Modelling with Polynomial Functions

Polynomial functions are not only abstract mathematical expressions. They can also be used as models to describe relationships between quantities in real situations.

A polynomial model can help us:

• Describe how one quantity changes with another.
• Estimate unknown values.
• Make predictions.
• Identify maximum or minimum values.
• Solve practical problems.
• Interpret patterns in data.

Polynomial models appear in areas such as physics, engineering, business, architecture, agriculture, and data analysis.

Learning Targets

• I can identify situations modelled by polynomial functions.
• I can interpret polynomial models in context.
• I can estimate values using polynomial graphs.
• I can evaluate polynomial functions for given inputs.
• I can solve real-world problems involving polynomial models.

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What Is a Mathematical Model?

A mathematical model is a mathematical representation of a real situation.

Models may use:

• Equations
• Functions
• Graphs
• Tables
• Diagrams
• Data

For example, suppose the height of an object above the ground is modelled by:

h(t) = −5t² + 20t + 2

where:

t = time in seconds

and:

h = height in metres

The polynomial connects two real quantities: time and height.

The equation allows us to investigate how the object's height changes with time.

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What Is a Polynomial Model?

A polynomial model uses a polynomial function to represent a relationship.

For example:

P(x) = 2x³ − 15x² + 24x + 10

could represent a quantity P that depends on x.

What P and x actually mean depends on the situation.

For example, x might represent:

• Time
• Distance
• Number of products
• Length
• Temperature
• Population
• Speed

P(x) might represent:

• Height
• Area
• Volume
• Cost
• Revenue
• Profit
• Energy

The mathematical expression becomes meaningful when its variables are connected to real quantities.

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Recognizing Polynomial Models

Consider:

A(x) = x² + 6x

This is a polynomial because the powers of x are non-negative whole numbers.

Now consider:

V(x) = x³ + 5x² + 6x

This is also a polynomial.

But:

f(x) = 1/x

is not a polynomial because:

1/x = x⁻¹

Similarly:

f(x) = √x

is not a polynomial because:

√x = x^(1/2)

A polynomial model uses powers such as:

x⁰, x¹, x², x³, x⁴, ...

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Why Use Polynomial Models?

Polynomial functions are useful because they can represent many different patterns.

A linear polynomial can model a constant rate of change.

A quadratic polynomial can model a relationship that rises and then falls, or falls and then rises.

Higher-degree polynomials can model relationships containing several changes in direction.

The appropriate polynomial depends on the pattern being modelled.

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Variables Have Meaning

When interpreting a polynomial model, always determine what the variables represent.

Suppose:

C(x) = 0.02x² + 5x + 100

models the cost of producing x items.

Then:

x

represents the number of items produced.

And:

C(x)

represents the production cost.

If cost is measured in dollars, then:

C(50)

means:

the cost of producing 50 items

This interpretation is just as important as the calculation.

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Evaluating a Polynomial Model

To evaluate a polynomial means to substitute a value for the variable and calculate the result.

Suppose:

P(x) = 2x² + 3x + 5

Find:

P(4)

Substitute:

x = 4

So:

P(4) = 2(4)² + 3(4) + 5

Calculate the exponent first:

P(4) = 2(16) + 12 + 5

Then:

P(4) = 32 + 12 + 5

Therefore:

P(4) = 49

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Interpreting an Evaluated Value

The number alone is not enough when working with a model.

Suppose:

C(x) = 2x² + 3x + 5

represents cost in dollars for producing x units.

If:

C(4) = 49

then the interpretation is:

The model predicts that producing 4 units will cost $49.

Always return to the context of the problem.

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Worked Example — Area

Suppose a rectangular garden has:

Length = x + 5

and:

Width = x + 2

The area is:

A = length × width

Therefore:

A(x) = (x + 5)(x + 2)

Expand:

A(x) = x² + 7x + 10

This is a polynomial model.

If:

x = 3 m

then:

A(3) = 3² + 7(3) + 10

A(3) = 9 + 21 + 10

A(3) = 40

Therefore:

Area = 40 m²

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Modelling Volume

Polynomial models commonly appear when calculating volume.

Suppose a rectangular box has dimensions:

Length = x

Width = x + 2

Height = x + 4

Then:

V(x) = x(x + 2)(x + 4)

Expand:

V(x) = x(x² + 6x + 8)

Therefore:

V(x) = x³ + 6x² + 8x

This is a cubic polynomial model.

