91Ó°ÊÓ

In mathematics, exponential functions describe situations where a quantity grows or decays at a rate proportional to its current value. The general form of an exponential function is given by \[ y = a \times b^x \], where \( a \) is the initial value, and \( b \) is the base of the exponential. Exponential growth occurs if \( b > 1 \), and the function takes the form \( g(x) = 2^x \). The value of the function rapidly increases as \( x \) becomes larger. Examples include population growth and compound interest.
Exponential decay, on the other hand, happens if \( 0 < b < 1 \). In our case, reflecting the function \( 2^x \) over the y-axis changes it to \( 2^{-x} \). The function \( h(x) = 2^{-x} \) is an example of exponential decay where the value of the function decreases rapidly as \( x \) increases.
Understanding these basic properties helps in grasping how exponential functions can model real-world phenomena like radioactive decay or cooling processes.

Reflecting a graph over the y-axis involves flipping it across the vertical axis, turning left into right and vice-versa. In the context of exponential functions, reflecting \( 2^x \) over the y-axis changes it into \( 2^{-x} \). Here's what you can expect:

• The exponential growth function \( 2^x \) becomes the exponential decay function \( 2^{-x} \).
• Points that were on the left side of the y-axis move to the right side and vice versa.
• The graph’s overall shape reflects symmetrically around the y-axis.

For our function problem, by transforming \( g(x) = 2^x \) into \( 2^{-x} \), we are essentially reversing the direction of growth, turning it into a decay function. This reflection is a fundamental transformation in understanding how to manipulate and analyze exponential functions.

A vertical shift moves the graph of a function up or down without changing its shape. In mathematical terms, if you have a function \( f(x) \) and you add a constant \( k \) to it, you shift the graph vertically by \( k \) units. For instance, the function \( f(x) + k \) results in positioning the graph of \( f(x) \) upward if \( k \) is positive.
The given function is \( f(x) = 5 - 2^{-x} \). To break this down:

• Start with the reflected function \( 2^{-x} \).
• The addition of \( 5 \) causes a vertical shift upwards by 5 units.

This transformation is straightforward but crucial because it modifies the y-values of the function directly. So, all the points on the original function \( 2^{-x} \) are lifted up by 5 units, resulting in our target function \( f(x) = 5 - 2^{-x} \). Visualizing this helps significantly in sketching and understanding the behavior of the graph.