Vertical Translations of Exponential Functions
Vertical translations of exponential functions shift the entire graph up or down by adding constant k to the parent function: f(x) = b^x + k. Grade 9 students in California Reveal Math (Unit 8: Exponential Functions) learn that adding k shifts the graph up k units, subtracting shifts it down, and the horizontal asymptote moves from y = 0 to y = k. For example, 2^x + 3 shifts the graph up 3 units with asymptote y = 3, and the y-intercept becomes (0, 4) instead of (0, 1).
Key Concepts
A vertical translation of an exponential function shifts the graph up or down by adding or subtracting a constant $k$ outside the base:.
$$f(x) = b^x + k$$.
Common Questions
How does a vertical translation affect the horizontal asymptote of an exponential function?
The asymptote shifts from y = 0 to y = k. For f(x) = 2^x + 3, the asymptote moves to y = 3. For f(x) = 2^x - 5, it moves to y = -5.
What is the y-intercept of f(x) = 2^x + 3?
At x = 0: f(0) = 2^0 + 3 = 1 + 3 = 4. The y-intercept is (0, 4) instead of the parent function's (0, 1) because k = 3 is added.
Does a vertical translation change the shape or rate of growth of an exponential?
No. The shape, base b, and general growth or decay behavior remain unchanged. Only the position of the graph shifts vertically along with the horizontal asymptote.
How do you identify a vertical translation from the function equation?
Look for a constant k added or subtracted outside the base: f(x) = b^x + k. The constant outside (not in the exponent) causes the vertical shift.
What is the horizontal asymptote of f(x) = (1/2)^x + 1?
The parent function (1/2)^x has asymptote y = 0. Adding k = 1 shifts it to y = 1. The y-intercept becomes (0, 2).