Graphs of Polynomial Functions

A polynomial function is a function made from variables raised to whole-number powers, multiplied by constants, and combined using addition or subtraction.

Examples include:

f(x) = 3x + 2

f(x) = x² − 4

f(x) = x³ − 2x

f(x) = 2x⁴ − 3x² + 1

Polynomial graphs can take many different shapes. However, their overall shape is strongly connected to the degree of the polynomial and the sign of its leading coefficient.

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Learning Targets

• I can identify the general shape of polynomial graphs.
• I can relate the degree of a polynomial to its graph.
• I can predict the end behaviour of polynomial functions.
• I can determine whether a graph has even or odd degree characteristics.
• I can sketch basic polynomial graphs.

What Is a Polynomial?

A polynomial is an algebraic expression containing terms such as:

axⁿ

where:

• a is a constant called a coefficient
• x is the variable
• n is a non-negative whole number

For example:

f(x) = 2x³ − 5x² + 4x − 7

is a polynomial.

The exponents:

3, 2, 1, 0

are all non-negative whole numbers.

What Is Not a Polynomial?

Some expressions are not polynomial functions.

For example:

f(x) = 1/x

is not a polynomial because x is effectively raised to the power −1.

Similarly:

f(x) = √x

is not a polynomial because:

√x = x^(1/2)

Polynomial variable exponents must be:

0, 1, 2, 3, 4, ...

They cannot be negative or fractional.

The Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable after the polynomial has been simplified.

For example:

f(x) = 5x³ − 2x + 7

has degree:

3

because the highest power of x is x³.

Similarly:

f(x) = 2x⁵ + 4x² − 1

has degree:

5

The degree provides important information about the possible shape of the graph.

Common Polynomial Degrees

Polynomials are often classified by degree.

Degree 0 → Constant

Example:

f(x) = 4

Degree 1 → Linear

Example:

f(x) = 2x + 1

Degree 2 → Quadratic

Example:

f(x) = x² − 4

Degree 3 → Cubic

Example:

f(x) = x³ − 3x

Degree 4 → Quartic

Example:

f(x) = x⁴ − 5x² + 4

Degree 5 → Quintic

Example:

f(x) = x⁵ − 2x³ + x

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Linear Functions

A polynomial of degree 1 produces a straight-line graph.

For example:

f(x) = 2x + 1

The highest exponent is:

1

so this is a linear function.

The graph has a constant slope and no curves or turning points.

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Quadratic Functions

A polynomial of degree 2 is called a quadratic function.

The basic quadratic function is:

f(x) = x²

Its graph is called a parabola.

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For:

f(x) = x²

both ends of the graph rise upward.

If the leading coefficient is negative:

f(x) = −x²

both ends point downward.

Cubic Functions

A polynomial of degree 3 is called a cubic function.

The basic cubic function is:

f(x) = x³

Its graph has an S-like shape.

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For:

f(x) = x³

the left end falls and the right end rises.

The two ends travel in opposite directions.

Quartic Functions

A polynomial of degree 4 is called a quartic function.

The simplest example is:

f(x) = x⁴

Like x², both ends rise upward.

However, quartic functions can have more complicated shapes than quadratics.

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For example, a quartic graph may contain several bends or turning points while still having both ends pointing in the same direction.

Higher-Degree Polynomials

As the degree increases, polynomial graphs can become increasingly complex.

A fifth-degree polynomial may have several bends.

A sixth-degree polynomial may have even more.

However, polynomial graphs remain smooth and continuous.

They do not suddenly break, jump, or form sharp corners.

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The Leading Term

The leading term is the term containing the highest power of x.

Consider:

f(x) = 3x⁴ − 5x² + 2x − 7

The leading term is:

3x⁴

The degree is:

4

and the leading coefficient is:

3

The leading term has a particularly strong effect on what happens to the graph far to the left and far to the right.

The Leading Coefficient

The leading coefficient is the coefficient of the highest-degree term.

For:

f(x) = 5x³ − 2x + 1

the leading coefficient is:

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For:

f(x) = −2x⁴ + 3x² − 7

the leading coefficient is:

−2

The sign of the leading coefficient helps determine the direction of the ends of the graph.

What Is End Behaviour?

