Problem 57 - Algebra and Trigonometry

Question

(a) Draw the graphs of the family of functions $$ f(x)=\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right) $$ (b) How does a larger value of \(a\) affect the graph?

Step-by-Step Solution

Answer

A larger value of \( a \) makes the graph wider and flatter.

Step 1: Understand the Function

The function \( f(x) = \frac{a}{2} \left( e^{x/a} + e^{-x/a} \right) \) is a hyperbolic cosine function scaled by \( a \). It can be rewritten as \( f(x) = a \cdot \cosh(\frac{x}{a}) \), where \( \cosh \) is the hyperbolic cosine function.

Step 2: Graph the Core Function

Plotting \( \cosh(x) \) shows a symmetrical shape about the \( y \)-axis. \( \cosh(x) \) is always positive, with its minimum value 1 at \( x=0 \) and increasing exponentially as \( |x| \) increases.

Step 3: Scale the Graph Vertically

For a general \( a \), the function \( f(x) = a \cdot \cosh(\frac{x}{a}) \) scales the graph vertically by a factor of \( a \). This means the minimum value at \( x=0 \) will be \( a \) instead of 1.

Step 4: Scale the Graph Horizontally

The term \( \cosh(\frac{x}{a}) \) compresses the function horizontally by a factor of \( a \). Wider values of \( a \) elongate the graph horizontally, making it less steep as compared to narrow ones.

Step 5: Explore Different Values of \(a\)

Graph functions for different values of \( a \), for example, \( a = 1, 2, 3 \). Observe how changes in \( a \) adjust the width (spread) of the graph.

Step 6: Analyze the Effect of Larger \(a\)

Larger values of \( a \) result in a graph that is wider and flatter. The exponential growth/decline on either side of \( x=0 \) occurs more slowly, indicating less steepness in the curve.

Key Concepts

Graphing Functions

Understanding how to graph functions is a vital math skill. It helps visualize equations and provides insight into their behavior. At its core, graphing a function involves plotting its values within a coordinate system. This means you assign points on a graph that correspond to functional output values for particular inputs. In this way, you observe how a function behaves across a specified domain. Graphing involves several key steps:

• Identify the function type. Here, we are dealing with a hyperbolic cosine function.
• Calculate key points. For instance, find where the function reaches its minimum, maximum, intersect, etc.
• Draw the curve. This involves smoothly connecting these points to reflect the function's continuity.

In this example, you graph the function by transforming a basic hyperbolic cosine function, resulting in symmetrically curved lines which offer a broad insight into its exponential behavior as it stretches along the graph.

Hyperbolic Cosine

The hyperbolic cosine function is often abbreviated as cosh. Formally, it is expressed as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). It's a critical mathematical function used in various applications, including physics, engineering, and hyperbolic geometry. Here are its fundamental characteristics:

• Symmetry: Hyperbolic cosine is even, meaning \( \cosh(x) = \cosh(-x) \).
• Minimum Value: At \( x=0 \), \( \cosh(x) \) reaches its minimum value of 1.
• Exponential Growth: As \( x \) moves away from zero in either direction, \( \cosh(x) \) increases rapidly.

Due to its symmetrical nature, the graph of the hyperbolic cosine rises identically on both sides of the y-axis. This unique trait makes it distinct from circular trigonometric functions, offering a different form of exponential growth profile.

Function Transformations

Function transformations alter the basic appearance of a graph, allowing it to stretch, compress, or shift within a coordinate plane. They are central concepts in understanding changes in graph behavior based on variable adjustments. For the given hyperbolic cosine function, the expression \( f(x) = a \cdot \cosh(\frac{x}{a}) \) applies both vertical and horizontal transformations:

• Vertical Stretch: Multiplying by \( a \) scales the graph vertically, meaning the peak at \( x=0 \) changes from 1 to \( a \).
• Horizontal Compression: Using \( \frac{x}{a} \) shortens the graph horizontally. Increasing \( a \) makes the graph wider and less steep.

Combined, these transformations not only change the appearance but also the steepness and spread of the graph. Larger \( a \) values result in broader, flatter curves, illustrating that function transformations are powerful tools in graph manipulation.