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Maths · Graphs

Quadratic graphs

Plotting \(y = ax^2 + bx + c\) from a table of values, reading off its roots, turning point, line of symmetry and \(y\)-intercept, and using the graph to solve quadratic equations.

5 key terms
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Last Lesson and Before

Answer each one, then check.

1

Work out \((-3)^2\).

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9
2

Work out \(-3^2\).

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\(-9\)
3

Substitute \(x = -1\) into \(x^2 - 4x + 3\).

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8
4

Factorise \(x^2 - 4x + 3\).

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\((x - 1)(x - 3)\)

Learning Objectives

1Complete a table of values for a quadratic and plot its graph.
2Recognise the shape of a quadratic graph.
3Find the roots, turning point, line of symmetry and \(y\)-intercept.
4Use a quadratic graph to solve equations.

Plotting a Quadratic

A quadratic has an \(x^2\) term and no higher power. Its graph is a smooth curve called a parabola.

Brackets for negatives
\((-2)^2 = 4\), so put negative values in brackets on your calculator.
Use the symmetry
The \(y\)-values repeat either side of the turning point - a quick check on your table.
Smooth curve
Join the points with a smooth curve, not straight lines, and don't flatten the bottom.
Shape
If the \(x^2\) term is positive, the curve is U-shaped; if it is negative, it is upside down (∩).

A Table of Values for y = x² − 4x + 3

\(-1\)
\(x^2\): 1. \(-4x\): 4. \(+3\): 3. y: 8
0
\(x^2\): 0. \(-4x\): 0. \(+3\): 3. y: 3
1
\(x^2\): 1. \(-4x\): \(-4\). \(+3\): 3. y: 0
2
\(x^2\): 4. \(-4x\): \(-8\). \(+3\): 3. y: \(-1\)
3
\(x^2\): 9. \(-4x\): \(-12\). \(+3\): 3. y: 0
4
\(x^2\): 16. \(-4x\): \(-16\). \(+3\): 3. y: 3
5
\(x^2\): 25. \(-4x\): \(-20\). \(+3\): 3. y: 8

The Features of a Parabola

The roots are where the curve crosses the \(x\)-axis: they solve \(x^2 - 4x + 3 = 0\). The turning point is the lowest point, halfway between the roots, on the line of symmetry \(x = 2\). The \(y\)-intercept is the number on its own: 3.

The U-shaped graph of y = x squared minus 4x plus 3, with its roots at 1 and 3, y-intercept 3, turning point at (2, −1) and line of symmetry x = 2 labelled.

Solving from the Graph

A graph gives the solutions of an equation as the \(x\)-coordinates where two things meet.

Equal to 0
The solutions of \(x^2 - 4x + 3 = 0\) are where the curve crosses the \(x\)-axis: \(x = 1\) and \(x = 3\).
Equal to a number
For \(x^2 - 4x + 3 = 3\), draw the line \(y = 3\): it meets the curve at \(x = 0\) and \(x = 4\).
Estimates
When the curve does not cross exactly on a grid line, give answers to 1 decimal place.
No solution
If the line misses the curve, there is no solution: \(x^2 - 4x + 3 = -2\) has none.

Reading Solutions

Use the graph of \(y = x^2 - 4x + 3\) to solve (a) \(x^2 - 4x + 3 = 0\) (b) \(x^2 - 4x + 3 = 8\).

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1 (a) Where the curve crosses the \(x\)-axis \(x = 1\) and \(x = 3\)
2 (b) Draw the line \(y = 8\) It meets the curve twice
3 Read the \(x\)-coordinates \(x = -1\) and \(x = 5\)
4 Check (b): \((-1)^2 - 4(-1) + 3 = 8\) Correct

Answer(a) \(x = 1\), \(x = 3\) (b) \(x = -1\), \(x = 5\)

Spot the Mistakes

A student made this table for \(y = x^2 + 2x - 1\): \(x = -3\) gives 2, \(x = -2\) gives \(-9\), \(x = -1\) gives \(-2\), \(x = 0\) gives \(-1\), \(x = 1\) gives 2. They plotted it and joined the points with straight lines. Find every mistake and describe the correct graph.

1. Check each value.

2. Look for symmetry.

3. Check how the points are joined.

A good answer shows: \(x = -2\) should give \(4 - 4 - 1 = -1\) (they worked out \(-2^2\) as \(-4\)). The rest are right. Correct values: 2, \(-1\), \(-2\), \(-1\), 2 - symmetrical about \(x = -1\), with minimum \((-1, -2)\). The points should be joined with a smooth curve.

