Problem 27 - College Algebra with Modeling and Visualization

Question

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. \(f(0)=5\) and for each unit increase in \(x,\) the output is multiplied by 1.5

Step-by-Step Solution

Answer

\( C = 5 \) and \( a = 1.5 \).

Step 1: Use the first condition

The function is given by \( f(x) = C a^x \) and satisfies \( f(0) = 5 \). Substituting \( x = 0 \) into the function yields \( f(0) = C a^0 = C = 5 \). Therefore, \( C = 5 \).

Step 2: Use the second condition

The condition states that for each unit increase in \( x \), the output is multiplied by 1.5. This means \( a = 1.5 \), as it represents the base of the exponential function that dictates the change rate for each increase in \( x \).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with variables and the rules for manipulating these variables. In the context of this exercise, we are working with an exponential function, which is expressed in an algebraic form as \( f(x) = C a^x \). Here, \( C \) and \( a \) are constants, while \( x \) is the variable. Understanding algebra involves recognizing how these variables and constants interact with each other. For instance, in the given function, the constant \( C \) represents the initial value or y-intercept when \( x = 0 \). It's the value at which the function starts. Algebra makes it easier to solve problems by allowing you to substitute known values into equations. In our solution, we substituted \( x = 0 \) into the function to find that \( C \), when multiplied by \( a^0 \) (which is always 1), equals 5. Algebraic manipulation like this helps break down problems step by step.

Function Transformation

Function transformation refers to the shifting, stretching, or compressing of a graph. In exponential functions like \( f(x) = Ca^x \), transformation often involves altering the graph's shape or position through changes in the function's constants. Specifically, changing \( a \) in an exponential function results in modifications to the growth rate of the graph. If \( a \) is greater than 1, such as in our exercise where \( a = 1.5 \), the function exhibits exponential growth and stretches vertically, making the graph steeper. Transformations can also move the graph along the axes when additional terms are introduced, although our current function focuses on vertical stretching due to the growth factor of \( a \). Understanding transformations helps in visualizing how changes to a function affect its graph, which is a cornerstone in the study of functions.

Exponential Growth

Exponential growth describes a situation where the rate of change of a quantity is proportional to its current value, leading to the quantity growing by a constant factor over equal increments. In our function \( f(x) = 5 \cdot 1.5^x \), the base \( a = 1.5 \) indicates that every time \( x \) increases by one unit, the output is multiplied by 1.5. This scenario exemplifies exponential growth, where as \( x \) increases, the value of \( f(x) \) increases at an accelerating pace.Exponential growth is commonly found in various real-world situations, such as population growth, compound interest, and certain biological processes. Recognizing and understanding exponential growth patterns allows us to predict trends and make projections more effectively.