Use the graph of f to describe the transformation that...

Use the graph of $f$ to describe the transformation that yields the graph of $g$. Then sketch the graphs of $f$ and $g$ by hand.$$f(x)=\left(\frac{2}{3}\right)^{x}, \quad g(x)=\left(\frac{2}{3}\right)^{x+2}-3$$.

Key Concepts

Graph Sketching Techniques

Graph sketching techniques involve plotting key points, identifying asymptotes, and noting intercepts along with other important features of the function. These techniques help in visualizing the effect of transformations and in accurately representing the behavior of a function on the coordinate plane.

Horizontal and Vertical Shifts

Horizontal shifts occur when the input variable is replaced by an expression that translates the x-values, moving the graph left or right. Vertical shifts involve adding or subtracting a constant to the entire function, moving the graph up or down. Both shifts are important for understanding how changes in the equation affect the location of the graph without changing its defining characteristics.

Exponential Functions

Exponential functions are equations in which a constant base is raised to a variable exponent. They are central in modeling growth and decay processes and have characteristic graphs that typically include a horizontal asymptote. An understanding of these functions provides insight into behaviors such as rapid changes and long-term trends.

Function Transformations

Function transformations refer to alterations made to the graph of a basic function to obtain a new graph. These can include translations (shifts), reflections, stretching, and compressions, all of which modify the position or shape of the original graph while preserving its overall structure.

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