End behaviour describes what happens to a graph as x becomes extremely large in the positive or negative direction.

We examine:

x → ∞

and:

x → −∞

In simpler language, we ask:

What happens on the far right of the graph?

and:

What happens on the far left of the graph?

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Even-Degree End Behaviour

Polynomials with an even degree have both ends travelling in the same direction.

Examples of even degrees include:

2, 4, 6, 8, ...

If the leading coefficient is positive:

Both ends rise.

If the leading coefficient is negative:

Both ends fall.

Positive Even Degree

Consider:

f(x) = x⁴ − 3x²

The degree is:

4

which is even.

The leading coefficient is:

positive

Therefore:

Left end → up

Right end → up

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The middle of the graph may have different shapes, but the two ends eventually rise.

Negative Even Degree

Consider:

f(x) = −2x⁴ + x² + 3

The degree is:

4

The leading coefficient is:

negative

Therefore:

Left end → down

Right end → down

The ends travel in the same direction because the degree is even.

They point downward because the leading coefficient is negative.

Odd-Degree End Behaviour

Polynomials with an odd degree have ends travelling in opposite directions.

Examples of odd degrees include:

1, 3, 5, 7, ...

If the leading coefficient is positive:

Left end → down

Right end → up

If the leading coefficient is negative:

Left end → up

Right end → down

Positive Odd Degree

Consider:

f(x) = 2x³ − 5x

The degree is:

3

which is odd.

The leading coefficient is positive.

Therefore:

Left end → down

Right end → up

This is the same general end behaviour as:

f(x) = x³

Negative Odd Degree

Consider:

f(x) = −x³ + 4x

The degree is:

3

The leading coefficient is negative.

Therefore:

Left end → up

Right end → down

The graph has been reversed vertically compared with a positive cubic.

A Useful End-Behaviour Pattern

The four possibilities can be summarized as:

Even degree + positive leading coefficient

Left: up

Right: up

Even degree + negative leading coefficient

Left: down

Right: down

Odd degree + positive leading coefficient

Left: down

Right: up

Odd degree + negative leading coefficient

Left: up

Right: down

This pattern is extremely useful when sketching polynomial graphs.

Recognizing Even-Degree Characteristics

Suppose you are shown a graph but not its equation.

If both ends travel in the same direction, the graph has the end-behaviour characteristics of an even-degree polynomial.

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Both ends up suggests:

Even degree with positive leading coefficient

Both ends down suggests:

Even degree with negative leading coefficient

However, end behaviour alone usually does not tell you the exact degree.

A graph with both ends up could be degree 2, 4, 6, or another even degree.

Recognizing Odd-Degree Characteristics

If the two ends travel in opposite directions, the graph has the end-behaviour characteristics of an odd-degree polynomial.

For example:

Left down and right up:

Positive leading coefficient

Left up and right down:

Negative leading coefficient

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Again, this does not necessarily identify the exact degree.

The graph could be degree 1, 3, 5, 7, or another odd degree.

Degree and Turning Points

A turning point occurs where a graph changes direction.

It may change from:

increasing → decreasing

or:

decreasing → increasing

A polynomial of degree n can have at most n − 1 turning points.

For example:

A degree 2 polynomial can have at most:

1 turning point

A degree 3 polynomial can have at most:

2 turning points

A degree 4 polynomial can have at most:

3 turning points

A degree 5 polynomial can have at most:

4 turning points

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Turning Points Give a Minimum Possible Degree

Suppose a graph has:

3 turning points

A polynomial with three turning points must have degree of at least 4.

Why?

Because:

Maximum turning points = degree − 1

Therefore:

degree ≥ turning points + 1

So:

degree ≥ 4

Turning points can help us estimate the minimum possible degree of a polynomial.

Worked Example — Predicting End Behaviour

Consider:

f(x) = 3x⁶ − 2x³ + 5

Identify the degree:

6

This is even.

Identify the leading coefficient:

3

This is positive.

Therefore:

Left end → up

Right end → up

We can predict this without calculating many points.

Worked Example — Negative Odd Degree

Consider:

f(x) = −4x⁵ + 2x² − 7

Degree:

5

The degree is odd.

Leading coefficient:

−4

The leading coefficient is negative.

