Problem 86 - Precalculus with Calculus Previews

Question

In Problems \(85-88\), discuss how the graph of the given function can be obtained from the graph of \(f(x)\) \(=\ln x\) by means of a rigid transformation (a shift or a reflection)? $$ y=\ln \frac{x}{4} $$

Step-by-Step Solution

Answer

The graph of \( y = \ln \frac{x}{4} \) is obtained by shifting \( f(x) = \ln x \) downward by \( \ln(4) \) units.

Step 1: Introduction to the Problem

The function given is \( y = \ln \frac{x}{4} \). This problem involves identifying how the transformation of the graph of \( f(x) = \ln x \) to \( y = \ln \frac{x}{4} \) can be described using rigid transformations.

Step 2: Understanding the Function Transformation

We can rewrite the function \( y = \ln \frac{x}{4} \) as \( y = \ln(x) - \ln(4) \) based on the properties of logarithms. This expression helps in understanding the transformation applied to the original function \( f(x) = \ln x \).

Step 3: Effect of Subtracting \( \ln(4) \)

Subtracting \( \ln(4) \) from \( \ln(x) \) results in a vertical shift of the graph. Specifically, it shifts the graph of \( \ln(x) \) downward by \( \ln(4) \) units. This is because subtracting a constant from a function translates the graph vertically downward by the constant amount.

Step 4: Conclusion

The transformation used is a vertical shift. The graph of \( f(x) = \ln x \) is shifted downward by \( \ln(4) \) units to obtain the graph of \( y = \ln \frac{x}{4} \). This irreversible transformation maintains the shape of the original graph but changes its position on the coordinate plane.

Key Concepts

Logarithmic Functions

Logarithmic functions are an important type of function used frequently in mathematics, particularly when dealing with exponential growth or decay. Essentially, a logarithmic function is the inverse of an exponential function. If you have an equation like

• if the exponential form is given by \( b^y = x \), then the logarithmic form would be \( y = \log_b(x) \).

Logaritmic functions are expressed using a base, usually noted as \( \log_b \). In most of the cases, as seen in your exercise, the natural logarithm \( \ln x \) is often used, which is simply a logarithm with the base \( e \) (Euler's number approximately 2.71828).
Logarithmic functions have some unique characteristics:

• The graph of a logarithm is a curve that rises slowly and continually with increasing \( x \), when \( x > 0 \).
• The function is undefined at \( x \le 0 \), which is why the graph only exists in the first quadrant with the x-values greater than zero.

Remembering these basics can often help simplify transformations and understand what's happening to \( \ln(x) \) when changes are applied.

Rigid Transformations

Rigid Transformations in mathematics involve moving a graph around the coordinate plane without altering its size or shape. When we talk about these transformations, we usually refer to either shifting or reflecting the graph.
In your problem with the logarithmic function \( y = \ln \frac{x}{4} \):

• To transform the original graph \( f(x) = \ln(x) \) to \( y = \ln(x) - \ln(4) \), we can use rigid transformations.

Reflecting a graph usually involves flipping it over an axis, but in this case, we use shifting. Shifting can be further divided into:

• Horizontal shifts: Move the graph left or right.
• Vertical shifts: Move the graph up or down.

A transformation that results from modifying only the input part of the function, such as in \( \frac{x}{4} \), typically involves a horizontal shift. However, because we re-express this as \( \ln(x) - \ln(4) \), it's actually a vertical influence due to subtracting \( \ln(4) \). Hence, it falls under vertical shifts in this context.

Vertical Shift

A Vertical Shift involves moving a function's graph up or down the y-axis without changing its shape.
A simple example is when you have a function \( f(x) \) and you add or subtract a constant value from it, which shifts its graph up or down respectively.

• For instance, \( y = f(x) + c \) shifts the graph of \( y = f(x) \) up by \( c \) units if \( c \) is positive.
• Conversely, \( y = f(x) - c \) shifts the graph down by \( c \) units if \( c \) is positive.

In the exercise given, the transformation takes the form \( y = \ln(x) - \ln(4) \). This translates to a vertical shift downward by \( \ln(4) \) units.
Subtracting \( \ln(4) \) effectively shifts every point of the graph of \( \ln(x) \) down by that constant amount, keeping the relative positioning and shape intact, while re-positioning where it crosses the y-axis. This is a clear factor in understanding how graphs are repositioned through simple constant modifications.