Graphs of Polynomials

SPEC 2.7 - ALGEBRA AND FUNCTIONS

Understanding shape, behaviour, roots, and turning points of polynomial functions

What is a Polynomial?

Definition

A polynomial is an expression of the form:

f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}

where

a_{n}, a_{n - 1}, \dots, a_{0}

are constants and

n

is a non-negative integer.

Key Terms

Degree: The highest power of x (e.g., x³ has degree 3)

Leading coefficient: The coefficient of the highest power term (aₙ)

Constant term: The term with no x (a₀)

DegreeNameExampleMax Turning Points

2Quadratic

f (x) = x^{2} - 4 x + 1

1

3Cubic

f (x) = x^{3} - 2 x^{2} + x

2

4Quartic

f (x) = x^{4} - 3 x^{2} + 2

3

End Behaviour

The end behaviour describes what happens to f(x) as x approaches positive or negative infinity.

Rule

End behaviour depends on TWO things:

Degree (odd or even)

Leading coefficient (positive or negative)

Leading Coeff.DegreeAs x → −∞As x → +∞Shape

PositiveEvenf(x) → +∞f(x) → +∞Both ends up

PositiveOddf(x) → −∞f(x) → +∞Down-up

NegativeEvenf(x) → −∞f(x) → −∞Both ends down

NegativeOddf(x) → +∞f(x) → −∞Up-down

Memory Tip

Even degree: Both ends go the SAME direction (like a parabola)

Odd degree: Ends go OPPOSITE directions (like a cubic)

Multiplicity of Roots

The multiplicity of a root tells you how many times that factor appears, which affects how the graph behaves at that point.

MultiplicityFactor FormGraph Behaviour

1 (Single)(x − a)Crosses the x-axis (straight through)

2 (Double)(x − a)²Touches and turns back (bounces off)

3 (Triple)(x − a)³Crosses with inflection (S-curve through)

Odd multiplicity: CROSSES

Even multiplicity: TOUCHES

Key Rule

Odd multiplicity (1, 3, 5...): Graph CROSSES the x-axis

Even multiplicity (2, 4, 6...): Graph TOUCHES and TURNS at the x-axis

Quadratic Functions (Review)

Recap

Quadratics have degree 2:

f (x) = a x^{2} + b x + c

Always produces a parabola

Exactly 1 turning point (vertex)

At most 2 real roots

a > 0, Two Roots

U-shape, 2 roots

a > 0, One Root

Touches x-axis

a > 0, No Roots

Above x-axis

a < 0, Two Roots

∩-shape, 2 roots

Key Features

Y-intercept: The constant term c (where x = 0)

X-intercepts: Solve ax² + bx + c = 0 using factorisation or the quadratic formula

Vertex: At x = −b/(2a), which is the axis of symmetry

Worked Example - Quadratic01

Sketch the graph of $y = - x^{2} + 4 x - 3$ .

1

End behaviour: Degree 2 (even), negative leading coefficient → both ends down (inverted parabola ∩)

2

Y-intercept: f(0) = −0 + 0 − 3 = −3 → point (0, −3)

3

X-intercepts: Solve −x² + 4x − 3 = 0

x² − 4x + 3 = 0 (multiply by −1)

(x − 1)(x − 3) = 0

x = 1, x = 3

4

Vertex: x = −b/(2a) = −4/(2×−1) = 2

y = −4 + 8 − 3 = 1 → vertex at (2, 1)

AnswerInverted parabola with maximum at (2, 1), crossing x-axis at 1 and 3, y-intercept at −3.

Cubic Functions

Key Properties

Cubics have degree 3:

f (x) = a x^{3} + b x^{2} + c x + d

End behaviour: Opposite ends (one up, one down)

Maximum turning points: 2

Roots: 1, 2, or 3 real roots

Always crosses the x-axis at least once

Down on left, Up on right (y = x³)

Up on left, Down on right (y = −x³)

Quick Check

For positive leading coefficient: think of y = x³ which goes from bottom-left to top-right.

For negative: it's flipped - from top-left to bottom-right.

