Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Find an exponential function of the form \(f(x)=b a^{-x}+c\) that has the given horizontal asymptote and \(y\)-intercept and passes through point \(P\). \(y=72 ; \quad y\)-intercept \(425 ; \quad P(1,248.5)\)

Step-by-Step Solution

Answer

The function is \(f(x) = 353 \cdot 2^{-x} + 72\).

Step 1: Understand the Problem

We need to find an exponential function of the form \(f(x) = b a^{-x} + c\) with several given constraints: a horizontal asymptote \(y = 72\), a \(y\)-intercept of \(425\), and the function passing through the point \(P(1, 248.5)\).

Step 2: Use Horizontal Asymptote to Find Constant c

The horizontal asymptote \(y = 72\) indicates that as \(x\) approaches infinity, the function \(f(x)\) approaches \(72\). Therefore, \(c = 72\).

Step 3: Determine the Parameters Using the y-intercept

Substituting the \(y\)-intercept into the function, we have \(f(0) = b a^{0} + c = 425\). Since \(a^0 = 1\), this simplifies to \(b + c = 425\). We know \(c = 72\), so \(b + 72 = 425\), thus \(b = 353\).

Step 4: Use the Point P(1,248.5) to Find Parameter a

Substitute the point \(P(1, 248.5)\) into the function: \(f(1) = b a^{-1} + c = 248.5\). This gives the equation \(353 a^{-1} + 72 = 248.5\). Simplify to find \(353 a^{-1} = 176.5\). Thus, \(a = \frac{353}{176.5}\), which simplifies to \(a = 2\).

Step 5: Write the Function

Now that we have determined \(b = 353\), \(a = 2\), and \(c = 72\), the function is \(f(x) = 353 \cdot 2^{-x} + 72\).

Key Concepts

Understanding Horizontal Asymptotes

In the context of exponential functions, the concept of a horizontal asymptote refers to a constant value that the function approaches as the input variable, often denoted as \(x\), heads towards infinity. For the given function form \(f(x) = b a^{-x} + c\), the horizontal asymptote is determined by the constant \(c\). This is because as \(x\) becomes very large, the term \(b a^{-x}\) diminishes towards zero, leaving \(f(x)\) to approach \(c\).

Therefore, understanding horizontal asymptotes is crucial when analyzing exponential functions because they provide insight into the long-term behavior of the function. In this exercise, the horizontal asymptote is \(y = 72\), which means that as \(x\) increases indefinitely, \(f(x)\) will get closer and closer to 72, but never actually reach it.

Identifying the Y-intercept

The \(y\)-intercept of a function is the point where the graph of the function crosses the \(y\)-axis, which occurs where \(x = 0\). For exponential functions like \(f(x) = b a^{-x} + c\), you find the \(y\)-intercept by substituting \(x = 0\) into the equation. This gives:

• \(f(0) = b a^{0} + c\)
• Since \(a^0 = 1\), this simplifies to \(f(0) = b + c\)

For this problem, the \(y\)-intercept is given as 425. By substituting \(b\) and \(c\) into the equation, we find that \(b + 72 = 425\). Solving gives \(b = 353\). Knowing how to find the \(y\)-intercept is fundamental for verifying correct parameters within exponential equations.

Solving Exponential Equations

Exponential equations involve expressions where the unknown variable appears in the exponent. Solving such equations often requires identifying the base of the exponent and using given values to find unknown parameters. In the function \(f(x) = b a^{-x} + c\), solving exponential equations allows us to determine parameters such as \(b\), \(a\), and \(c\).

In this exercise, after using the horizontal asymptote to find \(c\) and the \(y\)-intercept to solve for \(b\), the next step was to use the specific point \(P(1,248.5)\). By substituting this point into the equation \(f(1) = 248.5\), we got an exponential equation \(353 a^{-1} + 72 = 248.5\). Solving this provided us with \(a = 2\). Mastery in solving these helps in creating accurate models in various real-life scenarios.

Graphical Analysis of Exponential Functions

Graphically analyzing an exponential function involves understanding its slope, intercepts, asymptotes, and overall shape as described by its formula. Each parameter in the function \(f(x) = b a^{-x} + c\) affects its graph in different ways.

The parameter \(b\) influences the initial growth rate and vertical stretch or compression of the graph. The base \(a\) determines how quickly the function declines for negative \(x\) values, shaping its exponential decay. Finally, \(c\), the constant, shifts the graph vertically and determines the horizontal asymptote.

• A function with a greater value of \(b\) will start higher up on the \(y\)-axis for a given \(c\).
• If \(a\) is smaller, the graph approaches its asymptote more quickly.

Understanding these elements allows for a comprehensive graphical interpretation of the function, supporting the prediction of its behavior across different contexts.