Therefore:

Left end → up

Right end → down

Worked Example — Identifying Characteristics from a Graph

Suppose a graph has:

Left end → down

Right end → up

It also has two turning points.

The opposite end directions indicate an:

Odd degree

The right end rising indicates a:

Positive leading coefficient

Two turning points mean the degree must be at least:

3

A cubic polynomial would therefore be one possible type of polynomial producing this shape.

Roots and X-Intercepts

A root, or zero, is an x-value where:

f(x) = 0

Graphically, roots occur where the graph meets the x-axis.

For example, if a graph crosses the x-axis at:

x = −2

and:

x = 3

then:

−2 and 3

are roots of the function.

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The x-intercepts provide important information when sketching polynomial graphs.

Degree and Number of Roots

A polynomial of degree n can have at most n real roots.

For example:

A quadratic can have at most:

2 real roots

A cubic can have at most:

3 real roots

A quartic can have at most:

4 real roots

However, a polynomial does not necessarily have the maximum possible number of real roots.

Crossing and Touching the X-Axis

Polynomial graphs do not always cross the x-axis at a root.

Sometimes the graph touches the axis and turns around.

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For example:

f(x) = (x − 2)²

has a root at:

x = 2

The graph touches the x-axis at this point but does not cross it.

This behaviour becomes important when studying repeated roots and multiplicity.

Sketching a Basic Polynomial Graph

When making a basic sketch, start by examining the equation.

Ask:

• What is the degree?
• Is the degree even or odd?
• Is the leading coefficient positive or negative?
• What should the end behaviour be?
• Where are the x-intercepts, if known?
• Where is the y-intercept?
• How many turning points are possible?

You can then sketch a smooth curve that follows these conditions.

Worked Example — Sketching a Quadratic

Consider:

f(x) = x² − 4

Degree:

2

The degree is even.

Leading coefficient:

positive

Therefore:

Both ends rise.

Find the roots:

x² − 4 = 0

Factor:

(x − 2)(x + 2) = 0

Roots:

x = −2

and:

x = 2

The graph crosses the x-axis at:

(−2, 0)

and:

(2, 0)

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These features allow us to create a reasonable sketch.

Worked Example — Sketching a Cubic

Consider:

f(x) = x³ − 4x

Factor:

f(x) = x(x² − 4)

f(x) = x(x − 2)(x + 2)

Roots:

x = −2, 0, 2

Degree:

3

The degree is odd.

Leading coefficient:

positive

Therefore:

Left end → down

Right end → up

The graph crosses the x-axis at:

−2, 0, and 2

These features provide enough information for a basic sketch.

Polynomial Graphs Are Smooth

Polynomial functions produce smooth, continuous curves.

This means their graphs:

• Do not contain breaks
• Do not suddenly jump
• Do not contain holes
• Do not have sharp corners

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A graph with a sudden break or vertical asymptote cannot be the graph of a polynomial function over the real numbers.

Comparing Polynomial Graphs

Consider:

f(x) = x²

and:

g(x) = x⁴

Both have:

• Even degree
• Positive leading coefficients
• Both ends pointing upward

However, their exact shapes are different.

Now compare:

f(x) = x³

and:

g(x) = x⁵

Both have:

• Odd degree
• Positive leading coefficients
• Left end down
• Right end up

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This shows why end behaviour can identify general characteristics without necessarily identifying the exact degree.

Why the Leading Term Controls End Behaviour

Consider:

f(x) = 5x⁴ − 3x² + 2x − 7

When x becomes extremely large, the term:

5x⁴

becomes much larger than the lower-degree terms.

For this reason, the graph's far-left and far-right behaviour becomes increasingly similar to:

f(x) = 5x⁴

The lower-degree terms still affect the middle of the graph, but the leading term dominates the ends.

Polynomial Graphs in Real Situations

Polynomial functions can be used to model relationships in:

• Engineering
• Economics
• Physics
• Computer graphics
• Architecture
• Manufacturing
• Data analysis
• Optimization

For example, polynomial models may approximate the relationship between production levels and costs or describe curved paths over a limited interval.

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The graph helps us visualize how one quantity changes in relation to another.

