Transformations of Logarithmic Functions
Transformations of logarithmic functions is a Grade 11 algebra skill in Big Ideas Math. The parent function y = logₐ(x) can be transformed: y = logₐ(x − h) + k shifts h units right (or left if h < 0) and k units up/down, moving the vertical asymptote to x = h. Multiplying by a stretches (|a| > 1) or compresses (0 < |a| < 1) vertically; a < 0 reflects over the x-axis. For example, y = 2·log₂(x − 3) + 1 has a vertical asymptote at x = 3, is shifted up 1, and is vertically stretched by 2. The domain is (h, ∞) and the range remains all real numbers.
Key Concepts
The general form of a transformed logarithmic function is $f(x) = a\log b(x h) + k$, where $h$ represents horizontal shifts, $k$ represents vertical shifts, and $a$ represents vertical stretching or compression. The vertical asymptote moves from $x = 0$ to $x = h$, and the x intercept moves accordingly.
Common Questions
What is the parent function for logarithmic transformations?
y = logₐ(x), where a > 0 and a ≠ 1. It has a vertical asymptote at x = 0, passes through (1, 0) and (a, 1), and is defined only for x > 0.
How does y = logₐ(x − h) + k differ from the parent function?
Subtracting h from x shifts the graph h units right (vertical asymptote moves to x = h); adding k shifts k units up. Domain becomes (h, ∞).
How do you graph y = 2·log₂(x − 3) + 1?
Vertical asymptote: x = 3. Key points: shift parent y = log₂(x) right 3, up 1, and stretch vertically by 2. Plot (4, 1) → (4, 2·0+1=1), (5, 2·1+1=3), etc.
What is the domain of y = logₐ(x − h) + k?
Domain: x > h, written in interval notation as (h, ∞). Logarithms require positive arguments, so x − h > 0.
How does a negative coefficient affect the logarithmic graph?
y = −logₐ(x) reflects the graph over the x-axis. It is decreasing for a > 1 (whereas y = logₐ(x) with a > 1 is increasing).
What transformation moves the vertical asymptote of a logarithmic function?
A horizontal shift h in y = logₐ(x − h) moves the asymptote from x = 0 to x = h. The argument (x − h) must equal zero to find the asymptote.