Transformations of Exp/Log (AA SL)
Shifting, stretching, or reflecting an exponential or log graph follows the same rules as any other function, but the asymptote makes the effect easy to check: track where it lands and the rest of the curve follows. This page focuses on describing transformations and tracking the asymptote through them. It's part of the broader Exponentials & Logarithms topic.
10 questions on this sub-topic.
Exponent laws you'll lean on
Covered under IB syllabus reference SL1.5, which introduces logarithms with base 10 and \(e\), and the equivalence \(a^x=b \iff \log_a b = x\). Transformation questions often need these two exponent rules to simplify the transformed equation first.
Product rule
\(a^m \cdot a^n = a^{m+n}\)
Not in the formula booklet. Useful for rewriting a horizontal shift, like \(2^{x-3}\), as a vertical stretch: \(2^{x-3}=2^x\cdot2^{-3}\).
Power rule
\((a^m)^n = a^{mn}\)
Not in the formula booklet. Multiplies exponents rather than stacking them - relevant whenever a transformed exponential is raised to a further power.
Need the full syllabus wording and the log-base introduction? See Exponentials & Logarithms.
Worked examples
The graph of \(y=e^{x}\) is translated to give \(y=e^{x-2}+1\).
(a) Describe the transformations.
(b) State the equation of the horizontal asymptote.
Worked solution
(a) Comparing \(e^{x-2}+1\) with \(e^x\): \(x\to x-2\) shifts right 2. A1
The \(+1\) shifts up 1. A1
(b) The original asymptote \(y=0\) shifts up 1: \(y\). R1
\(=1\). A1
Describe the transformation taking \(y=2^{x}\) to \(y=2^{x-3}-1\).
Worked solution
Comparing \(2^{x-3}-1\) with \(2^x\): replacing \(x\) by \(x-3\) shifts right 3. A1
The \(-1\) shifts down 1. A1
Translation by \(\begin{pmatrix}3\\-1\end{pmatrix}\).
\(f(x)=x^2.\) Find the equation of \(y=f(x)\) after reflection in the \(x\)-axis, vertical stretch factor 3, then translation up 5; give it in expanded form.
Worked solution
\(y = -x^2.\) M1 A1
\(y = -3x^2.\) M1 A1
\(y = -3x^2 + 5.\) M1 A1
Common mistakes
- Sign errors with negative indices. \(a^{-n} = \dfrac{1}{a^n}\), not \(-a^n\) - important when a transformation introduces a negative exponent, since misreading the sign flips the wrong feature of the graph.
- Confusing \((a^m)^n\) with \(a^{(m^n)}\). A power of a power multiplies the exponents - \((2^3)^2 = 2^6 = 64\) - which is not the same as a tower of exponents, and matters when simplifying a stretched exponential.
- Forgetting the domain shifts with the graph. Translating \(y=\log_b x\) horizontally moves its vertical asymptote and domain restriction along with it - the new asymptote is not still \(x=0\) unless the shift was purely vertical.
Ready to practise properly?
10 exp/log transformation questions, marked instantly like the real exam.
Quick answers
How do you find the equation of a transformed exponential graph?
Apply each transformation in the order given to the base function: a reflection changes the sign in front, and a translation replaces \(x\) with \((x - h)\) and adds the vertical shift \(k\) outside.
What happens to the asymptote when you transform an exponential graph?
A vertical translation moves the horizontal asymptote by the same amount; a horizontal translation leaves it unchanged, since the asymptote is a \(y\)-value, not an \(x\)-value.