Multiple Transformations Revision Notes for HSC SSCE Mathematics Advanced

Multiple Transformations

What are multiple transformations?

When we work with functions, we often need to apply more than one transformation to a parent function. Multiple transformations combine different types of changes—translations (shifts), reflections (flips), and dilations (stretches or compressions)—to create a new, more complex function. Understanding how to apply these transformations in the correct order is essential for finding the equation of the transformed function and sketching its graph accurately.

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Multiple transformations allow us to create complex functions from simple parent functions by applying a sequence of geometric changes. Each transformation modifies the function in a specific way, and combining them creates powerful tools for modeling real-world situations.

The importance of order

When applying multiple transformations, the order in which you apply them makes a significant difference to the final result. Always follow the order specified in the question.

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Key rule: If no specific order is given, apply dilations and reflections first, then apply translations.

This default order ensures consistency and helps avoid errors when working with transformed functions.

Worked example 1: Transforming a reciprocal function

Let's transform the reciprocal function $f(x) = \frac{1}{x}$ by applying the following sequence:

Reflect over the y-axis
Dilate vertically by a factor of 2
Translate left by 3 units
Translate up by 1 unit

Finding the equation of the transformed function

We need to apply each transformation step by step, following the given order.

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Worked Example: Applying Multiple Transformations to a Reciprocal Function

Step 1: Write the parent function

$f(x) = \frac{1}{x}$

Step 2: Reflect over the y-axis (replace $x$ with $-x$ )

$f(-x) = \frac{1}{-x}$

Step 3: Dilate vertically by 2 (multiply the function by 2)

$2f(-x) = 2\left(\frac{1}{-x}\right)$

Step 4: Translate left by 3 (replace $x$ with $x + 3$ )

$2f(-(x + 3)) = \frac{2}{-(x + 3)} = \frac{-2}{x + 3}$

Step 5: Translate up by 1 (add 1 to the entire function)

$2f(-(x + 3)) + 1 = \frac{-2}{x + 3} + 1$

Step 6: Simplify the expression

$= -\frac{2}{x + 3} + 1$

The transformed function is:

$g(x) = -\frac{2}{x + 3} + 1$

Finding the vertical asymptote

A vertical asymptote occurs where the denominator of a rational function equals zero, as the function becomes undefined at these points.

For $g(x) = -\frac{2}{x + 3} + 1$ , we need to find where $x + 3 = 0$

x + 3 = 0 x = -3

The vertical asymptote is at $x = -3$ .

Finding the horizontal asymptote

For reciprocal functions, the horizontal asymptote is determined by the constant term that was added or subtracted during vertical translation.

The parent function $f(x) = \frac{1}{x}$ has a horizontal asymptote at $y = 0$ .

Since we translated the function up by 1 unit, the horizontal asymptote moves from $y = 0$ to:

$y = 0 + 1 = 1$

The horizontal asymptote is at $y = 1$ .

Determining the domain and range

Domain: The domain includes all real numbers except where the denominator equals zero (the vertical asymptote).

Domain: $x \neq -3$ , or in interval notation: $(-\infty, -3) \cup (-3, \infty)$

Range: The range includes all real numbers except the horizontal asymptote value.

Range: $y \neq 1$ , or in interval notation: $(-\infty, 1) \cup (1, \infty)$

Sketching the transformed graph

To sketch the graph accurately, we need to find the intercepts and plot key points.

Finding the y-intercept (substitute $x = 0$ ):

g(x) = -\frac{2}{x + 3} + 1 g(0) = -\frac{2}{0 + 3} + 1 = -\frac{2}{3} + 1 = \frac{1}{3}

The y-intercept is at $\left(0, \frac{1}{3}\right)$ .

Finding the x-intercept (set $g(x) = 0$ ):

g(x) = -\frac{2}{x + 3} + 1 0 = -\frac{2}{x + 3} + 1 -1 = -\frac{2}{x + 3} x + 3 = 2 x = -1

The x-intercept is at $(-1, 0)$ .

Finding an additional point: To help sketch the branch that doesn't intersect the x-axis or y-axis, we can substitute $x = -4$ :

g(-4) = -\frac{2}{-4 + 3} + 1 = -\frac{2}{-1} + 1 = 2 + 1 = 3

Another point on the graph is $(-4, 3)$ .

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The complete graph shows:

A hyperbola with two branches
Vertical asymptote: $x = -3$ (shown as a vertical dashed line)
Horizontal asymptote: $y = 1$ (shown as a horizontal dashed line)
Key points: $(-4, 3)$ , $(-1, 0)$ , and $\left(0, \frac{1}{3}\right)$
One branch curves from the upper left towards both asymptotes
The other branch curves from the lower right towards both asymptotes

The asymptotes have shifted from their original positions at $x = 0$ and $y = 0$ to $x = -3$ and $y = 1$ , confirming that our transformations were applied correctly.

