Problem 81 - Precalculus Mathematics for Calculus

Question

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? $$f(x)=\log (c x)$$

Step-by-Step Solution

Answer

The graphs are horizontal translations of each other, shifting left as \(c\) increases.

Step 1: Understanding the Function

We are given the family of functions defined as \(f(x) = \log(cx)\). Here, \(c\) serves as a parameter which impacts the graph of the function. We need to analyze and graph this function for different values of \(c\) (1, 2, 3, and 4).

Step 2: Plotting the Function for Different Values of c

For each given value of \(c\), substitute \(c\) into the function and plot it. Specifically, we plot \(f(x) = \log(x)\), \(f(x) = \log(2x)\), \(f(x) = \log(3x)\), and \(f(x) = \log(4x)\). These functions will be graphed on the Cartesian plane where the x-axis represents \(x \) values and the y-axis represents the function value, \(f(x)\) or \(\log(cx)\).

Step 3: Observing the Effect of c on Graphs

As \(c\) increases from 1 to 4, the graphs will shift to the left. This is because increasing \(c\) compresses the graph horizontally—more input \(x\) gives the same output \(f(x)\) value for larger \(c\). Hence, \(\log(4x)\) shifts more to the left compared to \(\log(3x)\), \(\log(2x)\), and \(\log(x)\).

Step 4: Identifying Relationships Among the Graphs

For each increase in \(c\), the graph of \(\log(cx)\) is a horizontal shift of the graph of \(\log(x)\) to the left. So, all the graphs are translations of \(\log(x)\) that move leftward as \(c\) increases. This means they all have the same shape: each is just horizontally shifted depending on \(c\).

Key Concepts

Graph Transformations

Graph transformations involve altering the original graph of a function in various ways to produce new graphs. These transformations can include shifts, reflections, stretches, and compressions. For the function \( f(x) = \log(cx) \), manipulating the parameter \( c \) results in a transformation of the graph of the logarithmic function.When we adjust \( c \), we primarily witness horizontal compressions or stretches. A larger \( c \) compresses the graph horizontally, meaning less change in the input \( x \) is needed to achieve the same result in \( f(x) \). Conversely, a smaller \( c \) allows more change in \( x \) for the same output. These changes do not affect the shape of the graph, only its position along the x-axis.

Horizontal Shifts

Horizontal shifts in graphs occur when the entire graph of a function moves either left or right on the coordinate plane. For logarithmic functions like \( f(x) = \log(cx) \), the parameter \( c \) plays a crucial role in determining the direction and degree of these shifts.As \( c \) increases in \( \log(cx) \), the graph shifts leftward. This leftward shift happens because a higher \( c \) compresses the x-values needed to reach the same y-value. Hence, every increment of \( c \) translates the graph a bit more to the left compared to the previous value. This results in noticeable shifts in position without altering the graph's shape or steepness.

Family of Functions

The concept of a family of functions refers to a group of functions that share a common mathematical form but differ by a parameter or coefficient, known as the family parameter. For the function \( f(x) = \log(cx) \), \( c \) is the varying parameter that groups different instances of \( \log(x) \) as a family.In this family, each individual function—defined by different values of \( c \)—shares the same basic logarithmic form but appears different due to shifts and compressions on the graph. This allows us to study how

• each graph in the family relates to the original function \( \log(x) \)
• changes in \( c \) affect the graph positioning and shape
• a unified approach can help analyze behaviors across the family

Thus, viewing functions as part of a family enriches our understanding of mathematical transformations and relationships.