Comparing Exponential Functions In these exercises we...

Comparing Exponential Functions In these exercises we compare the graphs of two exponential functions.
(a) Sketch the graphs of $f(x)=2^{x}$ and $g(x)=3\left(2^{x}\right)$
(b) How are the graphs related?

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 are characterized by rapid growth or decay and have a natural, smooth, and continuously increasing or decreasing curve. Their unique property is that they always remain positive for all real x and have a horizontal asymptote, usually the x-axis.

Graph Transformations

Graph transformations involve altering the basic graph of a function in specific ways, such as shifting, stretching, or reflecting. These transformations help in understanding how changes to the function’s formula affect its visual representation. They are fundamental for comparing different functions and analyzing their behavior on the coordinate plane.

Vertical Scaling

Vertical scaling is a transformation that multiplies the output of a function by a constant factor. When an exponential function is vertically scaled, its graph is stretched or compressed in the vertical direction. This transformation changes the amplitude of the function without affecting its overall exponential behavior, thus altering its rate of growth visually while keeping the intrinsic growth pattern intact.

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