Horizontal Dilation of Exponential Functions
Horizontal dilation of exponential functions is covered in Grade 9 Algebra 1 (California Reveal Math, Unit 8: Exponential Functions). A horizontal dilation of f(x) = b^x by factor 1/c produces g(x) = b^(cx) = (b^c)^x, which is equivalent to changing the base. When c > 1 the graph compresses horizontally (grows faster); when 0 < c < 1 it stretches (grows slower). The y-intercept stays at (0,1) and the horizontal asymptote remains y = 0. For example, 2^(3x) = 8^x grows much faster than 2^x.
Key Concepts
A horizontal dilation of an exponential function $f(x) = b^x$ by a factor of $\frac{1}{c}$ produces:.
$$g(x) = b^{cx}$$.
Common Questions
What is a horizontal dilation of an exponential function?
Multiplying the exponent by constant c gives g(x) = b^(cx). Since b^(cx) = (b^c)^x, this changes the effective base. Compresses graph when c > 1, stretches when 0 < c < 1.
How does f(x) = 2^x change when horizontally dilated by factor 1/3?
The result is g(x) = 2^(3x) = (2^3)^x = 8^x. The graph is steeper and grows faster than the original 2^x.
How does f(x) = 2^x change when horizontally dilated by factor 2?
The result is g(x) = 2^(x/2) = (2^(1/2))^x = sqrt(2)^x. The graph is wider and grows more slowly than the original.
Does a horizontal dilation change the y-intercept or asymptote of an exponential?
No. Both functions share the same y-intercept (0,1) and horizontal asymptote y = 0. Horizontal dilations only affect the rate of growth or decay, not these fixed features.
Why can every horizontal dilation of an exponential function be rewritten as a new base?
Because b^(cx) = (b^c)^x by exponent rules. The new base is b^c. This means horizontal dilation and base change are equivalent transformations for exponential functions.