Cubic Functions - Root Scenarios

A cubic can have 1, 2, or 3 real roots. Here are the shapes for positive and negative leading coefficients:

When a > 0 (positive leading coefficient):

Crosses 3 times

Crosses once, touches once

One crossing (no real quadratic roots)

When a < 0 (negative leading coefficient):

Crosses 3 times

Crosses once, touches once

One crossing (no real quadratic roots)

Worked Example - Cubic (3 roots)02

Sketch the graph of $y = x^{3} - 4 x$ , showing all intercepts.

1

End behaviour: Degree 3 (odd), positive leading coefficient → down on left, up on right

2

Y-intercept: f(0) = 0³ − 4(0) = 0 → passes through origin (0, 0)

3

X-intercepts: Solve x³ − 4x = 0

x(x² − 4) = 0

x(x − 2)(x + 2) = 0

x = −2, x = 0, x = 2

4

Multiplicity: All roots have multiplicity 1 → graph crosses at each

5

Turning points: 2 turning points - one local max between −2 and 0, one local min between 0 and 2

AnswerClassic cubic S-curve with 3 roots at x = −2, 0, 2.

Worked Example - Cubic (repeated root)03

Sketch the graph of $y = (x - 1)^{2} (x + 2)$ .

1

End behaviour: Expanding gives x³ term with positive coefficient → down on left, up on right

2

Y-intercept: f(0) = (−1)²(2) = 1 × 2 = 2 → point (0, 2)

3

X-intercepts: From factors: x = 1 and x = −2

4

Multiplicity:

x = 1 has multiplicity 2 → graph TOUCHES and TURNS

x = −2 has multiplicity 1 → graph CROSSES

5

Turning points: 2 - one at x = 1 (the touch point), one between −2 and 1

AnswerThe graph bounces off the x-axis at x = 1 (repeated) but crosses at x = −2.

Worked Example - Cubic (triple root)04

Sketch the graph of $y = - (x - 1)^{3}$ .

1

End behaviour: Degree 3 (odd), negative leading coefficient → up on left, down on right

2

Y-intercept: f(0) = −(0 − 1)³ = −(−1)³ = −(−1) = 1 → point (0, 1)

3

X-intercepts: −(x − 1)³ = 0 → x = 1 only

4

Multiplicity: x = 1 has multiplicity 3 (triple root) → graph CROSSES with an inflection point (S-curve shape)

5

Turning points: NO turning points! The curve has a point of inflection at (1, 0) but no local max/min.

AnswerReflected and translated y = x³ with inflection point at (1, 0). Goes from top-left to bottom-right.

Quartic Functions

Key Properties

Quartics have degree 4:

f (x) = a x^{4} + b x^{3} + c x^{2} + d x + e

End behaviour: Same direction on both ends (like quadratics)

Maximum turning points: 3

Roots: 0, 1, 2, 3, or 4 real roots

Both ends UP (W-shape with 4 roots)

Both ends DOWN (M-shape with 4 roots)

Remember

Quartics behave like "stretched" quadratics at the ends - even degree means both ends go the same direction!

Quartic Functions - Root Scenarios

A quartic can have 0, 1, 2, 3, or 4 real roots. Here are common shapes:

When a > 0 (positive leading coefficient):

W-shape crosses 4×

Touches at repeated root

Touches both roots

Above x-axis

When a < 0 (negative leading coefficient):

M-shape crosses 4×

Touches at repeated root

Touches both roots

Below x-axis

Worked Example - Quartic (4 roots)05

Sketch the graph of $y = x^{4} - 5 x^{2} + 4$ .

1

End behaviour: Degree 4 (even), positive leading coefficient → both ends UP

2

Y-intercept: f(0) = 0 − 0 + 4 = 4 → point (0, 4)

3

X-intercepts: Solve x⁴ − 5x² + 4 = 0

Let u = x²: u² − 5u + 4 = 0

(u − 1)(u − 4) = 0 → u = 1 or u = 4

x² = 1 → x = ±1; x² = 4 → x = ±2

Roots: x = −2, −1, 1, 2

4

Multiplicity: All roots have multiplicity 1 → crosses at all four points

5

Turning points: 3 - local max at x = 0 (at y = 4), local mins between roots

AnswerClassic W-shape with 4 x-intercepts at −2, −1, 1, 2 and local maximum at (0, 4).

