Sketch the graph of y=g(x) by starting with the graph of...

Sketch the graph of $y=g(x)$ by starting with the graph of $y=f(x)$ and using transformations. Track at least three points of your choice and the horizontal asymptote through the transformations. State the domain and range of $g$.
$f(x)=e^{x}, g(x)=8-e^{-x}$

Key Concepts

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. Exponential functions typically have a domain of all real numbers and a range that is determined by any vertical translations and reflections. Understanding how transformations affect the domain and range is essential for correctly describing the function's behavior.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches as x approaches infinity or negative infinity. Exponential functions often have horizontal asymptotes that change position after vertical shifts. Tracking the horizontal asymptote is important when performing graph transformations to ensure an accurate sketch.

Vertical Shifts

A vertical shift moves the graph of a function up or down without changing its shape. In this context, adding 8 to the function after reflecting the exponential term shifts the entire graph upward, which affects the horizontal asymptote and the overall position of the graph in the coordinate plane.

Graph Transformations

Graph transformations involve altering the graph of a function through operations such as reflections, translations (shifting), and scaling. Understanding how each transformation affects the original function’s graph is crucial for sketching the new graph and tracking key features like points and asymptotes.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where the base a is a positive constant. They have unique properties, such as rapid growth or decay depending on whether a > 1 or 0 < a < 1, and their graphs are continuous, smooth curves. Exponential functions are widely used in modeling phenomena in natural and social sciences.

Reflection and Negation

Reflection is a specific type of graph transformation where the graph is 'flipped' over a given axis. In the transformation from f(x) = e^x to g(x) = 8 - e^{-x}, the substitution of -x for x results in a reflection across the y-axis, and the negation of the exponential term results in a reflection across the x-axis.

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