Worked example 2: Transforming a quadratic function

Let's transform the quadratic function $f(x) = x^2$ by applying the following sequence:

Dilate horizontally by a factor of $\frac{1}{2}$
Reflect over the x-axis
Translate right by 2 units
Translate down by 1 unit

Finding the equation of the transformed function

We apply each transformation in the given order.

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Worked Example: Applying Multiple Transformations to a Quadratic Function

Step 1: Write the parent function

$f(x) = x^2$

Step 2: Dilate horizontally by $\frac{1}{2}$ (replace $x$ with $\frac{x}{\frac{1}{2}} = 2x$ )

Note: A horizontal dilation by factor $\frac{1}{2}$ means we replace $x$ with $\frac{x}{\frac{1}{2}}$ , which makes the graph wider.

f\left(\frac{x}{\frac{1}{2}}\right) = f(2x) = (2x)^2 = 4x^2 f(2x) = 4x^2

Step 3: Reflect over the x-axis (multiply by -1)

$-f(2x) = -4x^2$

Step 4: Translate right by 2 (replace $x$ with $x - 2$ )

$-f(2(x - 2)) = -4(x - 2)^2$

Step 5: Translate down by 1 (subtract 1 from the function)

$-f(2(x - 2)) - 1 = -4(x - 2)^2 - 1$

The transformed function is:

$g(x) = -4(x - 2)^2 - 1$

Finding the vertex

The transformed function is now in vertex form: $y = a(x - h)^2 + k$

In this form, the vertex is located at the point $(h, k)$ .

For $g(x) = -4(x - 2)^2 - 1$ :

$h = 2$ (remember the sign is opposite in the formula)
$k = -1$

The vertex is at $(2, -1)$ .

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In vertex form $y = a(x - h)^2 + k$ , the signs can be tricky! The vertex is at $(h, k)$ , but notice that in the equation, we have $(x - h)$ , not $(x + h)$ . So if you see $(x - 2)^2$ , then $h = 2$ (positive), not -2.

Determining the range

The range of a parabola depends on two factors:

The sign of $a$ (which determines whether the parabola opens upward or downward)
The value of $k$ (the y-coordinate of the vertex)

For $g(x) = -4(x - 2)^2 - 1$ :

$a = -4$ , which is negative
Since $a < 0$ , the parabola is concave down (opens downward)
The vertex represents the maximum point

Therefore, the range is: $(-\infty, -1)$ or $y \leq -1$

Sketching the transformed graph

To sketch an accurate graph, we need to find intercepts and additional points.

Finding the y-intercept (substitute $x = 0$ ):

g(x) = -4(x - 2)^2 - 1 g(0) = -4(0 - 2)^2 - 1 = -4 \times 4 - 1 = -16 - 1 = -17

The y-intercept is at $(0, -17)$ .

Finding the x-intercept (set $g(x) = 0$ ):

g(x) = -4(x - 2)^2 - 1 0 = -4(x - 2)^2 - 1 4(x - 2)^2 = -1 (x - 2)^2 = -\frac{1}{4}

Since $(x - 2)^2 \geq 0$ for all real values of $x$ , this equation has no solution.

There are no x-intercepts.

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The fact that there are no x-intercepts makes sense because the parabola opens downward with its maximum point (vertex) at $(2, -1)$ , which is below the x-axis. The entire parabola lies below the x-axis, so it never crosses it.

Finding an additional point: Since there's only one intercept, we need another point to help sketch the parabola accurately. Let's try $x = 3$ :

g(x) = -4(x - 2)^2 - 1 g(3) = -4(3 - 2)^2 - 1 = -4(1)^2 - 1 = -4 - 1 = -5

Another point on the graph is $(3, -5)$ .

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The complete graph shows:

An inverted (concave down) parabola
Vertex at $(2, -1)$ (the maximum point)
y-intercept at $(0, -17)$
No x-intercepts
Additional point at $(3, -5)$
The parabola is wider than the parent function $y = x^2$ due to the horizontal dilation

The vertex has shifted from the origin $(0, 0)$ to $(2, -1)$ , and the graph is now inverted and wider, confirming all our transformations have been applied correctly.

Remember!

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Key Points to Remember:

Order matters: Always apply transformations in the order specified. If no order is given, apply dilations and reflections before translations.
Build step by step: Apply one transformation at a time and write out each step clearly to avoid errors.
Check key features: After finding the transformed equation, verify your work by checking asymptotes (for rational functions), vertices (for quadratics), intercepts, domain, and range.
Sketch with landmarks: When drawing the transformed graph, always plot asymptotes (for rational functions), vertices (for quadratics), intercepts, and at least one additional point to ensure accuracy.
Connect to the parent function: Understanding how the transformations affect the parent function's key features helps you predict what the transformed graph should look like.

Multiple Transformations (HSC SSCE Mathematics Advanced): Revision Notes