Worked Example - Quartic (mixed roots)06

Sketch $y = x (x - 2)^{2} (x + 1)$ , identifying all key features.

1

Degree: 1 + 2 + 1 = 4 (quartic), positive leading coefficient → both ends UP

2

Y-intercept: f(0) = 0 × (−2)² × 1 = 0 → passes through origin

3

X-intercepts: x = 0, x = 2, x = −1

4

Multiplicity:

x = −1: multiplicity 1 → CROSSES

x = 0: multiplicity 1 → CROSSES

x = 2: multiplicity 2 → TOUCHES and TURNS

5

Turning points: 3 maximum - between −1 and 0, between 0 and 2, and at x = 2

AnswerQuartic crossing at x = −1 and 0, touching at x = 2, both ends pointing up.

Worked Example - Finding Equation07

A cubic graph passes through (−1, 0), (2, 0), and (4, 0), with a positive leading coefficient. The y-intercept is 8. Find its equation.

1

Form: Since roots are −1, 2, 4, the equation is:

y = a(x + 1)(x − 2)(x − 4)

2

Find a: Use y-intercept (0, 8):

8 = a(0 + 1)(0 − 2)(0 − 4)

8 = a(1)(−2)(−4)

8 = 8a

a = 1

3

Check: a = 1 is positive ✓

Answer: y = (x + 1)(x − 2)(x − 4)

Or expanded: y = x³ − 5x² + 2x + 8

Sketching Method Summary

Step-by-Step Process

Identify degree and leading coefficient → determines end behaviour

Find the y-intercept → substitute x = 0

Find the x-intercepts (roots) → solve f(x) = 0 or read from factors

Determine multiplicity of each root → cross (odd) or touch (even)?

Consider turning points → at most (degree − 1)

Draw a smooth curve through all points with correct behaviour

End Behaviour Quick Reference

Even degree: Both ends same direction

Odd degree: Opposite ends

Positive a: Right end UP

Negative a: Right end DOWN

Checklist

End behaviour marked?

Y-intercept labelled?

All x-intercepts shown?

Cross vs touch correct?

Practice

Question 1: Sketch the graph of

y = (x + 1) (x - 2) (x - 3)

, showing all intercepts.

End behaviour: Degree 3, positive leading coefficient → down left, up right

Y-intercept: f(0) = (1)(−2)(−3) = 6 → point (0, 6)

X-intercepts: x = −1, x = 2, x = 3 (all multiplicity 1, all cross)

Turning points: 2 - one local max between −1 and 2, one local min between 2 and 3

Sketch: Cubic crossing at −1, 2, 3 with y-intercept at 6.

Practice

Question 2: Sketch the graph of

y = - x^{4} + 4 x^{2}

, showing all intercepts and describing the end behaviour.

End behaviour: Degree 4 (even), negative leading coefficient → both ends DOWN

Y-intercept: f(0) = 0 → passes through origin

X-intercepts: −x⁴ + 4x² = 0 → −x²(x² − 4) = 0 → −x²(x−2)(x+2) = 0

Roots: x = 0 (double), x = −2, x = 2

Multiplicity: x = 0 has multiplicity 2 (touches), x = ±2 have multiplicity 1 (cross)

Shape: Inverted W (M-shape), touching at origin, crossing at ±2, max 3 turning points.

Practice

Question 3: A quartic graph has roots at x = −2, x = 0 (double root), and x = 3. The graph passes through (1, −12). Find the equation of the graph.

Form: y = a(x + 2)(x)²(x − 3) = ax²(x + 2)(x − 3)

Find a: Substitute (1, −12):

−12 = a(1)²(1 + 2)(1 − 3)

−12 = a(1)(3)(−2)

−12 = −6a

a = 2

Answery = 2x²(x + 2)(x − 3) or y = 2x⁴ − 2x³ − 12x²

Summary

Polynomial Graph Features

TypeDegreeMax RootsMax Turning PointsEnd Behaviour

Quadratic221Same direction

Cubic332Opposite directions

Quartic443Same direction

Exam Skills

Always state end behaviour first when sketching

Label all intercepts with coordinates

Clearly show whether curve crosses or touches at each root

For finding equations: use factored form with roots, then find 'a' using another point