91Ó°ÊÓ

In the world of exponential functions, the horizontal asymptote is a line that the graph approaches but never quite touches. This line represents the value that the function will get infinitely close to as the input values become very large in a positive or negative direction. For the function description given in the exercise, the horizontal asymptote was at \( y = 72 \). This helped us identify that the parameter \( c \) in the function \( f(x) = b a^{-x} + c \) equals 72.
This value essentially tells us the long-term behavior of the function. As \( x \) becomes very large, the term \( b a^{-x} \) shrinks towards zero, making the function value approach \( c \), which is 72 in this case.
Understanding horizontal asymptotes is crucial for graphing and analyzing the long-term trends of exponential functions.

The \( y \)-intercept of a function is the point where the graph crosses the \( y \)-axis. It is found by evaluating the function at \( x = 0 \). For the exercise, the \( y \)-intercept was given as 425, which indicates that when \( x = 0 \), \( f(x) = 425 \). This information allowed us to solve for \( b \) in the function \( f(x) = b a^{-x} + 72 \).
By setting \( x = 0 \) in the equation, we know:

• \( 425 = b \cdot a^{0} + 72 \)
• Since \( a^{0} = 1 \), the equation simplifies to \( 425 = b + 72 \)

From there, it is straightforward to solve for \( b \), verifying that \( b = 353 \). This intercept is a crucial piece in determining the specific form of our function.

Solving exponential equations involves finding unknown parameters that define the behavior of the function. In the given exercise, we knew the horizontal asymptote \( c \), the \( y \)-intercept which gave us \( b \), and had the point \( P(1, 248.5) \) which allowed us to solve for \( a \).
The pivotal step involved substituting point \( P \) into the equation:
\[ 248.5 = 353 \cdot 2^{-1} + 72 \]
This equation rearranges and solves for \( a \), verifying it through:

• Subtract 72 from 248.5 to isolate terms involving \( a \)
• Multiply both sides by \( a \) to remove the denominator
• Finally, result in \( a \approx 2 \)

These calculations ensure that the function satisfies all the given conditions.

Parameters in an exponential function define its shape and behavior. In our case, these parameters were \( b \), \( a \), and \( c \). Each one plays a unique role in constructing the specific function.
Let's quickly review them:

• \( b \): This represents the "amplitude" factor, shifting the function vertically and affecting how "steep" the function starts. We found \( b = 353 \) using the given \( y \)-intercept.
• \( a \): Known as the base of the exponent, \( a \) dictates the rate of exponential growth or decay. In this exercise, we calculated \( a \approx 2 \), implying it determines how fast the function values decrease as \( x \) increases.
• \( c \): The parameter \( c \) denotes the horizontal asymptote's value, dictating the level to which the graph eventually settles, which is 72 for this problem.

By understanding these parameters, we can sketch and analyze the behavior of exponential functions with confidence.