# Transformations of function graphs

> IB Mathematics AI SL · IB Math AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u2-transformations-of-function-graphs/

This module covers all four core types of function graph transformations tested in IB AI SL: translations, reflections, vertical and horizontal dilations. You will learn to relate algebraic changes to a function to geometric changes to its graph.

**Prerequisites:** [Basic function notation and graphing of elementary functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-introduction-to-functions/)

## Learning objectives

- Identify all core types of function graph transformations
- Apply transformations to find equations of transformed graphs
- Relate algebraic changes to geometric changes in graphs
- Correctly apply order of operations for combined transformations

## Vertical and Horizontal Translations

**Translation** — A rigid transformation that shifts every point on the graph of a function by a fixed distance, with no change to shape or size of the graph.

Translations follow consistent rules based on whether the constant is added inside or outside the function argument.

- For original function $y = f(x)$, a vertical translation by $a$ units is $y = f(x) + a$. Positive $a$ shifts up, negative $a$ shifts down.
- A horizontal translation by $b$ units is $y = f(x - b)$. Positive $b$ shifts right, negative $b$ shifts left.

> **warning**
>
> Horizontal translations are counter-intuitive: $f(x + 3)$ shifts 3 units left, not right. Always remember: $f(x - b)$ shifts towards positive $x$ for positive $b$.

**Worked example:** Original function is $f(x) = x^2$. Write the equation of $f(x)$ translated 2 units down and 4 units right.

1. Start with the original function:

   $$f(x) = x^2$$
2. Apply horizontal translation 4 units right: replace $x$ with $x - 4$:

   $$y = f(x - 4) = (x - 4)^2$$
3. Apply vertical translation 2 units down: add -2 outside the function:

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

> **Exam tip:** Always confirm your translation by checking a key point (like a vertex) to confirm shift direction.

## Reflections Across the Axes

**Reflection** — A transformation that flips the graph across a given line to create a mirror image. Shape is unchanged, but orientation is reversed.

Reflections across the x-axis and y-axis follow simple rules based on where the negative sign is placed.

- Reflection across the x-axis: $y = -f(x)$: every y-coordinate is multiplied by -1.
- Reflection across the y-axis: $y = f(-x)$: every x-coordinate is multiplied by -1.

**Worked example:** Original $f(x) = e^x$ has y-intercept at $(0, 1)$. Find the y-intercept of $y = -f(x)$ reflected across the x-axis.

1. Apply the reflection rule: all y-coordinates are multiplied by -1.
2. Original y-intercept is $(0, 1)$, so multiply the y-coordinate by -1:

   $$(0, 1 \times -1) = (0, -1)$$

## Vertical and Horizontal Dilations

**Dilation** — A non-rigid transformation that stretches or compresses a graph by a scale factor, changing size but not overall shape.

The location of the scale factor (inside or outside the function) tells you if the dilation is horizontal or vertical, just like translations.

- Vertical dilation by scale factor $k$: $y = k f(x)$, all y-coordinates multiplied by $k$.
- Horizontal dilation by scale factor $\frac{1}{c}$: $y = f(cx)$, all x-coordinates multiplied by $\frac{1}{c}$.

> **warning**
>
> Horizontal dilations are also counter-intuitive: $f(2x)$ compresses the graph horizontally by a factor of 2, not stretches it.

**Worked example:** Given $f(x) = \sin x$, find the equation after a vertical stretch by scale factor 3 and horizontal compression by scale factor 2.

1. Apply vertical stretch first: multiply the entire function by 3:

   $$y = 3 f(x) = 3 \sin x$$
2. Apply horizontal compression by 2: replace $x$ with $2x$:

   $$y = 3 \sin(2x)$$
3. Verify: original period of $\sin x$ is $2\pi$, new period is $\frac{2\pi}{2} = \pi$, which matches the compression.

> **Exam tip:** Dilations from the x-axis are vertical, dilations from the y-axis are horizontal.

## Combined Transformations

When multiple transformations are applied, order matters. We follow an order consistent with standard order of operations (BIDMAS/BODMAS):

1. First complete all horizontal transformations (inside $f(...)$): do dilations/reflections first, then translations.
2. Then complete all vertical transformations (outside $f(...)$): do dilations/reflections first, then translations.

