A2.2 Evaluate linear and quadratic functions in context

Ontario Grade 11 Mathematics

Quadratic Functions

Use an input, a function rule, and the meaning of the situation to find and explain an output.

A function rule can describe how one quantity depends on another. Evaluating a function means using a chosen input to find the matching output. In context, the calculation is only part of the answer: you also need to say what the result represents and use suitable units. This lesson focuses on evaluating linear and quadratic functions that are already given.

What you will learn

Identify the input and output in a real-world situation.
Evaluate a linear or quadratic function by substituting an input into its rule.
State an answer with units and explain what it means in context.
Check whether an input and its resulting output make sense for the situation.

1. Prerequisite bridge: inputs, outputs, and substitution

An input is the value you put into a function. An output is the value the function gives back. In a situation, the input might be time, distance, or the number of items. The output might be cost, height, or total distance.

A variable is a letter used to stand for a number. In the rule

C(n)=12+4n

, the letter

n

is the input and

C(n)

is the output. The notation

C(5)

means the value of the output when the input is

5

.

To evaluate a function, replace each occurrence of its input variable with the chosen value. Use brackets around a substituted negative number. Follow the usual order of operations: calculate powers before multiplication and addition. Then interpret the result using the situation.

C(5)=12+4(5)

Input: the value chosen for the independent quantity.
Output: the value produced by the function.
Evaluate: substitute an input and calculate the matching output.

2. Read a function in context

A linear function has a constant rate of change: equal increases in the input cause equal increases or decreases in the output. A quadratic function includes a squared input. Its rate of change is not constant, so its graph often curves. These descriptions help you recognize the kind of rule, but the evaluation process is the same for both: substitute the input and calculate.

The variables and units matter. If

t

is time in seconds and

h(t)

is height in metres, then an input such as

t=2

means two seconds. The output

h(2)

is a height in metres. Do not treat the function notation as multiplication:

h(2)

names the output for input

2

.

A function may only make sense for certain inputs. For example, a rule for the cost of buying items may use whole-number inputs, not fractions of an item. Time in a physical event may also be limited to the period when the event occurs. Check the situation before accepting a calculated answer.

f(x)=mx+b,\quad g(x)=ax^2+bx+c

A linear rule changes by a steady amount for equal input steps.
A quadratic rule has a squared input and may describe a curved relationship.
Use the context to identify meaningful inputs, outputs, and units.

3. Use equations, tables, and graphs

An equation gives a rule for calculating outputs. A table lists selected inputs with their matching outputs. A graph shows those input-output pairs as points. Each representation can help you evaluate a function.

For a table, find the row with the required input and read its output. On a graph, locate the input on the horizontal axis and read the corresponding vertical value. A graph reading may be approximate if the point falls between grid marks. An equation usually gives an exact result when its values are exact.

For either kind of function, keep the input and output roles clear. Substitute into the rule first; then attach units and describe the answer. A number without its meaning is an incomplete contextual answer.

x\longmapsto f(x)

Equation: calculate using the rule.
Table: match the input with its listed output.
Graph: read the output that corresponds to the input.

4. Application: calculate, interpret, and check

Suppose a question asks for a cost after a certain number of items, or a height at a specified time. First identify what the input represents and what value is requested. Next substitute the input into the function rule. Calculate carefully, including any squared term. Finally, state the result in a sentence with units.

A useful check is to compare the answer with the situation. A negative cost or a height far beyond the stated time period may signal a calculation error or an input outside the useful range of the model. A plausible answer is not automatically correct, but checking context can reveal problems.

When evaluating a quadratic rule, square only the input that is inside the squared term. For example, in

h(t)=-5t^2+20t+1

, the square applies to

t

before multiplication by

-5

. Brackets make the substitution clear, especially for negative inputs.

h(-2)=-5(-2)^2+20(-2)+1

Read the requested input and output from the context.
Substitute the input with brackets where needed.
Calculate, include units, and decide whether the result fits the situation.

