Transformations of Exponential and Logarithmic Graphs
Simple Explanation
Exponential and logarithmic graphs transform the same way as any other function: y=A·bˣ⁻ᶜ+D shifts the graph C units horizontally and D units vertically (this also moves the asymptote!), and scales/reflects it by A.
Why Do We Need It?
Real exponential and logarithmic relationships are rarely centered exactly at the origin — this lets you describe any shifted or scaled version of these graphs.
See It
Two exponential curves: the basic one, and a shifted version moved right by 1 and up by 1, with a correspondingly shifted horizontal asymptote
Formula
Transformations of Exponential and Logarithmic Graphs
y = A·bˣ⁻ᶜ + D (exponential); y = A·log_b(x−C) + D (logarithmic)
Shifting, stretching, or reflecting an exponential or logarithmic graph works the same way as for any other function — C shifts horizontally, D shifts vertically (moving the asymptote along with it), and A stretches/reflects.
- A
- — a vertical stretch/reflection factor
- C
- — the horizontal shift
- D
- — the vertical shift — also shifts the asymptote
When to use it: Whenever an exponential or logarithmic graph has been shifted or scaled from its basic form.
Worked Example
Describe a transformed exponential graph
Describe how the graph of y=2ˣ⁻¹+3 differs from y=2ˣ.
Why Does This Work?
Replacing x with (x−C) shifts any graph right by C (the function now reaches each output C units later); adding D to the whole output rigidly raises every point — including the asymptote itself — by D units.
Real-Life Example
Modeling cooling with a non-zero room temperature
An object cools exponentially toward room temperature, not toward zero.
The vertical shift D in the transformed exponential formula accounts for this non-zero final temperature, moving the asymptote to match.
Practice
Find the new horizontal asymptote of y=3ˣ+6 (compared to y=3ˣ, which has asymptote y=0).
MediumCommon mistake
Forgetting that a vertical shift also moves the horizontal asymptote by the same amount — the asymptote is not always y=0 after a transformation.
Quick Review
- y=A·bˣ⁻ᶜ+D: C shifts horizontally, D shifts vertically (and moves the asymptote).
- The same pattern applies to logarithmic graphs, with the vertical asymptote shifting instead.
- A scales and/or reflects the graph.