Reciprocal and cubic graphs

Lesson overview

Algebra — non-linear graphs

**Reciprocal graphs** have the form $y = \dfrac{k}{x}$ where $k$ is a constant. They have two **asymptotes**: the $x$ -axis ( $y = 0$ ) and the $y$ -axis ( $x = 0$ ). The curve approaches but never touches these lines. **Cubic graphs** have the form $y = ax^3 + bx^2 + cx + d$ where the highest power is $3$ . They can have up to two turning points and always extend from bottom-left to top-right (or vice versa). Key features to identify: - **Asymptotes** — lines the graph approaches but never reaches - **Turning points** — where the graph changes direction - **Roots** — where the graph crosses the $x$ -axis ( $y = 0$ )

Reciprocal graphs have two asymptotes

The graph of $y = \dfrac{1}{x}$ has asymptotes at $x = 0$ and $y = 0$ . As $x$ gets very large, $y$ approaches $0$ .

Vertical asymptote $x = 0$

the y-axis

Horizontal asymptote $y = 0$

the x-axis

The graph gets closer and closer to the axes but never touches them.

Cubic graphs have an S-shape

The graph of $y = x^3$ passes through the origin and extends from bottom-left to top-right.

Cubic graphs always have one continuous curve — they never have asymptotes.

Changing $k$ affects the reciprocal graph

The graph of $y = \dfrac{2}{x}$ is further from the origin than $y = \dfrac{1}{x}$ .

k = 1

$y = \dfrac{1}{x}$

closer to axes

k = 2

$y = \dfrac{2}{x}$

further out

k = -1

$y = -\dfrac{1}{x}$

reflected

Larger $|k|$ pushes the curve further from the origin. Negative $k$ reflects it.

Worked example

Sketch the graph of $y = \dfrac{3}{x}$ and state the equations of the asymptotes.

This is a reciprocal graph with $k = 3$ .
The asymptotes are $x = 0$ (the $y$ -axis) and $y = 0$ (the $x$ -axis).
Plot key points: when $x = 1$ , $y = 3$ ; when $x = 3$ , $y = 1$ ; when $x = -1$ , $y = -3$ ; when $x = -3$ , $y = -1$ .
Draw two smooth curves in the first and third quadrants, approaching but never touching the axes.

Try it

Identify the type of graph first (reciprocal or cubic), then look for key features.

Question 1

What are the asymptotes of the graph $y = \dfrac{5}{x}$ ?

Question 2

Which of these is a cubic function?

Question 3

The graph of $y = \dfrac{-2}{x}$ is a reflection of $y = \dfrac{2}{x}$ in which line?

Question 4

How many turning points can a cubic graph have at most?

Question 5

If $y = \dfrac{k}{x}$ and the point $(2, 3)$ lies on the graph, what is $k$ ?

Common mistakes

Watch for these when working through the lesson.

Drawing reciprocal graphs that touch or cross the asymptotes.
Forgetting that reciprocal graphs have two separate branches (not one continuous curve).
Confusing cubic graphs with quadratic graphs — cubics can have two turning points and always extend in opposite directions.

Reciprocal and cubic graphs

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here