Problem 508 - Precalculus

Question

Graph the function \(g(x)=\log (7 x+21)-4\)

Step-by-Step Solution

Answer

The graph of \(g(x)=\log (7x+21)-4\) has a vertical asymptote at \(x = -3\), shifts 3 units left and 4 units down, and has points (0, -2.678) and (1, -2.553).

Step 1: Identify the Transformations

Start with the basic function: the logarithm function \(f(x) = \log(x)\). For the given function \(g(x)=\log (7x+21)-4\), we see two transformations have occurred:1. **Horizontal transformation:** \(7x+21\) inside the logarithm shifts the graph left. Factor out the 7 to get \(\log[7(x+3)]\), indicating a horizontal shift of 3 units left.2. **Vertical transformation:** The \(-4\) outside the logarithm shifts the graph downward by 4 units.

Step 2: Find the Domain

The domain of the function is determined by the expression inside the logarithm: \(7x+21 > 0\). Solve for \(x\):\[7x + 21 > 0\]\[7x > -21\]\[x > -3\]Thus, the domain of \(g(x)\) is \(x > -3\).

Step 3: Determine the Vertical Asymptote

The vertical asymptote occurs where the logarithmic function is undefined. For \(g(x)\), the logarithmic part becomes undefined when \(7x + 21 = 0\):\[x = -3\]This means there is a vertical asymptote at \(x = -3\).

Step 4: Plot Key Points

Calculate a few key points to better visualize the function:1. **Point at \(x = 0\):** \[g(0) = \log(7 \times 0 + 21) - 4 = \log(21) - 4\] Approximate \(\log(21) \approx 1.322\), thus: \[g(0) \approx 1.322 - 4 = -2.678\]2. **Point at \(x = 1\):** \[g(1) = \log(7 \times 1 + 21) - 4 = \log(28) - 4\] Approximate \(\log(28) \approx 1.447\), thus: \[g(1) \approx 1.447 - 4 = -2.553\]These points, along with the asymptote at \(x = -3\), help to plot the graph.

Step 5: Sketch the Graph

Combine all the previous information to sketch the graph. Start with the vertical asymptote at \(x = -3\), plotting the key points calculated, and adjust for the transformations (3 units to the left and 4 units down). The graph approaches the asymptote from the right and decreases continuously without crossing it.

Key Concepts

Transformations

When exploring the function transformations for logarithmic functions, we aim to see how the changes inside and outside the function affect its graph. For the function \( g(x) = \log(7x + 21) - 4 \), two main transformations are observed:

• Horizontal transformation: This transformation occurs due to the expression \( 7x + 21 \) inside the logarithm. By factoring out a 7, we rewrite it as \( \log[7(x+3)] \), indicating that the logarithm is shifted to the left by 3 units. This is because the expression \( x + 3 \) moves the graph in the opposite direction of the sign.
• Vertical transformation: The subtraction of 4 from the logarithmic expression indicates a downward vertical shift. This means the entire graph of the logarithm is moved down by 4 units.

Understanding these transformations helps us visualize how the graph shifts and moves in the coordinate plane, providing insight into the function's behavior.

Domain of a Function

The domain of a function defines the set of all possible input values (\(x\) values) that the function can accept without resulting in undefined or inappropriate outcomes.
For logarithmic functions, the argument inside the logarithm must be greater than zero. In our case, \( 7x + 21 > 0 \).

• Simplifying this, we solve the inequality: \[7x + 21 > 0\]\[7x > -21\]\[x > -3\]
• Thus, the domain of \( g(x) = \log(7x + 21) - 4 \) is \( x > -3 \).

This domain dictates where the function is defined, directly affecting how we consider the behavior and the graph of \( g(x) \). Understanding the domain helps us to know where the log function has real number outputs.

Vertical Asymptote

A vertical asymptote is a line where the function approaches but never touches or crosses the line as \( x \) approaches a certain value.
For the function \( g(x) = \log(7x + 21) - 4 \), there exists a vertical asymptote where the logarithmic function is undefined.

• The logarithmic part of the function becomes undefined when \( 7x + 21 = 0 \).
• Solving for \( x \), \[7x + 21 = 0\]\[x = -3\]
• Therefore, the vertical asymptote is located at \( x = -3 \).

This asymptote represents a boundary that the graph of the logarithmic function cannot cross, visually indicated by a sharply rising curve close to the line \( x = -3 \). It guides us in predicting the behavior of the function close to this boundary.

Graphing Techniques

Graphing a logarithmic function involves using key features like transformations, domain, and asymptotes to sketch its curve accurately.

• Identify the Asymptote: Mark the vertical asymptote at \( x = -3 \) as a dotted line on the graph, indicating the boundary line the function cannot cross.
• Plot Key Points: Choose key points to understand the behavior of the function between domain limits and transformations. For instance, calculate \( g(0) \approx -2.678 \) and \( g(1) \approx -2.553 \) to see how the function behaves. This makes assembling the graph easier.
• Apply Transformations: Adjust the entire graph left or right due to horizontal shifts and up or down for vertical shifts. For \( g(x) \), shift any plotted points 3 units to the left and 4 units down to account for its transformations.

Begin sketching, ensuring the curve gets closer to the asymptote from the right without touching, finally plotting the major changes in direction and general shape of the curve. These methods will assist in drawing an accurate representation of a logarithmic function, providing a visual interpretation of the mathematical characteristics.