Range Changes in Vertical Translations
Translating an exponential function f(x) = aˣ vertically by k units to form g(x) = aˣ + k shifts the range from (0, ∞) to (k, ∞) — a transformation concept in enVision Algebra 1 Chapter 6 for Grade 11. For f(x) = 2ˣ with range (0, ∞), the function g(x) = 2ˣ + 3 has range (3, ∞) because every output is raised by 3. For h(x) = 3ˣ - 2, the range becomes (-2, ∞). The horizontal asymptote also shifts: instead of y = 0, it becomes y = k. This applies to both growth (a > 1) and decay (0 < a < 1) exponential functions.
Key Concepts
When an exponential function $f(x) = a^x$ is vertically translated by $k$ units to form $g(x) = a^x + k$: the range changes from $(0, \infty)$ to $(k, \infty)$ when $a 1$; the range changes from $(0, \infty)$ to $(k, \infty)$ when $0 < a < 1$.
Common Questions
How does adding k to an exponential function change its range?
Adding k raises every output by k units, shifting the range from (0, ∞) to (k, ∞). The lower bound of the range moves from 0 to k.
What is the range of g(x) = 2ˣ + 3?
The range is (3, ∞). Since 2ˣ > 0 for all x, adding 3 means g(x) > 3 for all x, and g(x) gets arbitrarily large but never equals 3.
What is the range of h(x) = 3ˣ - 2?
The range is (-2, ∞). Since 3ˣ > 0, subtracting 2 gives h(x) > -2 for all x.
How does a vertical translation change the horizontal asymptote?
The horizontal asymptote shifts from y = 0 to y = k. For g(x) = 2ˣ + 3, the asymptote is y = 3 because as x → -∞, g(x) → 3.
Does the vertical translation affect the domain of an exponential function?
No. The domain of exponential functions is all real numbers (-∞, ∞) regardless of vertical shifts. Only the range and asymptote change.