Problem 82 - Precalculus : Building Concepts and Connections

Question

The graph of the function \(f(x)=C a^{x}\) passes through the points (0,12) and (2,3). (a) Use \(f(0)\) to find \(C.\) (b) Is this function increasing or decreasing? Explain. (c) Now that you know \(C\), use \(f(2)\) to find \(a\). Does your value of \(a\) confirm your answer to part (b)?

Step-by-Step Solution

Answer

The constant \(C\) is equal to 12, and \(a\) is obtained through solving \(3 = 12 a^2\). This equation provides a value for \(a\) between 0 and 1 which confirms that the function is decreasing.

Step 1: Find C

Given that the function passes through the point (0,12), it's possible to substitute \(x = 0\) into the function to find \(C\). Since any nonzero number to the power of zero is 1, \(a^0 = 1\), therefore, \(f(0) = C*1 = C\). So, setting \(f(0) = 12\), we find that \(C = 12\).

Step 2: Determine if the function is increasing or decreasing

The function \(f(x)=C a^{x}\) increases if \(a>1\) and decreases if \(0

Step 3: Find a

With \(C\) confirmed as 12, it's possible to find \(a\) using the point (2,3). By substituting \(C = 12\), \(x = 2\) and \(f(2) = 3\) into the function equation \(f(x) = C a^x\), you get \(3 = 12 a^2\). Solving this equation will provide the value of \(a\).

Step 4: Confirm Direction

Once a value is found, compare it against the criteria confirmed in Step 2. If \(0

Key Concepts

Exponential Decay

Exponential decay is a pattern of data that shows how the size of a quantity decreases at a rate proportional to its current value. In functions, this is represented by an equation of the form \( f(x) = C \times a^x \), where \( 0 < a < 1 \). This condition for \( a \) ensures that as \( x \) increases, the function's value decreases, thus reflecting 'decay'.

An example of exponential decay from everyday life is radioactivity; a radioactive substance decays at a rate proportional to its current amount. In the exercise, we can determine that the function is experiencing exponential decay since the value decreases from 12 to 3 as \( x \) increases from 0 to 2. This characteristic is significant because it tells us about the nature of the process being modeled by the function.

Solving Exponential Equations

To solve an exponential equation, one must find the value of the variable that makes the equation true. In our exercise, the goal was to find the value of \( a \) in the function \( f(x) = C \times a^x \).

When solving such equations, it's common to isolate the term with the variable of interest — in this case \( a^x \) — and then use logarithms to solve for the variable. In the given exercise, however, we used the value of the function at specific points to deduce the value of \( a \) directly. This method is straightforward when the points given are at values of \( x \) that make the calculations manageable. Otherwise, logarithms might be necessary to solve more complicated exponential equations.

Function Transformation

Function transformation involves changing the graph of a function in various ways, such as shifting, stretching, compressing, or reflecting. In exponential functions, these transformations can change the rate at which exponentially increasing or decreasing behavior occurs.

For the function \( f(x) = C \times a^x \), the value of \( C \) is responsible for vertical shifts, while the base \( a \) affects the growth or decay rate. When the value of \( a \) is greater than 1, it indicates growth; when it's between 0 and 1, it indicates decay. Understanding these transformations helps in analyzing the behavior of exponential functions and their graphical representations.

Properties of Exponents

Properties of exponents are the rules that describe how to manipulate expressions involving powers. These rules are critical when working with exponential equations and functions. Some of the key properties include:

• The product of like bases: \( a^m \times a^n = a^{m+n} \).
• The power of a power: \( (a^m)^n = a^{mn} \).
• Division of like bases: \( a^m / a^n = a^{m-n} \), when \( a \eq 0 \).

These properties allow us to simplify expressions, solve equations, and understand how changes in the function's formula affect its graph. In our exercise's context, knowing that any non-zero number raised to the power of zero is one helped us to easily find the value of \( C \) when given that \( f(0) = 12 \).

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Key Concepts

Exponential Decay

Exponential decay is a pattern of data that shows how the size of a quantity decreases at a rate proportional to its current value. In functions, this is represented by an equation of the form \( f(x) = C \times a^x \), where \( 0 < a < 1 \). This condition for \( a \) ensures that as \( x \) increases, the function's value decreases, thus reflecting 'decay'.

An example of exponential decay from everyday life is radioactivity; a radioactive substance decays at a rate proportional to its current amount. In the exercise, we can determine that the function is experiencing exponential decay since the value decreases from 12 to 3 as \( x \) increases from 0 to 2. This characteristic is significant because it tells us about the nature of the process being modeled by the function.

Solving Exponential Equations

To solve an exponential equation, one must find the value of the variable that makes the equation true. In our exercise, the goal was to find the value of \( a \) in the function \( f(x) = C \times a^x \).

When solving such equations, it's common to isolate the term with the variable of interest — in this case \( a^x \) — and then use logarithms to solve for the variable. In the given exercise, however, we used the value of the function at specific points to deduce the value of \( a \) directly. This method is straightforward when the points given are at values of \( x \) that make the calculations manageable. Otherwise, logarithms might be necessary to solve more complicated exponential equations.

Function Transformation

Function transformation involves changing the graph of a function in various ways, such as shifting, stretching, compressing, or reflecting. In exponential functions, these transformations can change the rate at which exponentially increasing or decreasing behavior occurs.

For the function \( f(x) = C \times a^x \), the value of \( C \) is responsible for vertical shifts, while the base \( a \) affects the growth or decay rate. When the value of \( a \) is greater than 1, it indicates growth; when it's between 0 and 1, it indicates decay. Understanding these transformations helps in analyzing the behavior of exponential functions and their graphical representations.

Properties of Exponents

Properties of exponents are the rules that describe how to manipulate expressions involving powers. These rules are critical when working with exponential equations and functions. Some of the key properties include:

• The product of like bases: \( a^m \times a^n = a^{m+n} \).
• The power of a power: \( (a^m)^n = a^{mn} \).
• Division of like bases: \( a^m / a^n = a^{m-n} \), when \( a \eq 0 \).

These properties allow us to simplify expressions, solve equations, and understand how changes in the function's formula affect its graph. In our exercise's context, knowing that any non-zero number raised to the power of zero is one helped us to easily find the value of \( C \) when given that \( f(0) = 12 \).