If x is measured in centimetres, V(x) is measured in:

cm³

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Worked Example — Evaluating Volume

Using:

V(x) = x³ + 6x² + 8x

find the volume when:

x = 5 cm

Substitute:

V(5) = 5³ + 6(5²) + 8(5)

V(5) = 125 + 6(25) + 40

V(5) = 125 + 150 + 40

V(5) = 315

Therefore:

V = 315 cm³

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Polynomial Models in Motion

Quadratic functions are particularly important when modelling motion under approximately constant acceleration.

For example:

h(t) = −5t² + 20t + 2

might model the height of an object.

The graph shows how height changes with time.

The graph can provide information about:

• Initial height
• Maximum height
• Time of maximum height
• When the object reaches the ground

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Worked Example — Height

Suppose:

h(t) = −5t² + 20t + 2

Find the height after:

t = 2 s

Substitute:

h(2) = −5(2²) + 20(2) + 2

h(2) = −5(4) + 40 + 2

h(2) = −20 + 42

Therefore:

h(2) = 22 m

Interpretation:

According to the model, the object is 22 m above the ground after 2 seconds.

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Interpreting the Y-Intercept

The y-intercept often has an important meaning in a model.

For:

h(t) = −5t² + 20t + 2

set:

t = 0

Then:

h(0) = 2

So the y-intercept is:

(0, 2)

In context, this means:

The object begins 2 m above the ground.

The y-intercept often represents an initial value.

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Interpreting X-Intercepts

The x-intercepts occur when:

f(x) = 0

Their meaning depends on the situation.

For a height model, an x-intercept might represent the time when an object reaches ground level.

For a profit model, an x-intercept might represent a break-even point, where profit equals zero.

For an area model, a zero might represent a dimension for which the area becomes zero.

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Interpreting Turning Points

Turning points can also have important real-world meanings.

A turning point may represent:

• Maximum height
• Minimum cost
• Maximum profit
• Minimum distance
• Maximum area
• A change in a trend

For example, the highest point on a projectile's height-time graph represents its:

maximum height

A minimum point on a cost graph might represent the:

lowest possible cost

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Maximum and Minimum Values

Many real-world questions ask us to find a maximum or minimum.

Consider a quadratic graph that opens downward.

Its highest point is a:

maximum

A quadratic that opens upward has a:

minimum

The vertex of a quadratic can therefore have an important interpretation in a model.

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Worked Example — Maximum Height

Consider:

h(t) = −5t² + 20t + 2

The graph opens downward because the coefficient of t² is negative.

Therefore, the vertex represents the maximum height.

For a quadratic:

at² + bt + c

the horizontal coordinate of the vertex is:

t = −b/(2a)

Here:

a = −5

and:

b = 20

So:

t = −20 / (2 × −5)

t = 2

We already calculated:

h(2) = 22

Therefore:

Maximum height = 22 m

and it occurs after:

2 s

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Estimating Values from a Graph

Sometimes we do not need an exact calculation.

Instead, we can estimate a value directly from a graph.

Suppose a graph shows height against time.

To estimate the height at:

t = 3 s

we can:

• Locate 3 on the horizontal axis.
• Move vertically until reaching the curve.
• Move horizontally toward the vertical axis.
• Read the approximate height.

Because we are reading from a graph, the result may be approximate rather than exact.

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Exact Values and Estimates

An equation can often provide an exact calculated value.

A graph often provides an estimated value.

For example, an equation might give:

f(3) = 17.42

A graph might allow us to estimate:

f(3) ≈ 17

The symbol:

≈

means:

approximately equal to

Graphs are particularly useful when an approximate answer is sufficient or when we want to see the overall pattern.

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Interpolation

Interpolation means estimating a value within the range of known data.

Suppose measurements have been taken between:

0°C and 50°C

Using a model to estimate a value at:

35°C

is interpolation.

Because 35°C lies inside the measured range, the estimate may be reasonably supported by the existing data, depending on the quality of the model.

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Extrapolation

Extrapolation means predicting a value outside the range of known data.

Suppose the data only covers:

0°C to 50°C

but we use the model to predict what happens at:

100°C

That is extrapolation.

Extrapolation is generally less reliable because the real relationship may behave differently outside the measured range.