**Worked example:** Starting with $f(x) = x^2$, write the equation after: reflection across x-axis, horizontal shift 1 unit left, vertical shift 3 units up.

1. Start with original function:

   $$y = x^2$$
2. Apply horizontal transformation: shift 1 unit left: replace $x$ with $x + 1$:

   $$y = (x + 1)^2$$
3. Apply vertical transformation: reflection across x-axis, multiply by -1:

   $$y = -(x + 1)^2$$
4. Apply vertical translation 3 units up: add 3:

   $$y = -(x + 1)^2 + 3$$
5. Check: original vertex at $(0,0)$, new vertex at $(-1, 3)$, which matches the required transformations.

**Check your understanding**

Test your understanding of order:

1. What is the correct order to get $y = 2f(x + 3) - 1$ from $y = f(x)$?

   - A: Shift left 3, vertical stretch by 2, shift down 1
   - B: Vertical stretch by 2, shift left 3, shift down 1
   - C: Shift down 1, vertical stretch by 2, shift left 3

   *Why:* Correct! Horizontal transformations are done first, then vertical transformations with dilations before translations.

## Common pitfalls

- **Wrong:** Writing $f(x + 2)$ as a shift 2 units right
  - Why it fails: The rule for horizontal translation is $f(x - b)$ for shift $b$, so the sign is reversed
  - Correct: Recognize $f(x + c)$ shifts $c$ units left, $f(x - c)$ shifts $c$ units right for positive $c$
- **Wrong:** Stretching $f(2x)$ horizontally by factor 2
  - Why it fails: The scale factor for horizontal dilations is inverted
  - Correct: $f(cx)$ dilates horizontally by factor $\frac{1}{c}$, so $f(2x)$ compresses horizontally by factor 2
- **Wrong:** Doing vertical translation before vertical dilation
  - Why it fails: Order of operations requires multiplication (dilation) before addition (translation)
  - Correct: Always apply dilations and reflections before translations for vertical transformations
- **Wrong:** Interpreting $f(-x)$ as a reflection across the x-axis
  - Why it fails: The negative sign is inside the function argument, so it affects x-coordinates not y-coordinates
  - Correct: Negative outside $(-f(x))$ = x-axis reflection; negative inside $(f(-x))$ = y-axis reflection
- **Wrong:** Interpreting $y = f(x) + a$ as a horizontal shift left by $a$
  - Why it fails: Confusing the position of the constant inside vs outside the function
  - Correct: Constants outside the function change y-coordinates, so result in a vertical shift

## Cheatsheet

| Transformation | Rule for $y=f(x)$ | Effect on Graph |
| --- | --- | --- |
| Vertical translation $a$ | $y = f(x) + a$ | Shift up $a$ units ($a>0$) |
| Horizontal translation $b$ | $y = f(x - b)$ | Shift right $b$ units ($b>0$) |
| Reflection across x-axis | $y = -f(x)$ | Flip over x-axis, $y \to -y$ |
| Reflection across y-axis | $y = f(-x)$ | Flip over y-axis, $x \to -x$ |
| Vertical dilation $k$ | $y = k f(x)$ | Stretch vertically by factor $k$ |
| Horizontal dilation $c$ | $y = f(cx)$ | Compress horizontally by factor $c$ |

## What's next

Mastering function graph transformations is a foundational skill for nearly all graphing topics in IB Math AI SL, including quadratic functions, exponential and logarithmic models, and periodic trigonometric functions. Transformations allow you to quickly sketch complex graphs starting from simple parent functions, which saves critical time in both Paper 1 (non-calculator) and Paper 2 (calculator) exams. You will also apply transformations to model real-world phenomena, from shifting seasonal temperature models to scaling population growth curves. This knowledge directly prepares you for working with more complex functions and modeling problems that make up a large portion of your final exam.

- [Real-world modeling with functions](https://www.owlsprep.com/study/ib-math-ai-sl-u2-real-world-modeling-with-functions/)
- [Geometry and Trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-overview/)

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