Problem 30 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=\ln x\) by basic transformations: (a) \(y=\ln (1-x)\) (b) \(y=\ln (2+x)-1\) (c) \(y=-\ln (2-x)+1\)

Step-by-Step Solution

Answer

(a) Reflect y-axis; (b) Left 2, down 1; (c) Right 2, reflect y-axis, reflect x-axis, up 1.

Step 1: Identify the Basic Function

The base function given is \(y = \ln x\), which is the natural logarithm function.

Step 2: Transformation for (a)

To transform \(y = \ln x\) to \(y = \ln (1-x)\), apply a horizontal shift and reflection over the y-axis. The function \(y = \ln (1-x)\) can be seen as replacing \(x\) with \(1-x\), which involves a horizontal reflection due to the negative sign in front of \(x\).

Step 3: Transformation for (b)

For \(y = \ln (2+x) - 1\), start by considering \(y = \ln (x)\) and replace \(x\) with \((x-(-2))\), resulting in \(y = \ln (x+2)\), which is a leftward shift by 2 units. Then, apply a vertical shift down by 1 unit due to the \(-1\).

Step 4: Transformation for (c)

To obtain \(y = -\ln (2-x) + 1\), start with \(y = \ln x\). Replace \(x\) with \(2-x\) to get \(y = \ln (2-x)\), which includes a reflection over the y-axis and a shift right by 2 units. Next, apply a vertical reflection due to the negative sign before \(\ln\), and finally, shift the graph up by 1 unit due to the \(+1\).

Key Concepts

Function Transformations

Transformations of functions involve changing a graph's position or shape by altering its equation. When dealing with the natural logarithm base function, \( y = \ln x \), you can apply transformations to shift, reflect, compress, or stretch the graph. Transformations can be visualized by modifying the equation. Consider these important transformation types:

• Horizontal Shift: Moving the graph left or right.
• Vertical Shift: Moving the graph up or down.
• Reflection: Flipping the graph across an axis.
• Stretch/Compression: Making the graph narrower or wider (not commonly performed with logarithmic functions).

Each transformation modifies the function's graphical representation, impacting its domain and range accordingly.

Horizontal Shift

A horizontal shift involves moving the graph to the left or right. It occurs when you add or subtract a constant inside the logarithmic expression. For example, in the function \( y = \ln(2+x) \), the variable \( x \) is replaced by \( x+2 \).
This means every \( x \) value in \( \ln x \) is now 2 units less, effectively shifting the graph 2 units to the left.
Conversely, if \( x \) were replaced by \( x-2 \), the graph would shift 2 units to the right. This transformation leaves the graph's shape unchanged, except for its horizontal positioning.

Vertical Shift

Vertical shifts occur when a constant is added or subtracted outside the logarithmic function. When you see something like \( y = \ln (2+x) - 1 \), the \(-1\) indicates the whole graph is shifted downward by 1 unit.
In another case, if the expression was \( y = \ln (2+x) + 1 \), it would cause the graph to shift up by 1 unit. These types of shifts alter the y-values directly while maintaining the x-values, affecting the vertical position of the graph without changing its shape or orientation.

Reflection

Reflections flip the graph over a chosen axis. In the function \( y = \ln (1-x) \), we see a negative sign in front of \( x \), which causes the graph to reflect across the y-axis. This essentially mirrors the graph horizontally, swapping left and right.
Similarly, if the entire logarithmic expression is negated like in \( y = -\ln (2-x) + 1 \), it indicates a reflection over the x-axis. This vertical reflection inverts the graph's overall slope.
Reflections can be tricky but are essential in flipping graphs to meet specific conditions or graph characteristics visually.