Graph functions f and g in the same rectangular coordinate...

Graph functions $f$ and $g$ in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs.
$$f(x)=3^{x} \text { and } g(x)=\frac{1}{3} \cdot 3^{x}$$

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where the base a is a positive constant not equal to 1. They exhibit characteristic rapid growth or decay depending on whether the base is greater than or less than one. These functions are defined for all real numbers and are commonly used to model phenomena in science, economics, and other fields that involve rates of change.

Function Transformations

Function transformations involve operations such as translations, reflections, stretches, and compressions that modify the graph of a basic function. In the context of an exponential function, multiplying the function by a constant (vertical scaling) changes its steepness without affecting its fundamental structure, which is essential when comparing two similar exponential functions with different coefficients.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that a function's graph approaches as x tends to positive or negative infinity. For exponential functions of the form f(x) = a^x where a > 1 (or 0 < a < 1), the horizontal asymptote is typically the line y = 0 because the function’s values get arbitrarily close to zero as x approaches negative infinity, regardless of vertical scaling.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is used to graph functions and visualize their behavior in two dimensions. It is defined by a horizontal x-axis and a vertical y-axis, allowing for a clear depiction of function properties such as intercepts, asymptotes, and transformation effects, which are crucial when graphing and comparing multiple functions.

Please give Ace some feedback