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Models Have Valid Domains

A polynomial equation is mathematically defined for every real x-value, but a real-world model may not make sense for every x-value.

Suppose:

C = 2n² + 5n + 100

models the cost of producing n products.

Mathematically, we could calculate:

C(−10)

But producing:

−10 products

has no practical meaning.

The realistic domain might therefore be:

n ≥ 0

and n may need to be a whole number.

Context determines which inputs are meaningful.

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Worked Example — Choosing a Meaningful Domain

Suppose:

P(t)

models the population of bacteria during an experiment lasting 12 hours.

A reasonable domain might be:

0 ≤ t ≤ 12

Using the polynomial to predict the population after:

500 hours

may be mathematically possible but scientifically unreliable.

The model was designed for a much smaller interval.

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Polynomial Models and Data

Sometimes we collect experimental or observational data first and then find a polynomial that approximately matches the pattern.

This process is called polynomial regression or curve fitting.

The polynomial does not necessarily pass through every data point.

Instead, it may provide a curve that represents the overall trend.

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Models Are Approximations

A mathematical model is a simplified representation of reality.

For example, a projectile model might ignore:

• Air resistance
• Wind
• Shape of the object
• Changes in gravitational field strength

A business model may depend on assumptions about:

• Customer behaviour
• Production costs
• Prices
• Demand

Therefore, predictions should always be interpreted within the assumptions and limitations of the model.

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Worked Example — Profit Model

Suppose a company's profit is modelled by:

P(x) = −2x² + 120x − 1000

where:

x = number of units sold

and:

P(x) = profit in dollars

Find the predicted profit when:

x = 40

Substitute:

P(40) = −2(40²) + 120(40) − 1000

P(40) = −2(1600) + 4800 − 1000

P(40) = −3200 + 4800 − 1000

P(40) = 600

Therefore:

The model predicts a profit of $600 when 40 units are sold.

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Positive and Negative Values in Context

The sign of a polynomial's output can have an important meaning.

For a profit model:

P(x) > 0

means:

profit

while:

P(x) < 0

means:

loss

and:

P(x) = 0

means:

break-even

The mathematical result must always be translated back into the real situation.

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Solving a Real-World Polynomial Problem

Suppose the area of a rectangular enclosure is:

A(x) = x(20 − x)

Expand:

A(x) = 20x − x²

This is a quadratic polynomial.

Suppose we want an area of:

96 m²

Set:

A(x) = 96

So:

20x − x² = 96

Rearrange:

x² − 20x + 96 = 0

Factor:

(x − 8)(x − 12) = 0

Therefore:

x = 8

or:

x = 12

Both values can be checked against the original situation.

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Using Graphs to Solve Real Problems

A graph can often answer practical questions quickly.

Suppose:

P(x)

represents profit.

From its graph we might identify:

• Where profit is positive.
• Where the business experiences a loss.
• Break-even points.
• Maximum profit.
• The number of units associated with maximum profit.

This is one reason graphical representations are so useful in modelling.

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Evaluating Whether an Answer Makes Sense

After solving a polynomial model, ask:

• Does the answer fit the context?
• Is the value within the realistic domain?
• Are the units correct?
• Is a negative answer meaningful?
• Does the result agree with the graph?
• Is the answer exact or estimated?
• Are there multiple mathematical solutions?
• Are all mathematical solutions physically possible?

This final interpretation step is essential.

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Rejecting an Impossible Solution

Suppose solving a model gives:

x = 5

and:

x = −8

If x represents the length of a box, then:

x = −8 cm

is not physically meaningful.

Therefore, the appropriate solution is:

x = 5 cm

A mathematical equation may produce several solutions, but the context determines which solutions are valid.

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Units Matter

Always include appropriate units when interpreting a polynomial model.

Examples include:

Length → m

Area → m²

Volume → m³

Time → s

Speed → m/s

Money → $

Temperature → °C

A result such as:

V(4) = 120

is incomplete if the context requires:

V(4) = 120 cm³

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Choosing Between an Equation and a Graph

Equations and graphs provide different advantages.

An equation is useful for:

• Exact calculations
• Algebraic solving
• Substitution
• Finding precise values

A graph is useful for:

• Seeing trends
• Estimating values
• Identifying maxima and minima
• Finding approximate intercepts
• Comparing different parts of a model

Using both representations often gives the strongest understanding of a model.

