Quadratic graphs.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Quadratic graphs

Graphs

Lesson 6 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Graphs

Lesson 6 of 8

Answer each one, then check.

1

Work out (−3)².

2

Work out −3².

3

Substitute x = −1 into x² − 4x + 3.

4

Factorise x² − 4x + 3.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Graphs

Lesson 6 of 8

1

Work out (−3)².

9

2

Work out −3².

−9

3

Substitute x = −1 into x² − 4x + 3.

8

4

Factorise x² − 4x + 3.

(x − 1)(x − 3)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Graphs

Lesson 6 of 8

1

Complete a table of values for a quadratic and plot its graph.

2

Recognise the shape of a quadratic graph.

3

Find the roots, turning point, line of symmetry and y-intercept.

4

Use a quadratic graph to solve equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Plotting a Quadratic

Graphs

Lesson 6 of 8

A quadratic has an x² 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 x² term is positive, the curve is U-shaped; if it is negative, it is upside down (∩).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Table of Values for y = x² − 4x + 3

Graphs

Lesson 6 of 8

x

x²

−4x

+3

y

−1

1

4

3

8

0

0

0

3

3

1

1

−4

3

0

2

4

−8

3

−1

3

9

−12

3

0

4

16

−16

3

3

5

25

−20

3

8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Features of a Parabola

Graphs

Lesson 6 of 8

Roots, y-intercept, turning point and line of symmetry.

The roots are where the curve crosses the x-axis: they solve x² − 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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving from the Graph

Graphs

Lesson 6 of 8

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

Equal to 0

The solutions of x² − 4x + 3 = 0 are where the curve crosses the x-axis: x = 1 and x = 3.

Equal to a number

For x² − 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: x² − 4x + 3 = −2 has none.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reading Solutions

Graphs

Lesson 6 of 8

Use the graph of y = x² − 4x + 3 to solve (a) x² − 4x + 3 = 0 (b) x² − 4x + 3 = 8.

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Graphs

Lesson 6 of 8

Quadratic

An expression whose highest power of x is x².

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphs

Lesson 6 of 8

YOUR TASK

Spot the Mistakes

10 minutes

A student made this table for y = x² + 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.

WHAT A GOOD ANSWER SHOWS

x = −2 should give 4 − 4 − 1 = −1 (they worked out −2² 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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Graphs

Lesson 6 of 8

Complete a table of values, including negative x.

Plot a smooth quadratic curve.

Recognise U-shaped and ∩-shaped quadratics.

Find the roots from a graph.

Find the turning point and line of symmetry.

Find the y-intercept.

Solve ax² + bx + c = k using a graph.

Give estimates to 1 decimal place.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

A quadratic graph is a parabola: U-shaped for +x², ∩-shaped for −x².

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 = x² − 2x − 3 is drawn. Use it to find estimates for the solutions of x² − 2x − 3 = 2. (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.

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