# A Library of Graphs

> Recognise and sketch cubics, hyperbolas, exponentials, circles, absolute value and polynomial graphs.

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

### Story time: Spotting cars
Ethan's little brother can name a car from across the street. A ute, a hatchback, a four-wheel drive. He has seen so many that the shape alone tells him.

Good maths students do the same with graphs. Show them $y = \dfrac{1}{x}$ and they picture two curves in opposite corners straight away. Knowing the basic shapes by sight saves a lot of time in exams, because every harder graph is one of these shapes moved or stretched.

**In short:** The main graphs to know by shape are the line, parabola, cubic, hyperbola, exponential and circle. A hyperbola $y = \dfrac{k}{x}$ has asymptotes it never touches, and the circle with centre $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.

## Starting point: Years 9 and 10

| Equation | Name | Shape |
|---|---|---|
| $y = mx + c$ | linear | straight line |
| $y = x^2$ | parabola | U shape |
| $y = x^3$ | cubic | S shape through the origin |
| $y = \dfrac{k}{x}$ | hyperbola | two curves that never touch the axes |
| $y = 2^x$ | exponential | flat on the left, shoots up on the right |
| $x^2 + y^2 = r^2$ | circle | centre $(0, 0)$, radius $r$ |

*Diagram: Four small graphs: a cubic, a hyperbola, an exponential and a circle*
*Four shapes to know on sight.*

### Asymptotes

An **asymptote** is a line a graph gets closer and closer to but never reaches.

- $y = \dfrac{1}{x}$ has asymptotes $x = 0$ and $y = 0$, because you cannot divide by zero and $\dfrac{1}{x}$ is never zero.
- $y = 2^x$ has the asymptote $y = 0$ on the left.

### Hyperbolas and inverse variation

$y = \dfrac{k}{x}$ describes **inverse variation**: double $x$ and $y$ halves. If $k > 0$, the curves sit in the first and third quarters. If $k < 0$, they sit in the second and fourth.

### Worked example: Speed and time
A 120 km trip takes $t = \dfrac{120}{v}$ hours at speed $v$. At 60 km/h it takes 2 hours. At 120 km/h it takes 1 hour. Doubling the speed halves the time. That is a hyperbola.

### Circles

$x^2 + y^2 = r^2$ is a circle centred at the origin with radius $r$. So $x^2 + y^2 = 25$ has radius 5.

### Exam tip: Always check two points
To sketch, mark intercepts and one or two easy points. For $y = \dfrac{6}{x}$, the points $(1, 6)$, $(2, 3)$ and $(3, 2)$ give the curve quickly.

### Watch out: A circle is not a function
A circle fails the vertical line test. It is a relation. The top half $y = \sqrt{r^2 - x^2}$ on its own is a function.

**Mind map**

```text
Graph library
  Polynomials
    Line, parabola, cubic
    Roots from factors
  Hyperbola
    y = k/x
    Two asymptotes
  Exponential
    y = aˣ
    Asymptote y = 0
  Circle
    x² + y² = r²
    Centre (h, k) form
  Absolute value
    V shape
```

### Year 11: Circles with any centre
A circle with centre $(h, k)$ and radius $r$ is

$$(x - h)^2 + (y - k)^2 = r^2$$

If the equation is expanded, complete the square in $x$ and in $y$.

**Example.** $x^2 + 6x + y^2 - 4y = 3$ becomes $(x + 3)^2 - 9 + (y - 2)^2 - 4 = 3$, so $(x + 3)^2 + (y - 2)^2 = 16$. Centre $(-3, 2)$, radius 4.

### Year 11: Shifted hyperbolas and absolute value
$y = \dfrac{a}{x - b} + c$ is a hyperbola with vertical asymptote $x = b$ and horizontal asymptote $y = c$.

$y = |x|$ is a V shape with its corner at the origin, because $|x| = x$ when $x \geq 0$ and $|x| = -x$ when $x < 0$. The graph of $y = |x - 2| + 1$ has its corner at $(2, 1)$.

To solve $|2x - 1| = 5$, split it: $2x - 1 = 5$ or $2x - 1 = -5$, so $x = 3$ or $x = -2$.

**Horizontal asymptotes of fractions.** For a fraction of polynomials, look at large $x$. If the top and bottom have the same highest power, the asymptote is the ratio of their leading numbers: $y = \dfrac{2x^2 + x}{x^2 + 1}$ approaches $y = 2$. If the bottom has the higher power, the asymptote is $y = 0$. Vertical asymptotes are where the bottom is zero (and the top is not).

**Shifted exponentials.** $y = a \times 2^x + c$ has horizontal asymptote $y = c$ and $y$-intercept $a + c$. So an asymptote at $y = 1$ and $y$-intercept 4 give $c = 1$ and $a = 3$.

**Counting solutions with graphs.** To find how many solutions $x^2 = 2^x$ (or any hard equation) has, sketch $y = x^2$ and $y = 2^x$ on the same axes and count the crossings. Moving one side across the equals sign is fine: $e^x + x - 2 = 0$ has as many solutions as $e^x = 2 - x$ has crossings (here, one).

**Nearest and farthest points on a circle.** They lie on the line through the point and the centre. The origin is 5 units from the centre $(3, 4)$ of a circle of radius 2, so the nearest point is 3 units away and the farthest is 7 units away, both on the line $y = \frac{4}{3}x$.

### Year 12: Sketching polynomials from factors
For a polynomial in factored form, each factor gives a root.

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

- **Single** root, like $x = -2$ or $x = 3$: the graph cuts straight through.
- **Double** root, like $x = 1$: the graph touches the axis and turns back.
- **Triple** root: the graph flattens and passes through, like $x^3$ at the origin.

The **leading term** decides the ends. Here it is $x^4$, so both ends point up. Calculus then finds the exact turning points.

## Quick summary

- Know the shapes: line, parabola, cubic, hyperbola, exponential, circle.
- An asymptote is a line the graph approaches but never touches.
- $y = \dfrac{k}{x}$: inverse variation with asymptotes on both axes.
- $x^2 + y^2 = r^2$ is a circle of radius $r$ centred at the origin.
- Circle with centre $(h, k)$: $(x - h)^2 + (y - k)^2 = r^2$.

## Quiz

Try these questions from the chapter quiz.

1. Which equation gives an S-shaped curve through the origin?
   - A. $y = x^2$
   - B. $y = x^3$
   - C. $y = \dfrac{1}{x}$
   - D. $y = 2^x$
   - Answer: B. $y = x^3$. The basic cubic.

2. What is the radius of the circle $x^2 + y^2 = 49$?
   - A. 49
   - B. 24.5
   - C. 7
   - D. 14
   - Answer: C. 7. $r^2 = 49$, so $r = 7$.

3. What are the asymptotes of $y = \dfrac{4}{x}$?
   - A. $x = 4$ and $y = 0$
   - B. $x = 0$ and $y = 0$
   - C. $y = 4$ only
   - D. it has none
   - Answer: B. $x = 0$ and $y = 0$. You cannot divide by zero, and the output is never zero.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/graph-library/#quiz