Reciprocal Functions and Models Revision Notes for HSC SSCE Mathematics Standard

Reciprocal Functions and Models

Introduction to reciprocal functions

A reciprocal function is a type of mathematical function where the variable appears in the denominator. These functions create curved graphs called hyperbolas.

The general form of a reciprocal function is:

$y = \frac{k}{x}$

where $k$ is a constant number.

Common examples include $y = \frac{1}{x}$ , $y = \frac{2}{x}$ , and $y = -\frac{1}{x}$ .

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In reciprocal functions, as $x$ increases, $y$ decreases (or vice versa), showing an inverse relationship between the variables. This is the defining characteristic that makes reciprocal functions useful for modeling real-world situations where quantities work in opposite directions.

Key features of a hyperbola

When you graph a reciprocal function, you create a curve called a hyperbola. Hyperbolas have several distinctive characteristics that make them easy to identify.

Undefined at x = 0

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No value exists for $y$ when $x = 0$ because you cannot divide by zero. This creates a gap in the graph at the $y$ -axis.

This is a common source of errors - always remember that reciprocal functions are undefined at $x = 0$ .

Two branches

The curve has two separate parts called branches. Each branch is the same shape and size, making them symmetrical. The branches appear in opposite quadrants of the coordinate plane.

For $y = \frac{1}{x}$ : The branches appear in the first quadrant (top right) and third quadrant (bottom left).

For $y = -\frac{1}{x}$ : The branches appear in the second quadrant (top left) and fourth quadrant (bottom right).

Asymptotes

Asymptotes are lines that the curve approaches but never actually touches. For basic reciprocal functions, the asymptotes are:

The $x$ -axis (horizontal asymptote): $y = 0$
The $y$ -axis (vertical asymptote): $x = 0$

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The curve gets closer and closer to these axes but will never reach them, no matter how far you extend the graph. Think of asymptotes as invisible boundaries that guide the shape of the hyperbola.

Rectangular hyperbola

Because the asymptotes are at right angles to each other (the $x$ -axis and $y$ -axis are perpendicular), the curve is also called a rectangular hyperbola.

How to graph a hyperbola

To create an accurate graph of a reciprocal function, follow these steps:

Construct a table of values - Choose several $x$ values (both positive and negative) and calculate the corresponding $y$ values using the function
Draw a number plane - Set up your coordinate axes with appropriate scales
Plot the points - Mark each coordinate pair from your table on the graph
Join the points - Draw smooth curves through the points to form the two branches of the hyperbola

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Remember not to connect points across the $y$ -axis, as the function is undefined at $x = 0$ . The two branches must remain separate.

Worked example: graphing y = 2/x

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Worked Example: Graphing $y = \frac{2}{x}$

Let's graph the reciprocal function $y = \frac{2}{x}$ .

Step 1: Create a table of values

Choose $x$ values: $-4, -2, -1, -0.5, 0.5, 1, 2, 4$

Calculate $y$ values using $y = \frac{2}{x}$ :

$x$	$-4$	$-2$	$-1$	$-0.5$	$0.5$	$1$	$2$	$4$
$y$	$-0.5$	$-1$	$-2$	$-4$	$4$	$2$	$1$	$0.5$

Step 2: Draw the coordinate plane

Set up your axes with $x$ on the horizontal axis and $y$ on the vertical axis.

Step 3: Plot the points

Mark these coordinates: $(-4, -0.5)$ , $(-2, -1)$ , $(-1, -2)$ , $(-0.5, -4)$ , $(0.5, 4)$ , $(1, 2)$ , $(2, 1)$ , $(4, 0.5)$

Step 4: Join the points

Notice that no value exists when $x = 0$ , which creates two separate branches. Draw smooth curves through the points in each quadrant.

Worked example: graphing y = -2/x and identifying asymptotes

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Worked Example: Graphing $y = -\frac{2}{x}$ and Identifying Asymptotes

Now let's graph $y = -\frac{2}{x}$ and identify its asymptotes.

Step 1: Create a table of values

Choose $x$ values: $-4, -2, -1, -0.5, 0.5, 1, 2, 4$

Calculate $y$ values using $y = -\frac{2}{x}$ :

$x$	$-4$	$-2$	$-1$	$-0.5$	$0.5$	$1$	$2$	$4$
$y$	$0.5$	$1$	$2$	$4$	$-4$	$-2$	$-1$	$-0.5$

Step 2: Plot and draw the graph

Following the same process as before, plot the points and join them to form two branches.

Step 3: Identify the asymptotes

The curve approaches both the $x$ -axis and $y$ -axis but never touches them.