Using Technology to Explore Polynomial Graphs

Graphing software is especially useful for investigating higher-degree polynomials.

You can enter functions such as:

y = x³ − 4x

or:

y = x⁴ − 5x² + 4

and observe:

• End behaviour
• Roots
• Turning points
• Intercepts
• Overall shape

Changing individual coefficients allows you to investigate how the graph changes.

This is an excellent way to develop intuition about polynomial functions.

Common Mistakes

Mistake: Assuming Degree Tells the Exact Shape

Knowing that a polynomial has degree 4 does not tell you its exact graph.

Different quartic functions can have very different middle sections.

The degree tells us important limitations and general characteristics, not every detail.

Mistake: Ignoring the Leading Coefficient

Compare:

x⁴

and:

−x⁴

Both have degree 4, but their end behaviour is opposite.

Mistake: Thinking Even Degree Means Both Ends Up

Even degree means the ends travel in the same direction.

A negative leading coefficient makes both ends point downward.

Mistake: Thinking Odd Degree Always Rises to the Right

A positive odd-degree polynomial rises to the right.

A negative odd-degree polynomial falls to the right.

Mistake: Assuming Degree n Means Exactly n Turning Points

A degree n polynomial can have at most n − 1 turning points.

It may have fewer.

Mistake: Assuming Degree n Means Exactly n X-Intercepts

A degree n polynomial can have at most n real x-intercepts.

It may have fewer.

Mistake: Drawing Sharp Corners

Polynomial graphs should be drawn as smooth curves.

Key Terms

Polynomial – an expression involving variables raised to non-negative whole-number powers.

Polynomial function – a function defined using a polynomial expression.

Degree – the highest exponent in a polynomial.

Leading term – the term with the highest degree.

Leading coefficient – the coefficient of the leading term.

End behaviour – what happens to a graph as x becomes very large positively or negatively.

Even degree – a polynomial degree divisible by 2.

Odd degree – a polynomial degree not divisible by 2.

Turning point – a point where a graph changes from increasing to decreasing or vice versa.

Root – an x-value for which the function equals zero.

Zero – another name for a root.

X-intercept – a point where a graph meets the x-axis.

Check Your Understanding

1. Identify the degree of:

f(x) = 3x⁴ − 2x² + 7

2. Identify the degree and leading coefficient of:

f(x) = −5x³ + 2x − 1

3. Is:

f(x) = 2x⁶ − 4x + 3

an even- or odd-degree polynomial?

4. Predict the end behaviour of:

f(x) = 3x⁴ − x + 2

5. Predict the end behaviour of:

f(x) = −2x⁴ + 5x²

6. Predict the end behaviour of:

f(x) = 4x³ − 2x

7. Predict the end behaviour of:

f(x) = −x⁵ + 3x

8. A graph has both ends pointing upward. What can you determine about its degree characteristics and leading coefficient?

9. A graph has its left end pointing upward and its right end pointing downward. What can you determine?

10. A polynomial graph has three turning points. What is the minimum possible degree?

11. How many turning points can a degree 5 polynomial have at most?

12. Explain why the degree alone does not determine the exact shape of a polynomial graph.

13. Sketch the general shape of:

f(x) = x²

14. Sketch the general shape of:

f(x) = −x³

15. Explain how the degree and leading coefficient work together to determine end behaviour.

Key Takeaways

• Polynomial graphs are smooth and continuous.
• The degree is the highest exponent in a polynomial.
• Degree 1 polynomials are linear.
• Degree 2 polynomials are quadratic.
• Degree 3 polynomials are cubic.
• Degree 4 polynomials are quartic.
• The degree helps determine the possible shape of a polynomial graph.
• Even-degree polynomials have ends that travel in the same direction.
• Odd-degree polynomials have ends that travel in opposite directions.
• A positive even-degree polynomial has both ends pointing upward.
• A negative even-degree polynomial has both ends pointing downward.
• A positive odd-degree polynomial has its left end down and right end up.
• A negative odd-degree polynomial has its left end up and right end down.
• The leading term controls the graph's long-term end behaviour.
• A degree n polynomial can have at most n − 1 turning points.
• A degree n polynomial can have at most n real roots.
• Roots correspond to x-intercepts.
• Degree and leading coefficient reveal the general behaviour of a polynomial, but not its complete shape.

