C. Polynomial and Rational Functions · Polynomial Functions and Equations

Key Features of Polynomial Graphs: End Behaviour, Turning Points and Zeros

C1.3 C1.2About 70 minutes

Learning goals

I can predict the end behaviour of a polynomial function from its degree and the sign of its leading coefficient.
I can state the greatest possible number of turning points and zeros for a given degree, and which numbers are possible.
I can describe the domain, range, intercepts, intervals of increase and decrease, and local and absolute maximum and minimum values from a graph.
I can sketch graphs that match a description, or explain why no such graph exists.

Words to know

end behaviour
What the y-values do as x → −∞ (far left) and x → ∞ (far right).

turning point
A point where the graph changes from increasing to decreasing, or from decreasing to increasing.

local maximum / minimum
A turning point that is higher (or lower) than all nearby points on the graph.

absolute maximum / minimum
The greatest (or least) y-value of the whole function, if there is one.

zero (of a function)
A value of x where f(x) = 0. It is an x-intercept of the graph.

increasing / decreasing interval
An interval of x-values where the graph goes up (or down) as you move to the right.

Reading a graph from far away

If you zoom far out on any polynomial graph, the bumps in the middle shrink to nothing and you only see the two ends. Far from the origin, the leading term aₙxⁿ is so much bigger than all the other terms that it decides everything. So the end behaviour depends on just two things:

Is the degree n even or odd?
Is the leading coefficient a positive or negative?

We describe end behaviour with arrows: "as x → ∞, y → −∞" means "as x gets larger and larger, y gets more and more negative". A quick shorthand uses quadrants: "Q3 to Q1" means the graph comes up from quadrant 3 and leaves through quadrant 1.

The four possible end behaviours. Odd degree: the ends go in opposite directions. Even degree: both ends go the same way. A positive leading coefficient makes the right end go up.

Degree	Leading coefficient	As x → −∞	As x → ∞	Like
odd	a > 0	y → −∞	y → ∞	y = x, y = x³
odd	a < 0	y → ∞	y → −∞	y = −x, y = −x³
even	a > 0	y → ∞	y → ∞	y = x², y = x⁴
even	a < 0	y → −∞	y → −∞	y = −x², y = −x⁴

End behaviour at a glance.

Turning points and zeros

A polynomial function of degree n has:

at most n − 1 turning points. Also, the number of turning points is odd when n is even and even (possibly zero) when n is odd. A cubic has 0 or 2 turning points; a quartic has 1 or 3.
at most n zeros (x-intercepts). An odd-degree polynomial has at least one zero, because its ends go in opposite directions, so the graph must cross the x-axis. An even-degree polynomial may have no zeros at all.

Other key features:

Domain: all real numbers, for every polynomial function.
Range: all real numbers for odd degree. For even degree the graph has an absolute minimum (if a > 0) or absolute maximum (if a < 0), so the range is y ≥ min or y ≤ max.
y-intercept: the constant term, f(0).
Increasing and decreasing intervals: these change at the turning points.

Degree	Possible turning points	Possible number of zeros	Range
1	0	1	all real numbers
2	1	0, 1 or 2	y ≥ min or y ≤ max
3	0 or 2	1, 2 or 3	all real numbers
4	1 or 3	0, 1, 2, 3 or 4	y ≥ min or y ≤ max
5	0, 2 or 4	1, 2, 3, 4 or 5	all real numbers

What each degree allows. (Zeros are counted as different x-intercepts.)

The quartic f(x) = x⁴ − 5x² + 4 has 4 zeros (−2, −1, 1, 2), 3 turning points, both ends rising, and an absolute minimum of −2.25 at x = ±√10/2 ≈ ±1.58.

Worked example

Example 1: End behaviour from an equation

Describe the end behaviour of each function.

(a) f(x) = −2x⁵ + 3x² − 1 (b) g(x) = (3 − x)(x + 1)³ (c) h(x) = 0.1x⁶ − 50x⁵

(a) Degree 5 (odd), a = −2 < 0. As x → −∞, y → ∞; as x → ∞, y → −∞. Q2 to Q4.
(b) Degree 1 + 3 = 4 (even). Leading coefficient (−1)(1³) = −1 < 0. Both ends go down: as x → ±∞, y → −∞. Q3 to Q4.
(c) Degree 6 (even), a = 0.1 > 0. Both ends go up: Q2 to Q1. The −50x⁵ term is bigger than 0.1x⁶ for moderate x (for example h(100) = 10¹¹ − 5 × 10¹¹ < 0), but once x > 500 the leading term wins.

(a) Q2 to Q4 (b) Q3 to Q4 (c) Q2 to Q1

Worked example

Example 2: Describing a graph completely

Use the graph of f(x) = x⁴ − 5x² + 4 above. The local minimum points are (±√10/2, −9/4). Describe the key features.

Degree 4, a = 1 > 0, so both ends rise (Q2 to Q1). Domain: all real numbers.
Zeros: x = −2, −1, 1, 2 (since f(x) = (x² − 1)(x² − 4)). y-intercept: f(0) = 4.
Turning points: 3, which is the most a quartic can have. Local maximum 4 at x = 0. Local (and absolute) minimum −9/4 = −2.25 at x = ±√10/2 ≈ ±1.58.
Range: y ≥ −2.25.
Decreasing for x < −1.58 and for 0 < x < 1.58. Increasing for −1.58 < x < 0 and for x > 1.58.
Symmetry: the graph is a mirror image across the y-axis (only even powers of x appear; more in lesson 6).