Can I...?

1Complete a table of values, including negative \(x\).
2Plot a smooth quadratic curve.
3Recognise U-shaped and ∩-shaped quadratics.
4Find the roots from a graph.
5Find the turning point and line of symmetry.
6Find the \(y\)-intercept.
7Solve \(ax^2 + bx + c = k\) using a graph.
8Give estimates to 1 decimal place.

Summary & Exam Focus

A quadratic graph is a parabola: U-shaped for \(+x^2\), ∩-shaped for \(-x^2\).
Roots are where \(y = 0\); the \(y\)-intercept is the constant term.
The turning point lies on the line of symmetry, halfway between the roots.
To solve \(f(x) = k\), draw \(y = k\) and read the \(x\)-values.

Exam focus

The graph of \(y = x^2 - 2x - 3\) is drawn. Use it to find estimates for the solutions of \(x^2 - 2x - 3 = 2\). (2 marks) (2 marks)

When a question says "use the graph", draw the horizontal line on the graph and read off where it meets the curve. Calculated answers without the line drawn may not get the marks.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Quadratic: An expression whose highest power of \(x\) is \(x^2\).
Parabola: The U-shaped (or ∩-shaped) curve of a quadratic graph.
Roots: The \(x\)-values where a graph crosses the \(x\)-axis.
Turning point: The point where the curve changes direction: a minimum or maximum.
Line of symmetry: The vertical line through the turning point.

Practice questions

Have a go at each one before you open its answer.

Question 1 Non-calculator 2 marks

Complete the table of values for \(y = x^2 - 3x + 1\) for \(x = -1, 0, 1, 2, 3, 4\).
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Model answer

\(y = 5, 1, -1, -1, 1, 5\)

Mark scheme
- At least 4 correct values — B1
- All 6 correct — B1
Question 2 Non-calculator 5 marks

The graph of \(y = x^2 - 2x - 3\) is drawn for values of \(x\) from \(-2\) to 4. (a) Write down the roots of \(x^2 - 2x - 3 = 0\). (b) Write down the coordinates of the turning point. (c) Write down the equation of the line of symmetry. (d) Use the graph to find estimates for the solutions of \(x^2 - 2x - 3 = 2\).
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Model answer

(a) \(x = -1\) and \(x = 3\) (b) \((1, -4)\) (c) \(x = 1\) (d) Draw \(y = 2\): \(x \approx -1.4\) and \(x \approx 3.4\)

Mark scheme
- (a) \(-1\) and 3 — B1
- (b) \((1, -4)\) — B1
- (c) \(x = 1\) — B1
- (d) Line \(y = 2\) drawn, or reading at \(y = 2\) — M1
- (d) \(-1.4\) to \(-1.5\) and 3.4 to 3.5 — A1
Question 3 Non-calculator 2 marks

The graph of a quadratic crosses the \(x\)-axis at \(x = -2\) and \(x = 6\). Write down the equation of its line of symmetry, and explain how you know.
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Model answer

\(x = 2\). The line of symmetry is halfway between the roots: \(\frac{-2 + 6}{2} = 2\).

Mark scheme
- \(x = 2\) — B1
- Halfway between the roots — C1
Question 4 Non-calculator 2 marks

Sam says the graph of \(y = x^2 + 4\) crosses the \(x\)-axis twice. Is Sam right? Explain your answer.
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Model answer

No. \(x^2\) is never negative, so \(x^2 + 4\) is always at least 4. The graph never reaches \(y = 0\).

Mark scheme
- No — B1
- A reason, e.g. the minimum value is 4, or \(x^2 \ge 0\) — C1

Quick check

What shape is the graph of \(y = x^2 - 5\)?
1. AA U-shaped curve
2. BA straight line
3. CA ∩-shaped curve
4. DAn S-shaped curve
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A: A U-shaped curve

A positive \(x^2\) term gives a U-shaped parabola.
A quadratic graph has roots \(x = 1\) and \(x = 7\). What is the \(x\)-coordinate of its turning point?
1. A3
2. B6
3. C4
4. D8
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C: 4

The turning point is halfway between the roots: \(\frac{1 + 7}{2} = 4\).
What is the value of \(y = x^2 - 3x\) when \(x = -2\)?
1. A\(-10\)
2. B10
3. C\(-2\)
4. D2
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B: 10

\((-2)^2 - 3 \times (-2) = 4 + 6 = 10\).

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