# Graphs of Polynomials

> Sketch polynomials from their factors. End behaviour, multiple roots that touch or flatten, and reading equations from graphs.

From HSC Maths Extension 1 on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-extension-1/polynomial-graphs/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Reading a heart monitor
A nurse glances at a monitor and knows at once if something is wrong. She does not need numbers. The shape of the line tells her: where it rises, where it dips, where it flattens.

A polynomial graph can be read the same way. From the factors alone you can tell where the curve meets the axis, whether it crosses or just touches, and which way it heads at the far ends.

**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 $y = x^3$.

*Diagram: Graph of y equals x plus 2 times x minus 1 squared. It crosses the x-axis at minus 2, turns at minus 1, 4, cuts the y-axis at 2 and just touches the x-axis at 1.*
*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:

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

### Worked example: Sketch y = (x + 2)(x − 1)²
- Roots: $-2$ (single, crosses) and 1 (double, touches).
- y-intercept: $(2)(1) = 2$.
- Leading term $x^3$: starts low on the left, ends high on the right.

From the bottom left, cross at $-2$, rise over a hump, come down to touch at 1, then rise again. The hump is at $(-1, 4)$, which you can check by substitution.

### Worked example: Reading an equation from a graph
A cubic **crosses** the axis at $-3$, **touches** it at 2 and passes through $(0, 6)$. Find its equation.

So $P(x) = a(x + 3)(x - 2)^2$. Put in $(0, 6)$: $6 = a(3)(4)$, so $a = \dfrac{1}{2}$.

### Watch out: Count the degree
The multiplicities must add up to the degree. A cubic has 3 roots counted with multiplicity, though some may not be real.

**Mind map**

```text
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 11: Solving 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. $x^3 - 3x + 1$ has turning points at $x = \pm 1$ (where the derivative $3x^2 - 3$ is zero), with heights 3 and $-1$. One is above the axis and one below, so the curve crosses three times. For $x^3 - 3x + k = 0$ 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.

### Year 11: Polynomial inequalities from a sketch
Solve $(x + 2)(x - 1)^2 > 0$. The graph is above the axis to the right of $-2$, except at $x = 1$ where it touches zero. So $x > -2$, $x \ne 1$.

### Year 12: Double roots and the derivative
At a double root, the curve touches the axis, so the tangent there is flat: $P(a) = 0$ **and** $P'(a) = 0$. In Year 12 this gives a quick test for multiple roots and a way to find unknown coefficients.

**Stationary points of a factorised polynomial.** For $y = (x - 2)^3(x + 1)$, the product rule gives $y' = (x - 2)^2\big[3(x + 1) + (x - 2)\big] = (x - 2)^2(4x + 1)$. So $x = 2$ is a horizontal point of inflection (the triple root) and $x = -\frac{1}{4}$ is a minimum.

## 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

Try these questions from the chapter quiz.

1. Where does $y = (x - 4)(x + 1)$ cut the x-axis?
   - A. $-4$ and 1
   - B. 4 and $-1$
   - C. 4 and 1
   - D. $-4$ and $-1$
   - Answer: B. 4 and $-1$. Each factor is zero at one root.

2. The y-intercept of $y = (x - 1)(x + 2)(x - 3)$ is:
   - A. $-6$
   - B. 6
   - C. 0
   - D. 1
   - Answer: B. 6. $(-1)(2)(-3) = 6$.

3. At a double root, the graph:
   - A. crosses straight through
   - B. touches the axis and turns back
   - C. has an asymptote
   - D. flattens and crosses
   - Answer: B. touches the axis and turns back. A squared factor never changes sign, so the graph bounces off the axis.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-extension-1/polynomial-graphs/#quiz