Problem 74 - Algebra and Trigonometry

Question

\(73-74\) A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .\) (b) How are the graphs in part (a) related? $$ f(x)=c \log x $$

Step-by-Step Solution

Answer

The graphs are vertically stretched versions of \( \log x \), and they become steeper as \( c \) increases.

Step 1: Understanding the Function

The function given is \( f(x) = c \log x \), where \( c \) is a constant. This function represents a family of curves, each dependent on the value of \( c \). For different values of \( c \), the curve will have different steepness or vertical stretch.

Step 2: Graph for c=1

For \( c = 1 \), the function becomes \( f(x) = \log x \). This is the standard logarithmic curve, which passes through the point (1,0) and increases slowly as \( x \) increases. The curve is concave down and approaches negative infinity as \( x \) approaches 0 from the right.

Step 3: Graph for c=2

For \( c = 2 \), the function is \( f(x) = 2 \log x \). This graph is a vertically stretched version of the graph for \( c = 1 \). It will still pass through the point (1,0) but will rise more steeply.

Step 4: Graph for c=3

For \( c = 3 \), the function is \( f(x) = 3 \log x \). This graph is even more vertically stretched than \( c = 2 \). It follows the same general shape as the others, passing through (1,0) and rising more steeply.

Step 5: Graph for c=4

For \( c = 4 \), the function is \( f(x) = 4 \log x \). It has the greatest vertical stretch among the graphs for the given \( c \) values. Like the others, it passes through (1,0) and rises sharply.

Step 6: Relating the Graphs

All the graphs in part (a) have the same basic logarithmic shape. They all pass through the point (1,0) because at \( x = 1 \), \( \log 1 = 0 \), so \( f(1) = 0 \) regardless of the value of \( c \). As \( c \) increases, the graphs become more vertically stretched, which means they rise more quickly as \( x \) increases.

Key Concepts

Family of Functions

In mathematics, the concept of a "family of functions" refers to a set of functions that share common characteristics and are often defined by a parameter, like the constant \(c\) in our example function \(f(x) = c \log x\). Each member of this family is a variation of the base function, influenced by the parameter.

- The base function here is \(\log x\), which is the logarithmic function with the simplest form.
- By varying \(c\), we change the steepness or vertical stretch of the function, leading to different versions of the graph.
- For any given value of \(c\), such as 1, 2, 3, or 4, we get a different member of the function family, each visually distinct yet fundamentally connected.

Understanding a family of functions is crucial because it allows us to see how changes in parameters affect the function's behavior. For \(f(x) = c \log x\), higher \(c\) values result in steeper graphs, offering a dynamic way to explore mathematical relationships.

Graph Transformations

Graph transformations involve modifying the graph of a basic function by altering it through various operations, such as shifting, stretching, or compressing. In the context of our function \(f(x) = c \log x\), the transformation is achieved by vertically stretching the graph based on the parameter \(c\).

For this family of functions:

• When \(c = 1\), the transformation is nonexistent because it retains the standard shape of \(\log x\).
• For \(c > 1\), the graph becomes steeper. For example, at \(c = 2\), the graph is twice as steep, modifying how quickly \(f(x)\) increases compared to the base graph.
• Regardless of \(c\), the graph of \(\log x\) maintains its original properties, such as being concave down and crossing through the point \( (1, 0) \).

Transformations like these help students understand the flexibility of functions and how different manipulations affect their appearance on a graph.

Vertical Stretch

A vertical stretch is a specific graph transformation where the graph of a function is pulled away from the x-axis, making it appear taller or more elongated. This occurs when the constant \(c\) in our function \(f(x) = c \log x\) is any value greater than 1. The larger \(c\) is, the more significant the vertical stretch.

Consider these points:

• With \(c = 1\), there is no vertical stretch, and we see the basic logarithmic curve.
• \(c = 2\) means each point on the graph is multiplied by 2. The result is the curve rises more quickly and becomes noticeably steeper.
• As \(c\) increases to 3 or 4, the vertical stretch continues, demonstrating a pronounced effect on the curve's ascent.
• All vertically stretched graphs still pass through the point \( (1, 0) \) because any constant times 0 is 0, maintaining a critical anchor point.

Understanding vertical stretch helps students visualize and predict how a graph of a function changes as parameters vary, which is fundamental for mastering graph-related math concepts.