Did You Know?

The ends of a polynomial graph are controlled mainly by just one term—the leading term. Even if a polynomial contains many terms, the highest-degree term eventually becomes so much larger than the others that it dominates the function. This means that a complicated expression such as:

4x⁷ − 20x⁵ + 300x² − 900x + 5000

will eventually behave like:

4x⁷

far enough to the left and right of its graph.

Notice how the multiplicities explain what happens at each x-intercept.

Finding the Y-Intercept

The y-intercept can help make a sketch more accurate.

To find it, set:

x = 0

Consider again:

f(x) = (x + 2)(x − 1)²

Then:

f(0) = (0 + 2)(0 − 1)²

f(0) = 2(1)

f(0) = 2

Therefore, the y-intercept is:

(0, 2)

Now we know another point through which the graph must pass.

Worked Example — Three Different Multiplicities

Consider:

f(x) = (x + 3)(x − 1)²(x − 4)³

The zeros are:

x = −3, multiplicity 1

x = 1, multiplicity 2

x = 4, multiplicity 3

At:

x = −3

the graph crosses.

At:

x = 1

the graph touches and turns.

At:

x = 4

the graph crosses and flattens.

The degree is:

1 + 2 + 3 = 6

The degree is even.

The leading coefficient is positive.

Therefore:

Left end → up

Right end → up

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Reading Multiplicity from a Graph

Sometimes you are given the graph instead of the equation.

You can use the behaviour at each x-intercept to make conclusions about multiplicity.

If the graph crosses normally, the root likely has an odd multiplicity, often 1.

If the graph touches and turns, the root has an even multiplicity.

If the graph crosses while flattening, the root has an odd multiplicity greater than 1.

If the graph touches, turns, and becomes very flat, the root may have a higher even multiplicity.

However, the exact multiplicity is not always possible to determine from appearance alone unless additional information is provided.

A Graph Does Not Always Reveal Exact Multiplicity

Suppose a graph crosses the x-axis at:

x = 3

We can conclude that the zero has odd multiplicity.

But from a rough graph alone, we may not be able to determine whether the multiplicity is:

1, 3, 5, ...

Similarly, if a graph touches and turns at:

x = −2

we know the multiplicity is even, but it might be:

2, 4, 6, ...

This distinction is important when interpreting graphs.

Repeated Roots

A zero with multiplicity greater than 1 is often called a repeated root.

For example:

f(x) = (x − 5)²(x + 1)

has:

x = 5

as a repeated root.

The factor:

(x − 5)

appears twice.

Meanwhile:

x = −1

has multiplicity 1 and is not repeated.

Expanded Form Can Hide Multiplicity

Consider:

f(x) = x² − 6x + 9

It may not be immediately obvious that this polynomial has a repeated root.

Factor:

x² − 6x + 9 = (x − 3)²

Now we can see:

Zero: x = 3

Multiplicity: 2

Therefore, the graph touches the x-axis at:

(3, 0)

and turns around.

Factored form often reveals information about zeros and multiplicities much more clearly than expanded form.

Comparing Two Polynomials

Consider:

f(x) = (x − 2)(x + 1)

and:

g(x) = (x − 2)²(x + 1)

Both functions have zeros at:

x = 2

and:

x = −1

But their graphs behave differently at:

x = 2

For f(x), the multiplicity is 1, so the graph crosses.

For g(x), the multiplicity is 2, so the graph touches and turns.

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The locations of the zeros alone are therefore not enough to determine the shape of a polynomial graph.

We also need their multiplicities.

Why Multiplicity Matters

Multiplicity connects the algebraic form of a polynomial with its graphical behaviour.

The factor:

(x − a)ⁿ

tells us three important things:

The graph has a zero at:

x = a

The exponent n tells us:

the multiplicity

The parity of n tells us whether the graph:

crosses or touches the x-axis

This is a powerful connection between algebra and graphs.

Common Mistakes

Mistake: Confusing a Zero with an X-Intercept

If:

x = 4

is a zero, the x-intercept is:

(4, 0)

The zero is a number; the intercept is a point.