Matching inputs and outputs in the example

Function	Input	Output	Meaning
$C(n)=8+3n$	$n=4$ add-ons	$C(4)=20$	Estimated cost: CAD 20
$H(t)=-5t^2+20t+1$	$t=2$ seconds	$H(2)=21$	Estimated height: 21 metres

Worked example

Compare two quantities at a chosen input

A recreation centre estimates the cost of a drop-in visit with the linear function

C(n)=8+3n

, where

n

is the number of activity add-ons and

C(n)

is the cost in CAD. A ball’s height is modelled by

H(t)=-5t^2+20t+1

, where

t

is time in seconds and

H(t)

is height in metres. Evaluate both functions when

n=4

and

t=2

, and explain the results.

Identify the requested outputs
The cost input is four add-ons, so evaluate $C(4)$ . The height input is two seconds, so evaluate $H(2)$ . The units come from the descriptions of the functions.
$C(4),\quad H(2)$
Evaluate the linear function
Replace each $n$ in the cost rule with $4$ . Multiplication is done before addition, so four add-ons contribute CAD 12 to the starting cost.
$C(4)=8+3(4)=20$
Evaluate the quadratic function
Replace each $t$ with $2$ . Square the input first, then multiply by $-5$ and add the remaining terms.
$H(2)=-5(2)^2+20(2)+1=21$
Interpret and check
The first output is a cost, so it is measured in CAD. The second is a height, so it is measured in metres. Both inputs match the quantities described in the models.
$C(4)=20\text{ CAD},\quad H(2)=21\text{ m}$

Answer: Four add-ons have an estimated cost of CAD 20. At two seconds, the model gives the ball’s height as 21 metres.

Check: For the cost, the fixed CAD 8 plus four add-ons at CAD 3 each gives CAD 20. For the height,

-5(4)+40+1=21

.

Common mistakes and how to avoid them

Treating

C(4)

as

C

multiplied by

4

.

Correction:

C(4)

means the function’s output when its input is

4

. Substitute

4

into the rule.

Forgetting the square when evaluating a quadratic function.

Correction: Replace the input first, then apply the exponent to that substituted value before multiplying.

Giving a number without units or meaning.

Correction: Name the quantity and include its units, such as a cost of CAD 20 or a height of 21 metres.

Using an input that does not fit the situation.

Correction: Check what the input represents and whether that value is allowed or meaningful in the described context.

Lesson summary

Evaluating a function means substituting an input into its rule and calculating the output.
Linear and quadratic functions use the same evaluation process, even though their rules have different forms.
Use context to explain the output, add units, and check whether the result is sensible.

Check your understanding

Question 1

A delivery estimate is

D(k)=6+2.5k

, where

k

is the distance in kilometres and

D(k)

is the cost in CAD. What is the estimated cost for

4

kilometres?

CAD 10
CAD 16
CAD 17
CAD 26

Show answer and explanation

CAD 17

Substitute

4

for

k

:

D(4)=6+2.5(4)=6+10=16

. Correction: this calculation gives CAD 16, so the correct option is CAD 16.

Question 2

A height model is

p(t)=-2t^2+12t+3

, where

t

is time in seconds. What height does it give at

t=2

?

11 metres
19 metres
23 metres
35 metres

Show answer and explanation

19 metres

Substitute

2

and calculate the square first:

p(2)=-2(2)^2+12(2)+3=-8+24+3=19

. The model gives 19 metres.

Key terms

Input: The value supplied to a function, often representing the quantity you choose.
Output: The value a function gives for a chosen input.
Evaluate: Substitute an input into a function rule and calculate its output.
Linear function: A function whose output changes by a steady amount when the input increases by equal steps.
Quadratic function: A function rule that includes a squared input, such as $x^2$ .

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A2.2 · Evaluate linear and quadratic functions in context

Ontario Grade 11 Mathematics

Use an input, a function rule, and the meaning of the situation to find and explain an output.

What you will learn

1. Prerequisite bridge: inputs, outputs, and substitution

2. Read a function in context

3. Use equations, tables, and graphs

4. Application: calculate, interpret, and check

Matching inputs and outputs in the example

Compare two quantities at a chosen input

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through MCF3M

About this lesson