Problem 54 - Algebra and Trigonometry

Question

53–54 ? Draw graphs of the given family of functions for c = 0.25, 0.5, 1, 2, 4. How are the graphs related? $$ f(x)=2^{c x} $$

Step-by-Step Solution

Answer

The graphs are exponential, vary in steepness with c, and pass through (0,1).

Step 1: Understand the Function

We have the family of functions given by \( f(x) = 2^{cx} \). This means for each value of \( c \), we have a different function.

Step 2: Compute the Function Values for Different c

Let's substitute the given values of \( c \) (0.25, 0.5, 1, 2, and 4) to find the functions: \(- f(x) = 2^{0.25x}- f(x) = 2^{0.5x}- f(x) = 2^{1x} = 2^{x}- f(x) = 2^{2x}- f(x) = 2^{4x}\)

Step 3: Analyze the Effect of c on the Function

As \( c \) increases, the function \( 2^{cx} \) grows faster. The base of the exponential is constant (2), and \( c \) determines the rate of growth.

Step 4: Graph Each Function

Draw graphs for each of the functions:- For \( c = 0.25 \), the growth is slow and the graph is less steep.- For \( c = 0.5 \), the growth is moderate and slightly steeper.- For \( c = 1 \), the function is \( 2^x \), a standard exponential growth graph.- For \( c = 2 \), the growth becomes rapid and steeper.- For \( c = 4 \), the function grows very rapidly, with a steep graph.

Step 5: Identify Relationship Between Graphs

The graphs are all exponential with the same base but differ in steepness and growth rate due to different values of \( c \). As \( c \) increases, the function becomes steeper and grows faster. All graphs pass through (0,1) as \( 2^{cx} \) for any \( c \) at \( x=0 \) is 1.

Key Concepts

Graphing Exponential Functions

Understanding how to graph exponential functions can help in visualizing their behavior. An exponential function is known for its characteristic rapid increase or decrease. The equation \( f(x) = 2^{cx} \) is an exponential function where the base is 2, and it grows exponentially with changing values of \( c \).
To graph these functions, it's important to remember:

• Exponential functions have a constant base (here, it's 2).
• The exponent, in this case, is controlled by the variable \( c \).
• The graph passes through the point (0,1) since \( 2^{cx} \) equals 1 when \( x=0 \).

When drawing or analyzing these graphs:

• Start by plotting the point (0,1), as all graphs will intersect there regardless of \( c \).
• Observe the steepness and how quickly each graph rises as \( x \) increases.

Graphing visually demonstrates the differences and similarities amongst exponential functions with different values of \( c \).
Being comfortable with graphing these functions enhances understanding of their overall behavior.

Impact of Parameter on Function Graph

The parameter \( c \) in the function \( f(x) = 2^{cx} \) fundamentally affects its shape and growth rate. Let's dive deeper into how it influences the graph.
As \( c \) increases:

• The graph steepens significantly, indicating faster growth.
• Smaller increments in \( x \) yield much larger changes in \( f(x) \).

Understanding Variations with \( c \)

• For \( c = 0.25 \): The growth is quite slow, and the graph gently rises as \( x \) moves to the right.
• At \( c = 0.5 \): The function grows moderately faster, and the curve becomes noticeably steeper.
• With \( c = 1 \): We have the standard exponential curve where growth accelerates at a familiar and balanced pace.
• For \( c = 2 \): The graph rises quickly, indicating a rapid increase, steeper than \( c = 1 \).
• Finally, at \( c = 4 \): The function shows very rapid growth, with the graph rising sharply even for small increments of \( x \).

By recognizing these changes, students can predict how modifications in \( c \) will impact the overall curvature and steepness of the graph. This comprehension is crucial in fields that utilize exponential functions heavily, such as finance and biology.

Exponential Growth Rate

Exponential growth refers to a process where quantities grow at a consistent rate over time, and it's a vital concept in understanding exponential functions.
In the function \( f(x) = 2^{cx} \):

• The base of the exponential function, 2, dictates the core growth behavior.
• The multiplier, \( c \), adjusts the rate at which this growth occurs.

The exponential growth rate is key in various applications:

• When \( c \) is small, changes occur slowly, meaning the function takes longer to double in value.
• Larger values of \( c \) imply a faster growth rate, quickly amplifying the function’s value as \( x \) increases.

Understanding exponential growth:

• Exponential functions can model real-world phenomena, such as population growth, where the increase seems to accelerate over time.
• The rapid doubling and compounding of values are characteristic of exponential growth.

Comprehending how \( c \) impacts the exponential growth rate enhances our ability to apply these functions effectively, predicting long-term patterns and outcomes within various contexts.