The graph of an exponential function is given. Select the...

The graph of an exponential function is given. Select the function for each graph from the following options:
$$\begin{array}{l}
f(x)=3^{x}, g(x)=3^{x-1}, h(x)=3^{x}-1 \\
F(x)=-3^{x}, G(x)=3^{-x}, H(x)=-3^{-x}
\end{array}$$
(Graph can't copy)

Key Concepts

Reflections

Reflections in exponential functions occur when the function is multiplied by a negative coefficient. This results in the graph being flipped either across the x-axis or, in other cases, the y-axis if the exponent is negated. Reflections are essential for understanding how the sign of the function’s output or input transforms its graph.

Vertical Shifts

Vertical shifts involve adding or subtracting a constant to the output of an exponential function. This transformation results in the entire graph being moved up or down compared to the standard exponential curve. It is a key concept when analyzing graphs that do not intersect the usual horizontal asymptote.

Inversion through Negative Exponents

Using a negative exponent in an exponential function transforms it into a reciprocal function. For instance, replacing x with -x or using a negative coefficient in the exponent changes the growth behavior to decay or vice versa, essentially inverting the graph horizontally. This concept is important for distinguishing between functions that increase and those that decrease.

Exponential Functions

Exponential functions are mathematical functions in which a constant base is raised to a variable exponent. They are characterized by rapid growth or decay depending on the base, and they have unique properties such as a constant ratio of change and an asymptote, typically the horizontal axis. This concept is central to understanding many natural and economic processes.

Horizontal Shifts

A horizontal shift in an exponential function occurs when the variable x is replaced by x minus a constant. This transformation moves the graph left or right without altering its overall shape. Recognizing and applying horizontal shifts helps in matching a given graph with its corresponding algebraic function.

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