# Transformations of functions

> IB Mathematics AA HL · Unit 2: Functions
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u2-transformations-of-functions/

This module covers all core function transformations for IB AA HL: translations, reflections, stretches, and combined transformations. You will learn how transformations affect the equation and graph of a function, and master the correct order for multiple transformations.

**Prerequisites:** [Function notation and domain/range](https://www.owlsprep.com/study/ib-math-aa-hl-u2-introduction-to-functions/); Graphing basic parent functions

## Learning objectives

- Identify and apply all core types of function transformations
- Distinguish between the effects of horizontal and vertical transformations
- Find the equation of a transformed function from a parent function
- Sketch transformed graphs and label key feature correctly
- Apply multiple transformations in the correct order

## Vertical and Horizontal Translations

**Translation** — A translation shifts every point on the graph vertically or horizontally by a constant distance, with no change to shape or orientation.

*Notation:* Vertical: $y = f(x) + a$, Horizontal: $y = f(x+a)$

**Worked example:** Given $f(x) = x^2$, write the equation of $f(x)$ translated 3 units up and 2 units to the left.

1. Identify the effect of the vertical translation: 3 units up adds 3 to the entire function, giving $y = f(x) + 3$.
2. Identify the effect of the horizontal translation: 2 units left means we replace $x$ with $x+2$, giving $y = f(x+2) + 3$.
3. Substitute $f(x) = x^2$ to get the final equation:
4. $$y = (x+2)^2 + 3$$

> **tip**
>
> Horizontal translations always have the opposite sign of what is written: $f(x+a)$ shifts left by $a$, $f(x-a)$ shifts right by $a$.

> **Exam tip:** Direction of horizontal translations is the most common exam mistake, always double-check this.

## Reflections Across Axes

**Reflection** — A reflection flips the graph across a line (usually the x-axis or y-axis) to create a mirror image, preserving shape.

*Notation:* Over x-axis: $y = -f(x)$, Over y-axis: $y = f(-x)$

**Worked example:** Find the coordinates of the y-intercept of $y = -e^x + 2$, starting from $y = e^x$ which has y-intercept $(0,1)$.

1. Reflection over the x-axis ($y=-e^x$) flips the sign of all y-coordinates, so the original intercept $(0,1)$ becomes $(0,-1)$.
2. A vertical translation 2 units up adds 2 to the y-coordinate, so:
3. $$(0, -1 + 2) = (0, 1)$$

**Check your understanding**

What is the equation of $f(x) = \sin x$ reflected over the y-axis?

1. Choose the correct equation

   - $y = -\sin x$
   - $y = \sin(-x)$
   - $y = \sin(x-1)$

   *Answer:* $y = \sin(-x)$

   *Why:* Correct! Reflection over the y-axis replaces $x$ with $-x$.

## Stretches and Compressions

**Stretch and Compression** — A transformation that resizes a graph by a constant scale factor. Scale factor >1 stretches, 0<scale factor<1 compresses.

*Notation:* Vertical stretch by $a$: $y = a f(x)$, Horizontal stretch by $a$: $y = f\left(\frac{x}{a}\right)$

**Worked example:** What is the horizontal scale factor for $g(x) = (2x)^3$ from parent $f(x) = x^3$?

1. Rewrite $g(x)$ to match the standard form for horizontal stretches: $g(x) = f(2x) = f\left(\frac{x}{a}\right)$.
2. Equate the coefficients of $x$: $\frac{1}{a} = 2$, so solve for $a$:
3. $$a = \frac{1}{2}$$

> **tip**
>
> Again, horizontal transformations are inverse: $f(kx)$ gives a horizontal compression by factor $\frac{1}{k}$ for $k>1$.

> **Exam tip:** Always confirm if the question asks for vertical or horizontal scale factor, they are often mixed up.

## Combined Transformations

When applying multiple transformations, order matters. The IB exam standard order follows BIDMAS: do stretches/reflections first, then translations, for transformations inside the function (horizontal) and outside (vertical).

**Worked example:** Find the equation of $f(x) = x^2$ after these transformations in order: 1. Stretch vertically by scale factor 2, 2. Translate 1 unit down, 3. Translate 3 units right.

1. Apply vertical stretch first: multiply $f(x)$ by 2, giving $y = 2x^2$.
2. Next translate 1 unit down: subtract 1 from the function, giving $y = 2x^2 - 1$.
3. Finally translate 3 units right: replace $x$ with $x-3$, giving the final equation:
4. $$y = 2(x-3)^2 - 1$$

**Exam command terms**

- **Show that** — You must show every transformation step to get full marks, even if you know the final answer

- **Sketch** — You must label all key points (intercepts, turning points, asymptotes) of the transformed graph

## Common pitfalls

- **Wrong:** Claiming $f(x+2)$ shifts the graph 2 units right
  - Why it fails: Confuses direction of horizontal transformations, which act directly on the input $x$
  - Correct: $f(x+a)$ shifts left by $a$ units for $a>0$, and right for $a<0$
- **Wrong:** Doing translations before stretches when combining transformations
  - Why it fails: Reverses the order of operations, leading to an incorrect final equation
  - Correct: Always do stretches/reflections first, then translations for both horizontal and vertical transformations
- **Wrong:** Claiming $f(2x)$ is a horizontal stretch by scale factor 2
  - Why it fails: Ignores the inverse relationship for horizontal transformation coefficients
  - Correct: $f(kx)$ for $k>1$ is a horizontal compression by scale factor $\frac{1}{k}$
- **Wrong:** Stretching only positive y-values for a vertical stretch
  - Why it fails: Assumes stretching does not affect points below the x-axis
  - Correct: Every y-coordinate is multiplied by the scale factor, so all points are stretched regardless of position
- **Wrong:** Reversing the order of inside and outside transformations
  - Why it fails: Applies vertical transformations before all horizontal transformations are complete
  - Correct: All transformations inside the function (on $x$) are applied first, then transformations outside the function (on $y$)

## Cheatsheet

| Transformation | Equation | Effect on Graph |
| --- | --- | --- |
| Vertical translation | $y = f(x) + a, a>0$ | Shift up $a$ units |
| Vertical translation | $y = f(x) + a, a<0$ | Shift down $\|a\|$ units |
| Horizontal translation | $y = f(x+a), a>0$ | Shift left $a$ units |
| Horizontal translation | $y = f(x+a), a<0$ | Shift right $\|a\|$ units |
| Reflection x-axis | $y = -f(x)$ | Flip over x-axis, reverse sign of y |
| Reflection y-axis | $y = f(-x)$ | Flip over y-axis, reverse sign of x |
| Vertical stretch | $y = a f(x), a>1$ | Stretch vertically by $a$ |
| Vertical compression | $y = a f(x), 0<a<1$ | Compress vertically by $a$ |
| Horizontal stretch | $y = f(\frac{x}{a}), a>1$ | Stretch horizontally by $a$ |
| Horizontal compression | $y = f(a x), a>1$ | Compress horizontally by $\frac{1}{a}$ |

## What's next

Mastering transformations of functions is a foundational skill used throughout the entire IB AA HL course. You will apply transformations to build complex functions from basic parent functions, and they are critical for graphing polynomials, trigonometric functions, exponentials, logarithms, and reciprocal functions. Transformations frequently appear as part of larger questions on both Paper 1 and Paper 2, so mastering the order of operations and direction of horizontal changes is essential for exam success.

- [Linear and quadratic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-linear-and-quadratic-functions/)
- [Polynomial and Rational Functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-polynomial-and-rational-functions/)
- [Exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-hl-u2-exponential-and-logarithmic-functions/)

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