Y-Intercept Changes in Translated Exponential Functions
When exponential functions are translated, the y-intercept shifts in predictable ways studied in Grade 11 enVision Algebra 1 (Chapter 6: Exponents and Exponential Functions). For a vertical translation f(x) = aˣ + k, the y-intercept moves from (0, 1) to (0, 1 + k). For a horizontal translation f(x) = a^(x-h), substituting x = 0 gives the new y-intercept as (0, a^(−h)). When both translations are combined, the y-intercept becomes (0, a^(−h) + k). This skill is key for accurately graphing translated exponential functions.
Key Concepts
For vertical translations $f(x) = a^x + k$, the y intercept changes from $(0, 1)$ to $(0, 1 + k)$.
For horizontal translations $f(x) = a^{(x h)}$, the y intercept changes from $(0, 1)$ to $(0, a^{ h})$.
Common Questions
What is the y-intercept of f(x) = aˣ?
The y-intercept of the parent exponential function f(x) = aˣ is (0, 1), because any base raised to the power of 0 equals 1.
How does a vertical shift k affect the y-intercept of an exponential function?
For f(x) = aˣ + k, the y-intercept shifts from (0, 1) to (0, 1 + k). The entire graph moves up or down by k units.
How does a horizontal shift h affect the y-intercept of an exponential function?
For f(x) = a^(x−h), substitute x = 0 to get the new y-intercept: (0, a^(−h)). A positive h gives a fractional y-intercept.
What is the y-intercept if both horizontal and vertical shifts are applied?
For f(x) = a^(x−h) + k, the y-intercept is (0, a^(−h) + k).
Why does a horizontal shift affect the y-intercept but not the horizontal asymptote?
A horizontal shift moves the graph left or right, changing where x = 0 intersects the graph (the y-intercept), but the horizontal asymptote is determined by the vertical shift k, not h.
If f(x) = 2^(x−3), what is the y-intercept?
Substitute x = 0: f(0) = 2^(−3) = 1/8, so the y-intercept is (0, 1/8).