# Function Transformations (HL) | Maths with Lemon

Source: https://mathswithlemon.io/aa-function-transformations-hl.html


### Function Transformations — HL

IB Mathematics: Analysis and Approaches · AHL 2.16 . Learn how to sketch new graphs from a known function using absolute values, transformations of the input, squares and reciprocals. This page follows DPAAHL Lesson 2.6.

Lesson 2.6 worksheet (PDF) AA Functions overview


#### 1. Review: Translations and Stretches


##### What you need to know

- Function notation and coordinates on a graph.
- Horizontal and vertical translations.
- Reflections and stretches.
Start with the warm-up on page 1 of your worksheet.


##### Key Points

Changes outside the function affect output values; changes inside the function affect input values.

Up by \(k\) units when \(k>0\).

Right by \(h\) units when \(h>0\).

The horizontal scale factor is \(1/|a|\); a negative \(a\) also reverses horizontal orientation.

For \(y=f(2x+2)\), rewrite the input as \(2(x+1)\). This gives a horizontal compression by factor \(\frac12\) and a shift 1 unit left (when applied in that order).


#### 2. Absolute Value Transformations


##### What you need to know

- How to reflect a portion of a graph.
- Difference between absolute value outside and inside a function.
- Domain restrictions after transformations.
Use pages 1–4 of the worksheet to practise with cubic, square-root, logarithmic and reciprocal graphs.


##### Key Points

Keep the part above the \(x\)-axis; reflect the part below it upwards.

Keep the graph for \(x\ge0\) and mirror it across the \(y\)-axis.

Let \(f(x)=x^2-4\). For \(y=|f(x)|\), the section between \(x=-2\) and \(x=2\), where \(f(x)<0\), is reflected above the \(x\)-axis. The roots remain at \(x=\pm2\).


#### 3. Interactive Transformation Explorer


##### Try it yourself

- Choose a parent function.
- Select a transformation.
- Compare the original and transformed graphs.
- Predict what happens to roots, turning points and asymptotes.
The graphs are illustrative; discontinuities are not joined across asymptotes.


##### Compare the graphs


#### 4. Squaring a Function


##### What you need to know

- How output values change when squared.
- Which intercepts stay fixed.
- Why negative outputs become non-negative.
See page 5 of Lesson 2.6.


##### Key Points

Every output is squared. Zeros remain zeros; outputs of \(1\) or \(-1\) become \(1\). The new graph cannot lie below the \(x\)-axis.

If \(0<|f(x)|<1\), then

If \(|f(x)|>1\), then

For \(f(x)=x-2\), the squared graph is \(y=(x-2)^2\). The original crossing at \((2,0)\) becomes the vertex of a parabola touching the \(x\)-axis.


#### 5. Reciprocal Transformations


##### What you need to know

- Zeros and vertical asymptotes.
- Reciprocal values and their signs.
- Where the reciprocal is undefined.
See pages 6–8 of the worksheet.


##### Key Points

Whenever \(f(x)=0\), the reciprocal is undefined. A zero of \(f\) typically produces a vertical asymptote in \(1/f\).

Points with output \(1\) or \(-1\) remain fixed. Positive outputs stay positive; negative outputs stay negative.

If \(f(x)=x-3\), then

The graph has vertical asymptote \(x=3\) and horizontal asymptote \(y=0\).


#### 6. HL Challenge and Revision


##### From your worksheet

- Challenge 1: mapping \(f(2x+1)\) to \(f(3x)\).
- Challenge 2: translations and reflections of a tangent graph.
- Challenge 3: comparing transformations of exponential graphs.
Open the complete HL worksheet


##### Exam checklist

Before sketching a transformed graph, identify which features are preserved and which change: intercepts, turning points, asymptotes, domain and range.

Starting from \(y=f(x)\), sketch both

Explain why the order of the vertical translation and absolute-value transformation matters.


### IB Past Paper Problems

Select a topic and unit, then click the button below to get a new problem.

Press the button to start!

How was this problem?