At a double root, the graph:

touches the axis and turns back

Graphs of Polynomials | HSC Maths Extension 1

In short: The graph of a polynomial is set by its degree, its leading coefficient and its roots. Odd and even degrees have different ends, and at a single root the curve crosses the x-axis, at a double root it touches, and at a triple root it flattens as it crosses.

Starting point: what you already know

The roots of a factorised polynomial: $(x - 2) (x + 5) = 0$ gives $x = 2$ or $x = - 5$ .
The y-intercept is the value at $x = 0$ .
$y = x^{2}$ is a parabola that touches the axis at 0. $y = x^{3}$ passes through 0 and flattens there, then keeps going.
Factor theorem (previous chapter): if $P (a) = 0$ , then $(x - a)$ is a factor.

Step 1: the ends of the graph

For very large $x$ , only the leading term matters. So look at the degree and the sign of the leading coefficient.

Degree	Leading coefficient positive	Leading coefficient negative
even, like $x^{2}$	up on both ends	down on both ends
odd, like $x^{3}$	down on the left, up on the right	up on the left, down on the right

Step 2: what happens at each root

The power on a factor is called the multiplicity of that root.

Single root $(x - a)$ : the graph crosses the axis.
Double root $(x - a)^{2}$ : the graph touches the axis and turns back, like $y = x^{2}$ .
Triple root $(x - a)^{3}$ : the graph crosses but flattens as it passes, like $y = x^{3}$ .

P(x) = (x + 2)(x − 1)²: a single root crosses, a double root touches.

Step 3: put it together

To sketch $P (x)$ from its factors:

Mark the roots and note each multiplicity.
Find the y-intercept, $P (0)$ .
Use the leading term to start from the correct end.
Draw from left to right, crossing, touching or flattening at each root.

Polynomial graphs

Ends

Degree even or odd
Sign of leading coefficient

Roots

Single crosses
Double touches
Triple flattens

Sketch steps

Roots and multiplicity
y-intercept
Start from the correct end

From a graph

Write factors
Use a point to find a

Year 11Solving with a graph

How many solutions does $(x + 2) (x - 1)^{2} = k$ have? Draw $y = k$ across the sketch above. The hump is at height 4 and the touch point at height 0.

$k > 4$ or $k < 0$ : one solution.
$k = 4$ or $k = 0$ : two solutions.
$0 < k < 4$ : three solutions.

How many real roots? Look at the turning points. $x^{3} - 3 x + 1$ has turning points at $x = \pm 1$ (where the derivative $3 x^{2} - 3$ is zero), with heights 3 and $- 1$ . One is above the axis and one below, so the curve crosses three times. For $x^{3} - 3 x + k = 0$ to have three different roots, the line $y = - k$ must pass between the turning heights 2 and $- 2$ : $- 2 < k < 2$ .

Every cubic has a real root. One end goes down to $- \infty$ and the other up to $\infty$ , so the curve must cross the axis at least once.

Odd and even. An odd polynomial (only odd powers) is symmetrical about the origin, so roots come in pairs $\pm a$ and $P (0) = 0$ . An even one (only even powers) is symmetrical about the $y$ -axis.

Quick summary

The ends depend on the degree (odd or even) and the sign of the leading coefficient.
Single roots cross, double roots touch, triple roots flatten as they cross.
To sketch: roots with multiplicity, y-intercept, start from the correct end.
To find the equation from a graph, write the factors with an unknown $a$ and use a known point.

Quiz

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Try 3 questions now

Where does $y = (x - 4) (x + 1)$ cut the x-axis?
1. A $- 4$ and 1
2. B 4 and $- 1$
3. C 4 and 1
4. D $- 4$ and $- 1$
Show answer
B: 4 and $- 1$
Each factor is zero at one root.
The y-intercept of $y = (x - 1) (x + 2) (x - 3)$ is:
1. A $- 6$
2. B 6
3. C 0
4. D 1
Show answer
B: 6
$(- 1) (2) (- 3) = 6$ .
At a double root, the graph:
1. A crosses straight through
2. B touches the axis and turns back
3. C has an asymptote
4. D flattens and crosses
Show answer
B: touches the axis and turns back
A squared factor never changes sign, so the graph bounces off the axis.

Graphs of Polynomials