211r. Recognising Quadratic Graphs

Learning Intentions

• Identify quadratic functions from equations, tables and graphs.
• Describe the shape and symmetry of parabolas.
• Locate key features such as intercepts and turning points.

Pre-requisite Summary

• Know that a quadratic function contains a squared variable term such as .
• Know that linear functions have constant first differences, while quadratic functions often have constant second differences.
• Know that the graph of a quadratic function is called a parabola.
• Know that parabolas can open upward or downward and are symmetrical.
• Know that intercepts occur where a graph crosses an axis.
• Know that the turning point is where a parabola changes direction.

Worked Examples

Worked Example 1

Decide whether each equation represents a quadratic function.

a)

b)

c)

Worked Example 2

Use the table to decide whether the relationship is quadratic.

a) Find the first differences.

b) Find the second differences.

c) Decide whether the relationship is quadratic.

Worked Example 3

Describe the parabola represented by .

a) State whether the parabola opens upward or downward.

b) Describe the symmetry of the graph.

c) State the turning point.

Worked Example 4

Describe the parabola represented by .

a) State whether the parabola opens upward or downward.

b) State whether the turning point is a maximum or minimum.

c) Describe the symmetry of the graph.

Worked Example 5

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Worked Example 6

A parabola has turning point and opens upward.

a) State the axis of symmetry.

b) Explain how the graph is symmetrical.

c) State whether the turning point is a maximum or minimum.

Worked Example 7

Use the graph of a parabola to identify key features.

The graph crosses the -axis at and , crosses the -axis at , and has turning point .

a) State the intercepts.

b) State the turning point.

c) State the axis of symmetry.

Problems

Problem 1

Decide whether each equation represents a quadratic function.

a)

b)

c)

Problem 2

Use the table to decide whether the relationship is quadratic.

a) Find the first differences.

b) Find the second differences.

c) Decide whether the relationship is quadratic.

Problem 3

Describe the parabola represented by .

a) State whether the parabola opens upward or downward.

b) Describe the symmetry of the graph.

c) State the turning point.

Problem 4

Describe the parabola represented by .

a) State whether the parabola opens upward or downward.

b) State whether the turning point is a maximum or minimum.

c) Describe the symmetry of the graph.

Problem 5

For the graph of , identify:

a) the -intercepts

b) the -intercept

c) the turning point

Problem 6

A parabola has turning point and opens downward.

a) State the axis of symmetry.

b) Explain how the graph is symmetrical.

c) State whether the turning point is a maximum or minimum.

Problem 7

Use the graph of a parabola to identify key features.

The graph crosses the -axis at and , crosses the -axis at , and has turning point .

a) State the intercepts.

b) State the turning point.

c) State the axis of symmetry.

Exercises

Understanding and Fluency

Exercise 1

State whether each equation is linear or quadratic.

a)

b)

c)

d)

Exercise 2

Identify the quadratic equations.

a)

b)

c)

d)

Exercise 3

For each table, decide whether the relationship is quadratic.

a)

b)

Exercise 4

Find the first and second differences.

a)

b)

Exercise 5

Describe each parabola.

a)

b)

c)

d)

Exercise 6

State whether each turning point is a maximum or minimum.

a) A parabola opening upward with turning point

b) A parabola opening downward with turning point

c) A parabola opening upward with turning point

d) A parabola opening downward with turning point

Exercise 7

State the axis of symmetry for each parabola.

a) Turning point

b) Turning point

c) Turning point

d) Turning point

Exercise 8

Identify the intercepts for each description.

a) The graph crosses the -axis at and , and crosses the -axis at .

b) The graph crosses the -axis at and , and crosses the -axis at .

c) The graph touches the -axis at and crosses the -axis at .

Exercise 9

Identify the turning point and axis of symmetry.

a) A parabola has turning point .

b) A parabola has turning point .

c) A parabola has turning point .

Exercise 10

Describe the symmetry of each parabola.

a) A parabola with axis of symmetry

b) A parabola with axis of symmetry

c) A parabola with axis of symmetry

Reasoning

Exercise 11

Explain why a graph with a squared term is often a parabola.

Exercise 12

Explain why quadratic tables often have constant second differences.

Exercise 13

A student says that all parabolas open upward.

Explain why this is incorrect.

Exercise 14

A student says that the axis of symmetry is always the -axis.

Explain why this is incorrect.

Exercise 15

A parabola has turning point and axis of symmetry .

Explain how symmetry helps you sketch the graph.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) A quadratic graph can have at most two -intercepts.

b) Every parabola has a turning point.

c) A parabola can open downward.

Problem-solving

Exercise 17

A graph has the following features:

• opens upward
• turning point
• -intercepts at and

a) State the axis of symmetry.

b) State whether the turning point is a maximum or minimum.

c) Sketch the parabola using the information.

Exercise 18

The height of a ball is modelled by a quadratic graph.

The graph:

• starts at
• reaches a maximum at
• hits the ground at

a) State the -intercept.

b) State the turning point.

c) State the axis of symmetry.

d) Explain what each feature means in context.

Exercise 19

A business profit graph is a parabola with:

• -intercepts at and
• turning point

a) State the axis of symmetry.

b) Explain what the intercepts mean in context.

c) Explain what the turning point means in context.

Exercise 20

Create your own quadratic function investigation.

Your response must include:

• one quadratic equation
• a table of values
• a description of the parabola
• the intercepts
• the turning point
• the axis of symmetry
• a sketch or description of the graph shape

Potential Misunderstandings

• Students may identify any equation with variables as quadratic, even if there is no squared term.
• Students may confuse constant first differences with constant second differences.
• Students may think all curved graphs are quadratic.
• Students may think every parabola opens upward.
• Students may confuse the turning point with an intercept.
• Students may forget that parabolas are symmetrical about a vertical axis.
• Students may confuse -intercepts with the -intercept.
• Students may think the axis of symmetry is always the -axis.
• Students may incorrectly identify the turning point by choosing the highest or lowest visible point instead of the actual vertex.

Next: 212. Graphing Monic Quadratic