Mistake: Getting the Sign of a Zero Wrong

The factor:

(x − 5)

gives:

x = 5

But:

(x + 5)

gives:

x = −5

Mistake: Ignoring the Exponent

For:

(x − 3)⁴

the zero is not four different values.

There is one zero:

x = 3

with:

Multiplicity 4

Mistake: Assuming Every Root Crosses the X-Axis

Even-multiplicity roots touch the x-axis and turn around.

Mistake: Assuming Every Touching Root Has Multiplicity 2

A touching root has even multiplicity, which could be 2, 4, 6, or another even number.

Mistake: Ignoring End Behaviour When Sketching

Zeros tell you where the graph meets the x-axis, but they do not tell you everything about the overall graph.

Use the degree and leading coefficient as well.

Key Terms

Zero – an x-value that makes a function equal to zero.

Root – another name for a zero.

X-intercept – a point where a graph meets the x-axis.

Factor – an expression multiplied by another expression.

Multiplicity – the number of times a particular root occurs as a factor.

Repeated root – a root with multiplicity greater than 1.

Odd multiplicity – causes a polynomial graph to cross the x-axis.

Even multiplicity – causes a polynomial graph to touch the x-axis and turn around.

End behaviour – the behaviour of a graph far to the left and right.

Leading coefficient – the coefficient of the highest-degree term.

Check Your Understanding

1. Find the zeros of:

f(x) = (x − 3)(x + 5)

2. State the corresponding x-intercepts for the function above.

3. Find the zeros and their multiplicities:

f(x) = (x + 2)²(x − 4)

4. What happens to the graph at:

x = −2

in Question 3?

5. What happens to the graph at:

x = 4

in Question 3?

6. Find the zeros and multiplicities:

f(x) = (x − 1)³(x + 5)²

7. Does the graph cross or touch the x-axis at:

x = 1

in Question 6?

8. Does the graph cross or touch the x-axis at:

x = −5

in Question 6?

9. A graph crosses the x-axis at x = 4. What can you conclude about the multiplicity?

10. A graph touches the x-axis at x = −3 and turns around. What can you conclude about the multiplicity?

11. Determine the degree of:

f(x) = (x + 2)²(x − 1)³(x − 5)

12. Predict the end behaviour of the function in Question 11 if its leading coefficient is positive.

13. Factor:

x² − 8x + 16

and determine the zero and its multiplicity.

14. Explain why knowing only the locations of the zeros is not always enough to sketch a polynomial accurately.

15. Describe how you would use zeros, multiplicities, degree, leading coefficient, and the y-intercept to sketch a polynomial graph.

Key Takeaways

• A zero is an x-value that makes a polynomial equal to zero.
• Real zeros correspond to x-intercepts of the graph.
• A zero x = a corresponds to the x-intercept (a, 0).
• Factored form makes zeros particularly easy to identify.
• A factor (x − a)ⁿ gives a zero at x = a with multiplicity n.
• Multiplicity tells us how many times a root occurs.
• Odd multiplicity means the graph crosses the x-axis.
• Even multiplicity means the graph touches the x-axis and turns around.
• Higher multiplicities cause the graph to become flatter near the zero.
• A repeated root has multiplicity greater than 1.
• The sum of the multiplicities in a completely factored polynomial gives its degree.
• Zeros and multiplicities provide important information for sketching polynomial graphs.
• Degree and leading coefficient provide the end behaviour.
• The y-intercept can provide an additional useful point.
• A good polynomial sketch combines zeros, multiplicities, end behaviour, and intercepts.

Did You Know?

Multiplicity is one reason two polynomial graphs can have exactly the same x-intercepts but very different shapes. Knowing where a polynomial equals zero tells you where the graph reaches the x-axis, but multiplicity tells you what the graph does when it gets there—crossing through, touching and turning, or flattening as it passes through.

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.

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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.

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.

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.

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, ...

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.

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The appropriate polynomial depends on the pattern being modelled.

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.

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

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.

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

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

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.

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The graph can provide information about:

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

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.

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.

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

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

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

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

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.

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

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.

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.

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.

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

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.

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.

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.

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The polynomial does not necessarily pass through every data point.

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

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.

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.

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.

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.

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.

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

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.

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.

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

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.

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.

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.

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.

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.

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.