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.

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.

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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.

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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.

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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.

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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.

For:

f(x) = x²

both ends of the graph rise upward.

If the leading coefficient is negative:

f(x) = −x²

both ends point downward.

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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.

For:

f(x) = x³

the left end falls and the right end rises.

The two ends travel in opposite directions.

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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.

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

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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.

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The Leading Coefficient

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

For:

f(x) = 5x³ − 2x + 1

the leading coefficient is:

5

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.

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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.

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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

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

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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.

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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

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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

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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.

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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.

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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.

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.

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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

Again, this does not necessarily identify the exact degree.

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

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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.

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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.

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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

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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.

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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.

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

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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.

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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.

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.

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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.

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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)

These features allow us to create a reasonable sketch.

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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.

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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

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

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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

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

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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.

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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.

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

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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.

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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.

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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.

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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.

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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.