Reflections of Exponential Functions: X-Axis vs. Y-Axis
Reflections of exponential functions across the x-axis versus y-axis are distinct transformations in Grade 9 Algebra 1 (California Reveal Math, Unit 8). An x-axis reflection negates the output: g(x) = -b^x, flipping y-values to their opposites. A y-axis reflection negates the input: g(x) = b^(-x) = (1/b)^x, converting growth to decay or vice versa. The key distinction: a negative outside the base means x-axis reflection; a negative inside the exponent means y-axis reflection.
Key Concepts
Two distinct reflection rules apply to exponential functions:.
X axis reflection — Negate the entire function (multiply by $ 1$ outside):.
Common Questions
How do you reflect an exponential function across the x-axis?
Negate the entire function: g(x) = -b^x. The y-intercept shifts from (0,1) to (0,-1), and the graph opens downward. The negative is outside the base. For 2^x: -2^x reflects over x-axis.
How do you reflect an exponential function across the y-axis?
Negate the input: g(x) = b^(-x) = (1/b)^x. The y-intercept stays at (0,1) but growth becomes decay (or decay becomes growth). For 2^x: 2^(-x) = (1/2)^x reflects over y-axis.
How do you distinguish -3^x from 3^(-x)?
In g(x) = -3^x, the negative is outside, indicating x-axis reflection with a = -1. In h(x) = 3^(-x), the negative is inside the exponent, indicating y-axis reflection. Their y-intercepts are (0,-1) and (0,1) respectively.
What happens to exponential growth when reflected across the y-axis?
b^(-x) = (1/b)^x. A growth function with b > 1 becomes a decay function because 1/b < 1. The transformation swaps increasing and decreasing behavior across the y-axis.
Does an x-axis reflection change the horizontal asymptote?
No. The horizontal asymptote stays at y = 0 for both -b^x and b^x. The curve still approaches y = 0 from below after the x-axis reflection instead of above.