205. Recognising Quadratic Change

Learning Intentions

• Identify situations where change is not constant.
• Compare linear and quadratic patterns in tables and graphs.
• Describe key features of quadratic models in context.

Pre-requisite Summary

• Know that a constant rate of change means the output changes by the same amount each time the input increases by .
• Know that linear patterns have constant first differences.
• Know that quadratic patterns often have changing first differences but constant second differences.
• Know that linear graphs are straight lines, while quadratic graphs are curved parabolas.
• Know that a quadratic model can represent situations involving area, falling objects, curved paths or repeated growth in differences.
• Know that key features of a quadratic graph include its turning point, direction of opening and intercepts.

Worked Examples

Worked Example 1

Decide whether each situation has constant or non-constant change.

a) A taxi costs $ plus $ per kilometre.

b) The area of a square changes as its side length increases.

c) A ball is thrown upward and its height changes over time.

Worked Example 2

For each table, decide whether the pattern is linear, quadratic or neither.

a)

b)

Worked Example 3

Compare the first differences and second differences for the table.

Worked Example 4

Describe whether each graph would be linear or quadratic.

a) A straight line that rises from left to right.

b) A curved graph shaped like a U.

c) A curved graph that opens downward.

Worked Example 5

Compare the rules and using a table for .

Worked Example 6

A rectangular garden has width metres and length metres.

a) Write a rule for the area .

b) Decide whether the model is linear or quadratic.

c) Explain why the change in area is not constant.

Worked Example 7

A ball is thrown into the air. Its height is modelled by , where is height in metres and is time in seconds.

a) State whether the parabola opens upward or downward.

b) Interpret the vertical intercept.

c) Describe what the turning point means in context.

Worked Example 8

A square has side length centimetres and area .

a) Complete a table for .

b) Describe the first differences.

c) Explain why the area model is quadratic.

Problems

Problem 1

Decide whether each situation has constant or non-constant change.

a) A worker earns $ per hour.

b) The area of a circle changes as its radius increases.

c) A car travels at a constant speed of km/h.

Problem 2

For each table, decide whether the pattern is linear, quadratic or neither.

a)

b)

Problem 3

Compare the first differences and second differences for the table.

Problem 4

Describe whether each graph would be linear or quadratic.

a) A curved graph shaped like an upside-down U.

b) A straight line that falls from left to right.

c) A curved graph that opens upward.

Problem 5

Compare the rules and using a table for .

Problem 6

A rectangular garden has width metres and length metres.

a) Write a rule for the area .

b) Decide whether the model is linear or quadratic.

c) Explain why the change in area is not constant.

Problem 7

A ball is thrown into the air. Its height is modelled by , where is height in metres and is time in seconds.

a) State whether the parabola opens upward or downward.

b) Interpret the vertical intercept.

c) Describe what the turning point means in context.

Problem 8

A square has side length metres and area .

a) Complete a table for .

b) Describe the first differences.

c) Explain why the area model is quadratic.

Exercises

Understanding and Fluency

Exercise 1

Decide whether each situation has constant or non-constant change.

a) A phone plan costs $ plus $ per month.

b) The area of a square changes as the side length increases.

c) A candle burns down by cm each hour.

d) The height of a ball changes as it is thrown upward and falls back down.

Exercise 2

For each table, decide whether the pattern is linear, quadratic or neither.

a)

b)

c)

Exercise 3

Find the first differences for each table.

a)

b)

Exercise 4

Find the first differences and second differences for each table.

a)

b)

Exercise 5

Complete each table.

a)

---	---:	---:	---:	---:

b)

Exercise 6

For each rule, decide whether the model is linear or quadratic.

a)

b)

c)

d)

Exercise 7

Describe whether each graph would be linear or quadratic.

a) The graph is a straight line.

b) The graph is a U-shaped curve.

c) The graph is an upside-down U-shaped curve.

d) The graph has a constant gradient.

Exercise 8

For each quadratic rule, state whether the graph opens upward or downward.

a)

b)

c)

d)

Exercise 9

For each context, state whether a linear or quadratic model is more appropriate.

a) A person saves the same amount of money each week.

b) The area of a square changes as its side length changes.

c) A taxi fare increases by a fixed amount per kilometre.

d) The height of a thrown ball changes over time.

Exercise 10

Match each model to the most likely context.

a)

b)

c)

Contexts:

• the area of a square
• the height of a thrown object
• a cost with a starting fee and constant charge

Reasoning

Exercise 11

Explain why a table with constant first differences represents a linear pattern.

Exercise 12

Explain why a table with changing first differences may represent non-constant change.

Exercise 13

A student says this table is linear because the -values increase each time.

Explain the mistake.

Exercise 14

A student says is linear because it has a like .

Explain why this is incorrect.

Exercise 15

Explain why the graph of opens downward and what this means in the context of a thrown ball.

Exercise 16

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

a) A linear model has constant first differences.

b) A quadratic model always has a straight-line graph.

c) The turning point of a quadratic model can represent a maximum or minimum value.

Problem-solving

Exercise 17

A square has side length centimetres.

a) Write a rule for the area .

b) Complete a table for .

c) Explain why the area does not increase by a constant amount.

Exercise 18

A rectangular garden has width metres and length metres.

a) Write a rule for the area .

b) Expand the rule.

c) Explain why the model is quadratic.

Exercise 19

A ball’s height is modelled by , where is height in metres and is time in seconds.

a) Interpret the vertical intercept.

b) State whether the graph opens upward or downward.

c) Explain what the maximum point would represent in context.

Exercise 20

Create your own example of a quadratic model in context.

Your response must include:

• a real-world situation where change is not constant
• a quadratic rule
• a table of at least four values
• a sentence describing the first differences
• a sentence interpreting one key feature of the model

Potential Misunderstandings

• Students may think every increasing pattern is linear.
• Students may only check whether the -values increase, instead of checking whether the first differences are constant.
• Students may think non-constant change means there is no pattern.
• Students may confuse linear and quadratic graphs because both can increase over part of their domain.
• Students may not recognise that quadratic tables often have constant second differences.
• Students may think a curved graph cannot be represented by an algebraic rule.
• Students may interpret the vertical intercept as the rate of change instead of the starting value.
• Students may not recognise that the turning point of a quadratic graph can represent a maximum or minimum in context.
• Students may forget that a negative coefficient makes a parabola open downward.

Next: 206. Choosing Models