
A negative coefficient of \(x^2\) flips the parabola.
Because the coefficient of \(x^2\) is negative, the parabola opens downwards.
For \(y = x^2 - 4x + 3\), find the y-intercept.
Substitute \(x = 0\).
When \(x = 0\), \(y = 0^2 - 4(0) + 3 = 3\).
Find the roots of \(y = x^2 - 5x + 6\).
Factorise the quadratic.
\(x^2 - 5x + 6 = (x - 2)(x - 3)\). So the roots are \(x = 2\) and \(x = 3\).
For \(y = x^2 - 6x + 8\), find the roots. Give your answer as two x-values separated by a comma.
Factorise the quadratic.
\(x^2 - 6x + 8 = (x - 2)(x - 4)\). So the roots are \(x = 2\) and \(x = 4\).
A student says the y-intercept of \(y = x^2 + 3x - 7\) is 3. What mistake did they make?
The y-intercept is the value when \(x = 0\).
When \(x = 0\), the \(x^2\) and \(3x\) terms become 0, so \(y = -7\). The y-intercept is \(-7\), not 3.
For \(y = (x + 3)^2 - 2\), what is the turning point?
In \(y = (x + a)^2 + b\), the x-coordinate of the turning point is \(-a\).
\(y = (x + 3)^2 - 2\) has turning point \((-3, -2)\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What shape is a quadratic graph?
A parabola.
What are roots of a graph?
Points where the graph crosses the x-axis.
What is the turning point?
The highest or lowest point of the curve.