Translations and Dilations

Expert reviewed • 08 January 2025 • 7 minute read

HSC Maths Advanced Syllabus

apply transformations to sketch functions of the form 𝑦 = 𝑘𝑓(𝑎(𝑥 +𝑏))+𝑐, where 𝑓(𝑥) is a polynomial, reciprocal, absolute value, exponential or logarithmic function and 𝑎, 𝑏, 𝑐 and 𝑘 are constants
- examine translations and the graphs of 𝑦 = 𝑓(𝑥)+𝑐 and 𝑦 = 𝑓(𝑥 +𝑏) using technology
- examine dilations and the graphs of 𝑦 = 𝑘𝑓(𝑥) and 𝑦 = 𝑓(𝑎𝑥) using technology
- recognise that the order in which transformations are applied is important in the construction of the resulting function or graph

Note:

Video coming soon!

Translations and Reflections Review

Before we learn about dilations, it is important to review other graphical transformations.

To reflect a graph in the x-axis, a negative value is applied to the y-component of the function

y=-f(x)

To reflect a graph in the y-axis, a negative value is applied to the x-component of the function

y=f(-x)

To shift a graph vertically, the function changes to the form:

y=f(x)+k

It is important to note that a positive value of $k$ , will result in a shift upward, while a negative value will shift the graph downward.

To shift a graph horizontally, the function changes to the form:

y=f(x-h)

A positive value of $h$ , will result in a shift to the right, while a negative value will shift the graph to the left.

Vertical Dilations

Vertical dilations are stretches or compressions that change the vertical shape of a graph. To stretch a graph by a dilation factor of $a$ , the new function formula becomes:

y=af(x) \qquad or \qquad \frac{y}{a}=f(x)

when $a>1$ the graph stretches vertically, meaning that when $a<1$ the graph is compressed.

Horizontal Dilations

Horizontal dilations are stretches or compressions that alter the horizontal shape of a graph. To stretch a graph by a dilation factor of $b$ , the new function formula becomes:

y=f(\frac{x}{b})\qquad

when $b>1$ the graph stretches horizontally, meaning that when $b<1$ the graph is compressed.

Now it is important to note, that when a graph is dilated by a negative factor Eg. $-a$ , the graph is reflected in either the x or y-axis (depending on the dilation), by a factor of $a$ .

Practice Question 1

What are the vertical and horizontal dilations factors in the following equation: $(2x)^2+(\frac{y}{4})^2=1$ , from the original graph of $x^2+y^2=1$

Applying the formula for horizontal dilations:

HD=\frac{x}{b}\\=2x\\=\frac{x}{\frac{1}{2}}\\b=\frac{1}{2}

Applying the formula for vertical dilations:

VD=\frac{y}{a}\\=\frac{y}{4}\\a=4

$\therefore$ The function has a horizontal dilation factor of $\frac{1}{2}$ and a vertical dilation factor of 4

The Order of Transformations

The order in which transformations are applied to functions and their graphs is crucial. This is because each transformation can affect the subsequent ones, leading to different final results. Thus, when graph sketching, transformations must be completed in the following order:

Vertical translations
Horizontal translations
Vertical dilations
Horizontal dilations

Graphing: Key Concepts and Revision

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Graph Sketching Made Easy

Return to Module 2: Graphs and Equations