Combining Transformations

Apply multiple transformations to functions in sequence. Grade 12 Math 30-1.

Lesson 1.8 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Transformation Summary. Review all transformation types: translations, stretches, compressions, and reflections—understand how each affects graphs.
• Combined Transformations. Learn how to apply multiple transformations together using y = a·f(b(x-h)) + k and understand the order of operations.
• Applying Combined Transformations. Practice transforming points and sketching graphs with multiple transformations applied simultaneously.

Inside the lesson — a free preview

Overview of All Transformations

This comprehensive table summarizes all transformation types we've learned: translations, reflections, stretches, and compressions.

Each transformation has a specific function form, point mapping rule, and invariant points that remain unchanged.

How to Use This Table

Quick Reference Guide.

Key Insights.

Important Reminders

Common Mistakes to Avoid.

The Complete General Form

y = a · f(b(x - h)) + k

This form contains all possible transformations:

Parameters.

Order of Transformations (SRT)

When describing transformations from y = a · f(b(x - h)) + k, it is IMPORTANT to follow the order:

SRT Order.

Within Each Category:.

Important. When you follow this order, you are working from left to right in the equation. Use the order SRT unless otherwise specified.

Mapping Notation

The mapping notation for transformations based on y = a · f(b(x - h)) + k is:

(x, y) → ((1)/(b) · x + h, a · y + k)

This shows how each point moves from the parent function to the transformed function.

How to Identify Each Parameter

Step 1: Look outside the function for a and k.

Step 2: Look inside the function for b and h.

What Each Parameter Does

a (vertical scaling & reflection):.

b (horizontal scaling & reflection):.

h (horizontal shift):.

k (vertical shift):.

Using Mapping Notation

When we know the transformation equation, we can use mapping notation to find where points move.

Mapping Formula. For y = a · f(b(x - h)) + k, points transform as:

(x, y) → ((1)/(b) · x + h, a · y + k)

How to Use It.

Worked example: Example 1

y = -2 · f(3(x - 4)) + 5

Identify:

a = -2 b = 3 h = 4 k = 5

Transformations (in SRT order):

S — Stretches:.

R — Reflections:.

T — Translations:.

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