Transformations of Exponential Functions
A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.
Exponential transformations change the asymptote, direction, and scale without changing the repeated-multiplier idea. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Shift, stretch, and reflect exponential graphs
- Use transformations of exponential functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. The graph of y = 3^x + 3 is the parent y = 3^x shifted up 3. What is the value of c for its horizontal asymptote y = c?
- The parent y = 3^x has asymptote y = 0.
- Shifting up 3 units raises the asymptote by 3.
- So the asymptote is y = 3, giving c = 3.
Answer: 3
Practice problems
1. The graph of y = 3^x + 3 is the parent y = 3^x shifted up 3. What is the value of c for its horizontal asymptote y = c?
Show solution
- The parent y = 3^x has asymptote y = 0.
- Shifting up 3 units raises the asymptote by 3.
- So the asymptote is y = 3, giving c = 3.
Answer: 3
2. For y = 2^x - 4, the asymptote is y = c. Find c.
Show solution
- Start from the parent asymptote y = 0.
- Subtracting 4 shifts the graph down 4.
- So the asymptote is y = -4, giving c = -4.
Answer: -4
3. The parent function y = 3^x has horizontal asymptote y = c. Find c.
Show solution
- As x decreases, 3^x approaches 0 but never reaches it.
- The curve hugs the x-axis on the left.
- So the asymptote is y = 0, giving c = 0.
Answer: 0
4. Compared with y = 2^x, the graph of y = -2^x is:
Choices: reflected across the x-axis · reflected across the y-axis · shifted down 2 · unchanged
Show solution
- The negative sign multiplies every output by -1.
- Negating outputs flips the graph top-to-bottom.
- That is a reflection across the x-axis.
Answer: reflected across the x-axis
5. Compared with y = 3^x, the graph of y = 3^(-x) is:
Choices: reflected across the y-axis · reflected across the x-axis · shifted left 3 · vertically stretched
Show solution
- Replacing x with -x swaps inputs left-to-right.
- Each point (x, y) moves to (-x, y).
- That is a reflection across the y-axis.
Answer: reflected across the y-axis
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