Quartic, Q2 to Q1, 4 zeros, 3 turning points, range y ≥ −2.25, symmetric about the y-axis.

Worked example

Example 3: Is it possible?

Can you sketch (a) a cubic with exactly one turning point? (b) a quartic with a < 0 and no zeros? (c) a quintic with no zeros?

(a) A cubic has 0 or 2 turning points: no. If it turned once, both ends would go the same way, which is even-degree behaviour.
(b) Yes. Both ends go down and the absolute maximum can be below the x-axis, for example y = −x⁴ − 1.
(c) No. Odd degree means the ends go in opposite directions, so the graph must cross the x-axis at least once.

(a) impossible (b) possible (c) impossible

Turning points vs zeros

Turning points are where the graph changes direction; zeros are where it meets the x-axis. They are different things. In Calculus and Vectors (MCV4U) you will find turning points exactly using derivatives. For now, read them from a graph (graphing technology helps) or use the information you are given.

Just for fun

Why don't polynomials ever get lost on long trips?

Their leading term always tells them which way the ends are going.

Hands-on activity

String graphs

You need: a piece of string or yarn about 50 cm long, a sheet of grid paper with x- and y-axes drawn on it, and a pencil.

For each description, lay the string on the axes to make a graph that fits, then trace it. If a description is impossible, explain why.

Cubic, a > 0, three zeros.
Quartic, a < 0, two zeros, three turning points.
Quartic, a > 0, one turning point, no zeros.
Cubic with one turning point.
Quintic, a < 0, one zero, four turning points.

What to look for: 1 rises from Q3 to Q1 with two turns and crosses three times. 2 is an M shape with both ends down; for exactly two zeros, keep the middle dip above the x-axis (so the graph crosses only on the far left and far right). 3 is a U shape sitting above the x-axis. 4 is impossible (a cubic has 0 or 2 turning points). 5 comes down from Q2 and leaves through Q4 with four turns, but crosses the x-axis only once (for example, the bumps before the crossing stay above the axis and the ones after it stay below).

Your turn

Describe the end behaviour of f(x) = 4x³ − x + 9.

Describe the end behaviour of f(x) = −x⁴ + 3x³.

Describe the end behaviour of f(x) = (2x − 1)(x + 3)(1 − x).

For a polynomial of degree 5, give the greatest number of turning points, the greatest number of zeros and the least number of zeros.

How many turning points can a quartic function have?

f(x) = −x⁴ + 2x² can be written −(x² − 1)² + 1. State its range.

True or false: every cubic function has at least one x-intercept. Explain.

More practice: warm-up

Even degree, a > 0. Where do the ends go?

What is the domain of every polynomial function?

What is the greatest number of turning points for degree 6?

What is the range of an odd-degree polynomial function?

What is the y-intercept of f(x) = 3x⁴ − 2x + 5?

More practice: on your own

Describe a graph of a cubic function with a < 0 and three zeros: end behaviour and number of turning points.

f(x) = x³ − 3x has turning points (−1, 2) and (1, −2). Give the intervals of increase and decrease and the local extrema.

Explain why a quartic function always has either an absolute maximum or an absolute minimum.

g(x) = −(x + 1)²(x − 3)(x − 5). Give the degree, the end behaviour and the y-intercept.

A polynomial's graph rises from Q3, has exactly 2 turning points, and leaves through Q1. What is the smallest possible degree?

Challenge

A student sketches a polynomial graph whose ends both go up and which has exactly 4 turning points. Explain why the sketch cannot be right.

For h(x) = 0.1x⁶ − 50x⁵, show that h(100) < 0 but h(1000) > 0, and explain how this fits the end behaviour.

Sketch (describe) a quartic with a > 0 and exactly two zeros: one version with 3 turning points and one with 1 turning point.

Check your understanding

Quiz

What is the end behaviour of f(x) = −3x⁵ + x²?

What is the greatest number of turning points a quartic function can have?

Which number of x-intercepts is impossible for a cubic function?

A polynomial of even degree has a negative leading coefficient. What is true about its range?

What is the end behaviour of f(x) = (x − 2)(3 − 2x)(x + 1)²?

A quartic has 3 distinct zeros. What is the possible number of turning points?

For parents and teachers

Big idea (C1.3): far from the origin, the leading term decides where the graph goes. The degree also limits the number of turning points (at most n − 1, with the right parity) and zeros (at most n; at least one for odd degree).

How to help: Draw a quick graph shape and ask, "Odd or even degree? Positive or negative leading coefficient? What is the smallest degree that could make this?" Then reverse it: name a degree and a sign and ask for a sketch.

Common difficulties:

Using the first coefficient written instead of the leading coefficient (in 5 + 2x − x³ the leading coefficient is −1).
Thinking a quartic must have 3 turning points or 4 zeros; those are maximums, not requirements.
Mixing up turning points and zeros.
Writing intervals of increase in terms of y instead of x.