Identify polynomial graph features and end behaviour

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Identify key features and describe end behaviour

A polynomial graph has features you can identify, such as where it meets the axes, where it changes direction, and what happens at its far ends. A zero is an input that gives an output of zero. An intercept is a point on an axis. These ideas help describe a graph without needing to know every point on it. This lesson reviews those features and shows how the degree and leading coefficient of a polynomial determine its end behaviour.

What you will learn

Identify zeros, intercepts, and visible turning points on a polynomial graph.
Describe a polynomial graph’s behaviour at its far-left and far-right ends.
Use the degree and leading coefficient to predict a polynomial’s end behaviour.
Distinguish a feature read from a graph from one determined using an equation.

1. Prerequisite bridge: coordinates and intercepts

A graph connects an input, usually called

x

, to an output, usually called

y

. A point is written as an ordered pair, such as

(3,0)

. The first number is the input; the second is the output.

An

x

-intercept is a point where the graph meets the

x

-axis. At any such point, the output is

0

. The input value at that point is a zero of the function. For example, if the graph passes through

(3,0)

, then

3

is a zero and

(3,0)

is an

x

-intercept.

A

y

-intercept is where the graph meets the

y

-axis. At that point, the input is

0

. If the graph passes through

(0,-4)

, then

(0,-4)

is its

y

-intercept. A zero is an input value; an intercept is a point.

f(x)=0

At an $x$ -intercept, the output is $0$ .
At a $y$ -intercept, the input is $0$ .
A zero and its corresponding $x$ -intercept are related but not the same kind of information.

2. Plain language: turning points and end behaviour

A turning point is a point where the graph changes direction. For example, a graph may rise before the point and fall after it, or fall before the point and rise after it. A local maximum is a turning point higher than nearby graph points. A local minimum is a turning point lower than nearby graph points. Here, “local” means nearby, rather than compared with every point on the graph.

End behaviour describes what a graph does far to the left and far to the right. Far to the left means at very negative input values; far to the right means at very positive input values. This describes the ends of the graph, not what happens near an intercept or a turning point.

Polynomial graphs have no breaks or holes. Their ends follow one of four patterns: both rise, both fall, the left falls while the right rises, or the left rises while the right falls. For a polynomial, the degree and leading coefficient determine which pattern applies.

A turning point is where the graph changes direction.
End behaviour describes the far-left and far-right ends.
The degree and leading coefficient determine a polynomial’s end behaviour.

3. From equation to end behaviour

The degree of a polynomial is the greatest exponent of its variable when the polynomial is written as a sum of powers. The leading coefficient is the coefficient of the term with that greatest exponent. In

-2x^5+3x^2-1

, the degree is

5

and the leading coefficient is

-2

.

For an even degree, the two ends point in the same direction. A positive leading coefficient means both ends rise; a negative leading coefficient means both ends fall. For an odd degree, the ends point in opposite directions. With a positive leading coefficient, the left end falls and the right end rises. With a negative leading coefficient, the left end rises and the right end falls.

A polynomial in factored form can also show zeros. For example,

(x-2)(x+1)

has zeros

2

and

-1

, because setting each factor equal to zero gives those input values. To find the

y

-intercept from an equation, substitute

0

for the input. Turning points are identified from a graph when they are shown. Do not infer a turning point just from a zero or from the end behaviour.

f(x)=ax^n+\cdots,\quad a\ne 0

Even degree gives ends in the same direction; odd degree gives ends in opposite directions.
The sign of the leading coefficient determines whether the right end rises or falls.
Factored form can show zeros; substituting zero for the input gives the $y$ -intercept.

4. Read features with care

When a graph is provided, record its visible intercepts and turning points. Then describe the left and right ends separately. A graph may show exact points, such as an intercept, and broad patterns, such as an end rising. Keep those descriptions distinct.

When an equation is provided, identify its degree and leading coefficient to determine end behaviour. If it is factored, set each factor equal to zero to find the zeros. Substitute zero for the input to find the

y

-intercept. These steps identify features supported by the equation; they do not give turning-point coordinates.

