D3.3 Model applications by reasoning with functions

Ontario Grade 12 Mathematics

Characteristics of Functions

Connect a situation to a function, then use its values and graph to interpret what happens.

A function describes how one quantity depends on another. In an application, the function is more than a formula: its inputs and outputs represent quantities in a situation. Reasoning with a function means connecting its equation, table, and graph to the context. This lesson uses values and graph features, including numerical rates of change, to interpret models. It does not use derivative rules or integration.

What you will learn

Identify the input, output, and reasonable domain in an application.
Choose and use a function model that matches a situation.
Interpret function values, graph features, and numerical rates of change in context.
Check whether a model’s predictions make sense for the situation.

1. Review the function basics

A variable is a letter that stands for a quantity that can change. The input is the quantity you choose or observe. The output is the quantity determined by that input. A function assigns exactly one output to each allowed input.

For example, if

t

is time in hours and

d(t)

is distance in kilometres, then

d(2)

means the distance after two hours. The notation names the output for a particular input; it does not mean

d

multiplied by

2

.

The domain is the set of inputs that make sense for the function and the situation. A formula may accept many numerical inputs, but a context may restrict them. If

t

represents time since an event, negative values may not be meaningful. Always state a reasonable domain when applying a model.

y=f(x)

Name what each variable measures and include units.
Use the situation, not just the formula, to choose the domain.
An output must have units that fit the quantity being modelled.

2. Match a function to the situation

A mathematical model is a function used to represent a real situation. Start by asking how the output changes as the input changes. A steady increase by the same amount over equal input intervals may be modelled by a linear function. A changing rate of increase or decrease may suggest a nonlinear function, such as a polynomial, rational, trigonometric, logarithmic, or composed function. The context and available information determine whether a model is suitable.

A composed function is built by using one function’s output as another function’s input. For instance, a rule might first convert time into an angle and then use a trigonometric function to represent a repeating height. The model must still match the situation and its domain.

A table lists selected input-output pairs. A graph shows the pattern between those values. An equation gives a rule for calculating outputs. These representations can support each other: use a table to inspect values, a graph to see overall behaviour, and an equation to calculate or explain specific outputs.

For any model, distinguish a prediction from a known measurement. A model may fit observed values well but become unrealistic outside the measured range. For example, a model for a machine’s output may be useful only for operating times tested in a study.

Choose a function type based on the pattern and context.
Use equation, table, and graph as connected representations.
Limit predictions to inputs for which the model is meaningful.

3. Read changes and features in context

A function value answers a question about one input. For example,

f(4)

gives the modelled output when the input is

4

. A zero is an input where the output is zero. An intercept is where a graph crosses an axis. These features can answer contextual questions, such as when a measured quantity reaches zero or what its starting value is.

A maximum is the greatest output over the part of the model being considered; a minimum is the least. A graph may show where output rises, falls, or repeats. Interpret each feature using the units and the domain. A maximum outside the relevant domain does not answer a question about the situation.

A rate of change compares the change in output with the change in input. The average rate of change between two inputs is calculated from the two corresponding function values. It describes the average change over that interval, not necessarily the change at every point. Its units are output units per input unit.

A graph can help estimate a rate by comparing two points. A table can give a more direct numerical calculation. If the rate is positive, the output increased on average over the interval; if negative, it decreased on average. If the rate is zero, the two endpoint outputs are equal, though the function may still change between them.

\frac{f(b)-f(a)}{b-a}

Translate zeros, intercepts, maxima, and minima into words about the situation.
Use two points to calculate an average rate over an interval.
State the interval and units when reporting a rate.

4. Build, test, and communicate a model

A clear modelling process begins with the question. Identify the input and output, their units, and the reasonable domain. Then choose or build a function using the information provided. Check its outputs against known data or the shape expected from the situation.

Next, use the model to answer the question. A calculated value should be described in context, including units. If the question asks how the output changes, compare values over the stated interval or inspect the graph. Do not claim that a model is exact merely because it gives a numerical answer.

Finally, check reasonableness. Does the result fit the domain? Does its size make sense? Does the graph’s behaviour match what is known about the situation? If not, the model, input, or interpretation may need revision. A useful answer explains both the calculation and what the result means.

State assumptions and the meaningful input range.
Check model outputs against context and available information.
Report conclusions in complete sentences with units.

