Maths with Lemon

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.

Vertical shift
\[y=f(x)+k\]

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

Horizontal shift
\[y=f(x-h)\]

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

\[y=f(ax+b)=f\!\left(a\left(x+\frac{b}{a}\right)\right),\quad a\ne0\]

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

Worked example

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

Absolute value of the output
\[y=|f(x)|\]

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

Absolute value of the input
\[y=f(|x|)\]

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

The order matters: \(|f(x)+2|\) and \(|f(x)|+2\) usually give different graphs.
Worked example: a quadratic

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

Original f(x)Transformed function

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

\[y=[f(x)]^2\]

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

\[[f(x)]^2<|f(x)|\]

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

\[[f(x)]^2>|f(x)|\]
Worked example

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

\[y=\frac{1}{f(x)}\]

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.

A zero of \(f(x)\) does not automatically create an asymptote if the function has special behaviour near that zero. Check the one-sided limits when sketching.
Worked example

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

\[y=\frac{1}{x-3}\]

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.

Try this

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

\[y=|f(x)-2|\qquad\text{and}\qquad y=|f(x)|-2\]

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

For the full set of graph-based exercises, use the original Lesson 2.6 worksheet rather than treating these examples as a replacement.

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