Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\log _{2}\left(x^{3}\right) $$

Step-by-Step Solution

Answer

Stretch the graph of \( \log_{2}(x) \) vertically by a factor of 3.

Step 1: Understand the Transformation

The function given is \( f(x) = \log_{2}(x^3) \). This can be rewritten using the logarithmic identity \( \log_b(a^c) = c \log_b(a) \) as \( f(x) = 3 \log_{2}(x) \). This means the graph of \( \log_{2}(x) \) will be vertically stretched by a factor of 3.

Step 2: Sketch the Base Function

Begin with the graph of the base function \( \log_{2}(x) \). The graph of \( y = \log_{2}(x) \) is a curve that passes through (1,0), where \( x = 1 \) is the x-intercept, and increases slowly, never touching the y-axis, as it is undefined for \( x \leq 0 \).

Step 3: Apply the Vertical Stretch

To get the graph of \( f(x) = 3 \log_{2}(x) \), stretch the graph of \( y = \log_{2}(x) \) vertically by a factor of 3. This means that for any point (x, y) on the original graph, the corresponding point on the new graph will be (x, 3y).

Key Concepts

Logarithmic Transformations

Let's explore the concept of logarithmic transformations, a key step in graphing complex logarithmic functions. Normally, we understand logarithms as the inverse of exponential functions, which allow us to work with multiplicative relationships easily. In this exercise, we encounter a transformation where we deal with the function \( f(x) = \log_{2}(x^3) \).
By using the identity \( \log_b(a^c) = c \log_b(a) \), we simplify the original function into \( f(x) = 3 \log_{2}(x) \).
This transformation involves the exponentiation within the logarithmic function and converts it into a simpler expression that's easier to graph. This highlights how logarithmic identities can simplify complex functions, making them more accessible for analysis.

• Important: The transformation is crucial for understanding the scaling or change in the function.
• Remember: Always try to simplify using logarithmic identities.

This step is a fundamental precursor to applying other graphical transformations, like stretching or shrinking.

Vertical Stretching

The vertical stretching of a graph is a type of transformation that alters the appearance of a function's graph, by multiplying all y-values by a constant. Here, we need to stretch the function \( f(x) = 3 \log_{2}(x) \). Vertical stretching by a factor occurs when we multiply the entire function by a constant, in this case, 3.
This constant affects the steepness without changing the domain.
This means that every output value of the logarithmic function will be tripled.

• For every point \((x, y)\) on the original \( \log_{2}(x) \) graph, its vertical stretch will be seen as \((x, 3y)\).
• A vertical stretch affects the "height" of the graph rather than its horizontal position.

This effect gives us a more exaggerated, amplified version of the base graph, which helps showcase transformations in amplitude without altering the x-intercept or the shape's x-stretch.

Base Logarithmic Function

Understanding the base logarithmic function \( \log_{2}(x) \) is essential before applying any transformations. This function forms the foundation upon which other modifications like stretching are made. The graph of \( y = \log_{2}(x) \) starts at the point (1,0), because \( \log_{2}(1) = 0 \).
The function only exists for \( x > 0 \) and increases very slowly as \( x \) increases.
The curve never touches the y-axis, staying undefined for negative x-values.

• Key Feature: The logarithmic curve passes through (1,0). This point is crucial as it is the x-intercept.
• Approach: The curve approaches infinity as \( x \to \infty \), showing the unbounded nature of logarithms.

Recognizing these characteristics of the base function helps in analyzing how additional transformations, such as the vertical stretch, impact the overall graph.