Using Transformations to Sketch Graphs Revision Notes for VCE SSCE Mathematical Methods

Using Transformations to Sketch Graphs

Introduction

When faced with a complicated-looking function, we can often sketch its graph by viewing it as a transformed version of a simpler base function. By identifying the sequence of transformations applied to the base function, we can construct the graph step by step. This approach works for many types of functions, including rational functions, square root functions, and reciprocal functions.

infoNote

This method is particularly powerful because it allows you to build complex graphs from simple ones you already know well. Instead of plotting numerous points, you transform key features of the base function.

Identifying transformations algebraically

To find the transformations that map one function onto another, we can use an algebraic method. This involves rearranging the transformed function to match the form of the base function.

lightbulbExample

Worked Example: Transforming the Reciprocal Function

Question: Identify a sequence of transformations that maps the graph of $y = \frac{1}{x}$ onto the graph of $y = \frac{4}{x+5} - 3$ . Use this to sketch the graph, stating the equations of asymptotes and the coordinates of axis intercepts.

Solution:

First, we rearrange the transformed equation to have the same 'shape' as the base function $y = \frac{1}{x}$ :

\frac{y' + 3}{4} = \frac{1}{x' + 5}

where $(x', y')$ are the coordinates of the image of $(x, y)$ .

This gives us $x = x' + 5$ and $y = \frac{y' + 3}{4}$ .

Rearranging these equations:

$x' = x - 5$

$y' = 4y - 3$

The mapping is $(x, y) \to (x - 5, 4y - 3)$ .

From this mapping, we can identify the sequence of transformations:

A dilation of factor 4 from the x-axis
A translation of 5 units in the negative direction of the x-axis
A translation of 3 units in the negative direction of the y-axis

Applying the transformations

We start with the base function $y = \frac{1}{x}$ , which passes through the points $(1, 1)$ and $(-1, -1)$ .

Step 1: Dilation from the x-axis by factor 4

This multiplies all y-coordinates by 4:

$(1, 1) \to (1, 4)$

$(-1, -1) \to (-1, -4)$

Step 2: Translation 5 units in negative x-direction

This shifts the graph $5$ units to the left:

$(1, 4) \to (-4, 4)$

$(-1, -4) \to (-6, -4)$

The vertical asymptote moves from $x = 0$ to $x = -5$ .

Step 3: Translation 3 units in negative y-direction

This shifts the graph $3$ units down:

$(-4, 4) \to (-4, 1)$

$(-6, -4) \to (-6, -7)$

The horizontal asymptote moves from $y = 0$ to $y = -3$ .

chatImportant

When applying multiple transformations, track both the key points AND the asymptotes through each step. This ensures your final graph is accurate.

Finding axis intercepts

To complete our sketch, we need to find where the graph crosses the axes.

y-intercept (when $x = 0$ ):

y = \frac{4}{0 + 5} - 3 = \frac{4}{5} - 3 = -2\frac{1}{5}

The $y$ -intercept is at $(0, -2\frac{1}{5})$ .

x-intercept (when $y = 0$ ):

\frac{4}{x + 5} - 3 = 0

\frac{4}{x + 5} = 3

4 = 3(x + 5)

4 = 3x + 15

3x = -11

x = -\frac{11}{3}

The $x$ -intercept is at $\left(-\frac{11}{3}, 0\right)$ .

Final graph

The transformed graph has:

Vertical asymptote: $x = -5$
Horizontal asymptote: $y = -3$
x-intercept: $\left(-\frac{11}{3}, 0\right)$
y-intercept: $\left(0, -2\frac{1}{5}\right)$

Quick identification of transformations

Once you understand the basic process, you can identify transformations more quickly by carefully observing the rule of the transformed function and relating it to the base function in its family. The following examples demonstrate this quicker approach.

infoNote

With practice, you can identify transformations directly from the function's form without needing to go through the full algebraic rearrangement. Look for patterns in how the function differs from its base form.

lightbulbExample

Worked Example: Square Root Function

Question: Sketch the graph of $y = -\sqrt{x - 4} - 5$ .

Solution:

This is a transformation of the base function $y = \sqrt{x}$ .

The transformations are:

A reflection in the x-axis (due to the negative sign in front of the square root)
A translation of 5 units in the negative direction of the y-axis (the $-5$ at the end)
A translation of 4 units in the positive direction of the x-axis (the $-4$ inside the square root means we shift right)

The graph starts at the point (4, -5) and extends to the right, curving downward.

lightbulbExample

Worked Example: Reciprocal Squared Function

Question: Sketch the graph of $y = \frac{3}{(x-2)^2} + 5$ .

Solution:

This is a transformation of the base function $y = \frac{1}{x^2}$ .

The transformations are:

A dilation of factor 3 from the x-axis (the numerator $3$ multiplies the function)
A translation of 5 units in the positive direction of the y-axis (the $+5$ )
A translation of 2 units in the positive direction of the x-axis (the $-2$ in the denominator means we shift right)

The graph has:

Vertical asymptote: $x = 2$
Horizontal asymptote: $y = 5$
y-intercept: $\left(0, 5\frac{3}{4}\right)$ (found by substituting $x = 0$ : $y = \frac{3}{4} + 5 = 5\frac{3}{4}$ )

chatImportant

Common Mistake Alert!

When you see $(x - h)$ inside a function, the graph shifts h units to the RIGHT, not left. Similarly, $(x + h)$ shifts the graph h units to the LEFT. The direction is opposite to the sign!

bookmarkSummary

Key Points to Remember:

To sketch a transformed function, identify the base function first, then determine the sequence of transformations applied to it.
Use algebraic rearrangement to find the mapping $(x, y) \to (x', y')$ , which reveals the transformations clearly.
Apply transformations in order: typically dilations first, then horizontal translations, then vertical translations.
For rational functions in the form $y = \frac{a}{x - h} + k$ , the vertical asymptote is at $x = h$ and the horizontal asymptote is at $y = k$ .
Always calculate and mark axis intercepts and asymptotes on your sketch to make it complete and accurate.

Using Transformations to Sketch Graphs (VCE SSCE Mathematical Methods): Revision Notes