Selected values from the tank model

Time $t$ (min)	Level $h(t)$ (cm)
0	10
2	18
4	18
6	10

Worked example

Using a quadratic model for a water level

A tank’s water level, in centimetres, is modelled during the first six minutes by

h(t)=-t^2+6t+10

, where

t

is time in minutes and

0\leq t\leq6

. Find the starting level, the level after four minutes, and the average rate of change from minute two to minute four. Interpret the results.

Identify the quantities
The input $t$ measures time in minutes. The output $h(t)$ measures water level in centimetres. The stated domain limits conclusions to the first six minutes.
Evaluate the model
Substitute $t=0$ to find the starting level. Substitute $t=4$ to find the level after four minutes. The function rule gives one output for each of these allowed inputs.
$h(0)=10, h(4)=18$
Calculate the average change
Compare the level at minute four with the level at minute two, then divide by the two-minute interval. The result is an average rate, so it summarizes the change across the interval rather than at each instant.
$h(2)=18, \frac{h(4)-h(2)}{4-2}=0$
Interpret the result
The model gives a starting level of $10$ centimetres and a level of $18$ centimetres after four minutes. The average rate from minute two to minute four is $0$ centimetres per minute because the endpoint levels are equal. This does not mean the level stayed constant throughout those two minutes; the model’s values can rise and then fall.

Answer: The starting level is

10

centimetres, and the modelled level at four minutes is

18

centimetres. The average rate from minute two to minute four is

0

centimetres per minute.

Check: The model gives

h(2)=18

and

h(4)=18

, so the net change over that interval is zero. Both inputs are within the stated domain.

Common mistakes and how to avoid them

Reporting an answer without units or without naming what the input represents.

Correction: State the quantity and units, such as a level of

18

centimetres at four minutes.

Assuming equal endpoint values mean the function stayed constant between them.

Correction: An average rate of zero means only that the net change over the interval is zero. Inspect intermediate values or the graph to see what happened between the endpoints.

Using a model outside the context’s stated domain.

Correction: Check the input against the meaningful range before interpreting a prediction.

Treating an average rate as the rate at every point in the interval.

Correction: Describe it as the average change across the stated interval. A two-point calculation does not give the rate at every input.

Lesson summary

A function connects an input to one output.
A model’s domain comes from the situation, not only from its equation.
Equations, tables, and graphs show different views of the same function.
Interpret values and graph features using context and units.
A numerical average rate compares output change with input change over an interval.
Check that predictions are sensible and within the model’s useful range.

Check your understanding

Question 1

For

p(t)=3t+5

, where

t

is time in hours and

p(t)

is distance in kilometres, what does

p(2)

represent?

The modelled distance after two hours
The time when the distance is two kilometres
The increase in distance for every two hours
The starting distance multiplied by two

Show answer and explanation

The modelled distance after two hours

The input is time, so

p(2)

is the output distance when time is two hours.

Question 2

A model gives

q(1)=7

and

q(5)=19

. What is the average rate of change from input

1

to input

5

?

$3$ output units per input unit
$4$ output units per input unit
$12$ output units per input unit
$26$ output units per input unit

Show answer and explanation

3

output units per input unit

The output increases by

12

over an input increase of

4

, giving an average rate of

3

output units per input unit.

Question 3

A function is used to model the number of visitors to an event, where the input is hours after the doors open. Which input is outside the natural domain for this context?

$-2$ hours
$0$ hours
$1$ hour
$3$ hours

Show answer and explanation

-2

hours

Negative time would be before the doors open, so it is outside the stated context of hours after opening.

Key terms

Function: A rule that assigns exactly one output to each allowed input.
Input: The quantity supplied to a function.
Output: The quantity determined by a function for a given input.
Domain: The set of inputs that are allowed or meaningful.
Model: A mathematical representation of a situation, often using a function.
Average rate of change: The change in output divided by the change in input over an interval.

Continue through MHF4U

View the complete MHF4U Ontario Grade 12 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D3.3. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

D3.3 · Model applications by reasoning with functions

Ontario Grade 12 Mathematics

Connect a situation to a function, then use its values and graph to interpret what happens.

What you will learn

1. Review the function basics

2. Match a function to the situation

3. Read changes and features in context

4. Build, test, and communicate a model

Selected values from the tank model

Using a quadratic model for a water level

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review