Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln x-5$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)= \ln x - 5\) is obtained by shifting the graph of \(f(x)=\ln x\) downwards by 5 units.

Step 1: Identify the Transformation

The difference between \(f(x)\) and \(g(x)\) lies in the '-5' term in the function \(g(x)\). This term is not multiplied with x, hence it doesn't affect the slope of the function. Instead, it shifts the function vertically. In specific, '-5' indicates a downward shift by 5 units.

Step 2: Describe the Transformation

For any given x-value, the y-value of the function \(g(x)\) will be 5 units less than the y-value for the same x in function \(f(x)\). To put it in other words, the graph for the function \(g(x)=\ln x - 5\) is a vertical translation of the graph of \(f(x) = \ln x\) downwards by 5 units.

Step 3: Specify Direction of Shift

The term '-5' suggests that the transformation shifts the original function, \(f(x)\), downwards. If this term was '+5', the shift would have been upwards by 5 units.

Key Concepts

Vertical Shift

A vertical shift in graph transformations refers to moving the entire graph of a function up or down on the coordinate plane. When working with functions, this type of shift is achieved by adding or subtracting a constant value to the function's output (the y-value). Here's how it works:

For the function transformation from \(f(x)\) to \(g(x)\), the expression \(g(x) = f(x) + c\) indicates a vertical shift.

• If \(c > 0\), the graph of \(f(x)\) shifts upwards by \(c\) units.
• If \(c < 0\), the graph shifts downwards by \(c\) units.

In our specific example, we have \(g(x) = \ln x - 5\). Here, the \(-5\) means the graph of \(f(x) = \ln x\) shifts down a total of 5 units. This process does not affect the shape or the horizontal position of the graph. It only affects where the baseline of the graph is located relative to the y-axis.

Function Translation

Function translation refers to any shift in a graph's position without altering its shape. Translations can be vertical, horizontal, or both. Understanding function translation is crucial for graph transformations.

The effects translate as follows for vertical and horizontal translations:

• Vertical Translation: This moves the graph up or down and is achieved by adding or subtracting a constant to the function. For instance, \(g(x) = f(x) - 5\) shows a vertical shift down by 5 units, as seen in our example.
• Horizontal Translation: Moves the graph left or right and is achieved by adding or subtracting a constant directly to the function's input, \(x\). For example, \(f(x - h)\) moves the graph to the right by \(h\) units if \(h > 0\) or to the left if \(h < 0\).

In general, translations allow us to manipulate the position of graphs in the coordinate plane while preserving their overall shape and orientation, which is vital in understanding and sketching functions accurately.

Logarithmic Function

Logarithmic functions are a type of mathematical function that are the inverse of exponential functions. They are represented in the form \(f(x) = \ln x\), where \(\ln x\) is the natural logarithm of \(x\). Logarithmic functions have distinctive characteristics in their shape and behavior:

Key characteristics include:

• The graph passes through the point (1, 0) since \(\ln 1 = 0\).
• It has a vertical asymptote at \(x = 0\) because the logarithm is undefined for non-positive values.
• The function increases slowly and is defined only for \(x > 0\).

When transforming the graph of a logarithmic function, such as through a vertical shift, the shape of the graph retains its distinct properties, including the asymptote and its slow increasing nature. For example, in our transformation, \(g(x) = \ln x - 5\), the entire curve just moves 5 units downward, maintaining its original form and asymptote behavior.