Problem 29 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=\ln x\) by basic transformations: (a) \(y=\ln (x-1)\) (b) \(y=-\ln x+1\) (c) \(y=\ln (x+3)-1\)

Step-by-Step Solution

Answer

(a) Shift 1 unit right. (b) Reflect across x-axis and shift up 1. (c) Shift 3 units left and 1 down.

Step 1: Understanding the Parent Function

The given parent function is \(y = \ln x\). This is the natural logarithm function, and it is defined for all \(x > 0\) with the graph having a vertical asymptote at \(x=0\) and passing through the point (1,0). The domain is \((0, \infty)\) and range is \((-\infty, \infty)\).

Step 2: Transformation for (a): Horizontal Shift Right

For the function \(y = \ln(x-1)\), compare it with the parent function \(y = \ln x\). The transformation here is a horizontal shift. The expression \((x-1)\) indicates that the graph of the parent function should shift 1 unit to the right. This is because the input \(x\) is being replaced with \((x-1)\).

Step 3: Transformation for (b): Reflection and Vertical Shift

For the function \(y = -\ln x + 1\), first, observe \(-\ln x\). The negative sign indicates a reflection across the x-axis. Adding 1 shifts the entire graph upward by 1 unit. Combining both transformations, the graph of \(y = \ln x\) is first reflected and then shifted up.

Step 4: Transformation for (c): Horizontal Shift Left and Vertical Shift Down

For the function \(y = \ln(x+3) - 1\), the term \((x+3)\) implies a shift to the left by 3 units since \(x\) is replaced by \(x + 3\). The \(-1\) indicates a downward shift by 1 unit. Thus, the transformation involves shifting the graph of \(y = \ln x\) 3 units to the left and then 1 unit down.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( y = \ln x \), is a fundamental mathematical function. It represents the logarithm base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is used to solve equations involving exponential growth and decay. This function is defined for all positive numbers \( x > 0 \), making its domain \( (0, \infty) \).

Its graph is characteristic for having a vertical asymptote at \( x = 0 \) and it always passes through the point \( (1, 0) \). This intersection at \( (1, 0) \) shows where the logarithm of 1 equals 0, one of its essential properties. The range of the natural logarithm is \( (-\infty, \infty) \), allowing it to generate any real number as output.

This makes the natural logarithm invaluable in calculus, especially when dealing with limits and integrals, because of its smooth and continuous curve.

Horizontal Shift

A horizontal shift occurs when we adjust the \( x \)-value of a function by adding or subtracting a constant value. This changes where the function's graph appears along the \( x \)-axis. For logarithmic functions like \( y = \ln(x - k) \), the horizontal shift moves the graph according to the value of \( k \).

Here's a simple way to understand it:

• If \( k \) is positive, \( x - k \) results in the graph shifting \( k \) units to the right.
• If \( k \) is negative, \( x + |k| \) results in shifting \( |k| \) units to the left.

For example, \( y = \ln(x - 1) \) shifts the parent graph \( y = \ln x \) one unit to the right, because the expression inside the logarithm is \( x - 1 \). This shift modifies the input requirement, but the overall shape remains the same, maintaining the vertical asymptote relative to the change.

Vertical Shift

Vertical shifts move a graph up or down along the \( y\)-axis. For the logarithmic function \( y = \ln x \), adding or subtracting a constant value outside the logarithm will result in a vertical shift.

To identify how the graph will shift:

• Adding a constant (like \( + c \)) shifts the graph upwards by \( c \) units.
• Subtracting a constant (like \( - c \)) shifts it downwards by \( c \) units.

For instance, \( y = \ln(x) - 1 \) moves the graph of \( y = \ln x \) down by 1 unit. This adjustment vertically translates all the points on the graph without affecting their spacing horizontally. Vertical shifts are crucial for fine-tuning the position of the graph on the \( y \)-axis so that it can meet specific values and conditions.

Reflection Across x-axis

A reflection across the \( x \)-axis means the graph of the function is inverted. For a natural logarithm function like \( y = -\ln x \), this reflection flips the graph upside down.

Here's what happens during a reflection:

• If the function is positive before reflection, it becomes negative.
• The y-value of each point is multiplied by \(-1\), mirroring it across the horizontal axis.

This transformation can be observed in the function \( y = -\ln x + 1 \). First, the graph of \( y = \ln x \) is reflected to become \( y = -\ln x \), flipping its orientation upside down.

Reflections are particularly useful for altering the direction of growth or decay of a function, offering a different perspective on the same set of values.