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A General Modelling Process

When solving a problem with a polynomial model:

• Identify what each variable represents.
• Identify the units.
• Determine the meaningful domain.
• Write or identify the polynomial model.
• Substitute values when necessary.
• Solve the equation if an unknown input is required.
• Use the graph to interpret behaviour.
• Check whether the solution is realistic.
• State the answer in context with appropriate units.

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Common Mistakes

Mistake: Giving Only a Number

If:

h(3) = 12

do not simply write:

12

Explain what it means:

The object is 12 m above the ground after 3 seconds.

Mistake: Forgetting Units

An area of:

50

should usually be reported with square units, such as:

50 m²

Mistake: Using Every Mathematical Solution

A polynomial equation may produce solutions that are impossible in the real situation.

Negative lengths, negative numbers of objects, or times outside the model's domain may need to be rejected.

Mistake: Treating an Estimate as Exact

A value read from a graph should usually be described as approximately:

≈

Mistake: Extrapolating Too Far

A polynomial may fit existing data well but produce unrealistic predictions far outside the measured range.

Mistake: Ignoring the Meaning of Intercepts

Intercepts often represent important real events, such as:

• Initial conditions
• Ground level
• Break-even points
• Zero area
• Zero profit

Mistake: Ignoring the Domain

The mathematical function may allow many inputs that are meaningless in the real-world situation.

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Key Terms

Mathematical model – a mathematical representation of a real situation.

Polynomial model – a model that uses a polynomial function.

Evaluate – substitute an input into a function and calculate the output.

Interpret – explain what a mathematical result means in its real-world context.

Domain – the possible or meaningful input values.

Range – the possible or meaningful output values.

Interpolation – estimating within the range of known data.

Extrapolation – predicting outside the range of known data.

Polynomial regression – fitting a polynomial curve to data.

Maximum – the greatest value of a function over a specified domain.

Minimum – the smallest value of a function over a specified domain.

Break-even point – a point where profit equals zero.

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Check Your Understanding

1. A model is given by:

P(x) = 3x² + 2x + 5

Calculate:

P(4)

2. The height of an object is modelled by:

h(t) = −5t² + 25t + 2

Calculate its height when:

t = 2 s

3. In Question 2, what does:

h(0)

represent?

4. Explain what an x-intercept could represent on a height-time graph.

5. A company's profit is modelled by:

P(x) = −x² + 50x − 300

What does:

P(20)

represent?

6. Calculate:

P(20)

for the model in Question 5.

7. A box has dimensions:

x, x + 2, and x + 5

Write a polynomial model for its volume.

8. Calculate the volume in Question 7 when:

x = 3 cm

9. Explain the difference between interpolation and extrapolation.

10. Why might extrapolation using a polynomial model be unreliable?

11. A model predicts that a box should have a length of:

−4 cm

Explain why this solution should be rejected.

12. A graph gives:

f(5) ≈ 27

Explain what the symbol:

≈

tells you about the value.

13. Why is it important to identify the domain of a real-world polynomial model?

14. Give an example of a real situation that could be represented by a quadratic polynomial.

15. Explain why checking whether a calculated answer makes sense is an important final step in mathematical modelling.

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Key Takeaways

• Polynomial functions can model many real-world relationships.
• Variables in a model represent real quantities and should be interpreted carefully.
• Evaluating a polynomial means substituting an input and calculating the output.
• Every calculated value should be interpreted in its real-world context.
• Graphs can be used to estimate values and understand overall behaviour.
• X-intercepts and y-intercepts often have important practical meanings.
• Turning points may represent maximum or minimum values.
• Polynomial models can represent area, volume, motion, cost, revenue, profit, and many other quantities.
• Interpolation estimates values within known data.
• Extrapolation predicts outside known data and is generally less reliable.
• Real-world models often have a restricted domain.
• Mathematical solutions are not automatically realistic solutions.
• Units should always be included when appropriate.
• Polynomial models are simplifications of reality and have limitations.
• Equations are useful for precise calculations, while graphs are useful for seeing patterns and estimating values.

Did You Know?

A polynomial can fit a set of measured data extremely well and still make terrible predictions outside the data range. Higher-degree polynomials can change direction dramatically beyond the observed points. This is why scientists, engineers, and data analysts consider both how well a model fits the existing data and whether there is a good reason to expect the model to remain valid when making predictions.