Asymptotes are: $x = 0$ and $y = 0$

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Exam Tip: When asked to identify asymptotes, always write them as equations (e.g., $x = 0$ and $y = 0$ ), not just as "the axes" or "horizontal and vertical lines."

Reciprocal models

Reciprocal modelling occurs when we use a reciprocal function to describe a real-world situation mathematically. These models are particularly useful when quantities have an inverse relationship - as one increases, the other decreases.

A reciprocal model has the form:

$y = \frac{k}{x}$

where $k$ is a constant and the variables represent practical quantities like time, speed, cost, or number of people.

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Reciprocal models typically show fast growth or decay and are useful for situations involving:

Speed and time relationships
Sharing costs among people
Concentration and dilution
Intensity and distance

Worked example: road trip time and speed model

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Worked Example: Road Trip Time and Speed Model

The time taken ( $t$ ) in hours for a road trip at speed ( $s$ ) in km/h is given by the reciprocal function:

$t = \frac{2000}{s}$

a) Construct a table of values for $s$ and $t$

Choose appropriate speed values: $s = 10, 20, 40, 50, 60, 80, 100$

Calculate time values using $t = \frac{2000}{s}$ :

$s$	$10$	$20$	$40$	$50$	$60$	$80$	$100$
$t$	$200$	$100$	$50$	$40$	$33$	$25$	$20$

b) Draw the graph of $t = \frac{2000}{s}$

Set up axes with $s$ (speed) on the horizontal axis
Place $t$ (time) on the vertical axis
Plot the coordinate pairs from the table
Join the points to form one branch of a hyperbola

c) How long did the road trip take at a speed of 70 km/h?

Read the graph at $s = 70$ . The corresponding $t$ value is approximately $30$ hours.

Answer: Approximately 30 hours

d) Why is it impossible to complete the road trip in 10 hours?

To find the required speed:

Substitute $t = 10$ into the equation
$10 = \frac{2000}{s}$
Solve for $s$ : $s = \frac{2000}{10} = 200$ km/h

Answer: The speed required to complete the trip in 10 hours is 200 km/h, which is above the speed limit on Australian roads, making it impossible to complete legally.

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Learning Point: This example shows how reciprocal models can help us understand practical limitations in real-world situations. Always check whether your mathematical answer makes sense in the real-world context.

Worked example: pizza cost sharing model

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Worked Example: Pizza Cost Sharing Model

The cost per person ( $C$ ) in dollars when sharing a pizza depends on the number of people ( $n$ ) eating it. The reciprocal equation is:

$C = \frac{24}{n}$

a) Describe the possible values for $n$

Since $n$ represents the number of people sharing a pizza, it must be:

A positive whole number (you can't have negative or fractional people)
Likely less than 10 (for a typical pizza sharing scenario)

Answer: $n$ is a positive whole number, likely less than 10

b) Construct a table of values for $n$ and $C$

Choose values: $n = 1, 2, 3, 4, 5, 6, 7, 8$

Calculate $C$ using $C = \frac{24}{n}$ :

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$C$	$24$	$12$	$8$	$6$	$4.8$	$4$	$3.4$	$3$

c) Draw the graph of $C = \frac{24}{n}$

Set up axes with $n$ (number of people) on the horizontal axis
Place $C$ (cost per person) on the vertical axis
Plot the points from the table
Join them to form one branch of a hyperbola

d) What is the cost per person if six people are sharing a pizza?

Read the value from the table or graph when $n = 6$ .

Answer: Cost per person is $4

e) How many people shared a pizza if the cost was $2.40 per person?

Substitute $C = 2.4$ into the equation: $2.4 = \frac{24}{n}$
Rearrange to solve for $n$ :

$n = \frac{24}{2.4}$

Calculate: $n = 10$
Check this is reasonable (10 people sharing one pizza is possible)

Answer: The number of people sharing the pizza was 10

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Exam Tip: When solving reciprocal equations, remember to rearrange the formula by multiplying both sides by the denominator variable, then dividing to isolate the variable you're solving for.

Summary

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Key Points to Remember:

A reciprocal function has the form $y = \frac{k}{x}$ where $k$ is a constant
Reciprocal functions create hyperbolas with two branches in opposite quadrants
The graph is undefined at $x = 0$ because you cannot divide by zero
Asymptotes are the $x$ -axis ( $y = 0$ ) and $y$ -axis ( $x = 0$ ) - the curve approaches but never touches them
Reciprocal models describe real-world inverse relationships where as one quantity increases, the other decreases
To graph a hyperbola: create a table of values, plot points, and join them with smooth curves (keeping branches separate)
In practical problems, always check that your answer makes sense in the real-world context

Reciprocal Functions and Models (HSC SSCE Mathematics Standard): Revision Notes