Use precise language. A zero is an input value. An intercept is a point. A turning point marks a change in direction. End behaviour describes what happens far to the left and far to the right.

Use the graph to identify visible turning points.
Use degree and leading coefficient to determine end behaviour.
Describe only the features supported by the graph or equation.

End behaviour from degree and leading coefficient

Degree	Leading coefficient	Left end	Right end
Even	Positive	Rises	Rises
Even	Negative	Falls	Falls
Odd	Positive	Falls	Rises
Odd	Negative	Rises	Falls

Worked example

Identify zeros, an intercept, and end behaviour

For

f(x)=-(x+2)^2(x-1)

, identify its zeros, find its

y

-intercept, and state its end behaviour.

Find the zeros
A zero makes the function’s output equal to zero. Set each factor equal to zero: $x+2=0$ gives $x=-2$ , and $x-1=0$ gives $x=1$ .
$x=-2,\quad x=1$
Find the y-intercept
The $y$ -intercept occurs when the input is zero. Substitute $x=0$ into the equation; the output is $4$ .
$f(0)=-(0+2)^2(0-1)=4$
Determine the end behaviour
The highest-power parts of the factors give the leading term $-x^3$ . The degree is odd, so the ends point in opposite directions. The leading coefficient is negative, so the left end rises and the right end falls.
$-(x^2)(x)=-x^3$

Answer: The zeros are

-2

and

1

. The

y

-intercept is

(0,4)

. The left end rises and the right end falls.

Check: The leading term has degree

3

and a negative coefficient. An odd degree gives opposite end directions, and the negative coefficient makes the right end fall. This confirms the stated end behaviour.

Common mistakes and how to avoid them

Calling the zero itself an intercept.

Correction: A zero is an input value. The corresponding

x

-intercept is the point

(x,0)

.

Using only the leading coefficient to determine both ends.

Correction: First check whether the degree is even or odd. Then use the leading coefficient to determine whether the right end rises or falls.

Assuming a zero is also a turning point.

Correction: A zero tells you where the graph meets the

x

-axis. A turning point is where the graph changes direction; identify it from the graph when it is shown.

Confusing the

y

-intercept with a zero.

Correction: The

y

-intercept occurs at input

0

. A zero is an input that makes the output

0

.

Lesson summary

A zero is an input that gives an output of zero; its $x$ -intercept is a point.
Substitute $0$ for the input to find the $y$ -intercept from an equation.
A turning point is where the graph changes direction.
Degree parity and the leading coefficient determine end behaviour.
Identify turning points from a graph when they are shown.

Check your understanding

Question 1

A polynomial has even degree and a negative leading coefficient. What happens at its ends?

Both ends rise.
Both ends fall.
The left end rises and the right end falls.
The left end falls and the right end rises.

Show answer and explanation

Both ends fall.

Even degree means both ends point in the same direction. A negative leading coefficient makes both ends fall.

Question 2

A graph has an

x

-intercept at

(4,0)

. What is the corresponding zero?

$0$
$4$
$(0,4)$
There is no zero.

Show answer and explanation

4

The zero is the input coordinate of the

x

-intercept, so it is

4

.

Question 3

A polynomial has degree

5

and a positive leading coefficient. What is its end behaviour?

Both ends rise.
Both ends fall.
The left end falls and the right end rises.
The left end rises and the right end falls.

Show answer and explanation

The left end falls and the right end rises.

The degree is odd, so the ends point in opposite directions. A positive leading coefficient means the right end rises, so the left end falls.

Key terms

Polynomial: An expression built from constants and non-negative integer powers of a variable, combined using addition, subtraction, and multiplication.
Degree: The greatest exponent of the variable in a polynomial written as a sum of powers.
Leading coefficient: The coefficient of the term with the greatest exponent.
Zero: An input value that makes a function’s output equal to zero.
Intercept: A point where a graph meets one of the axes.
Turning point: A point where a graph changes direction.
End behaviour: A description of what a graph does far to the left and far to the right.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C1.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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