Problem 54 - College Algebra

Question

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$g(x)=-2 \log _{2} x$$

Step-by-Step Solution

Answer

The x-intercept of the graph of the function \(g(x) = -2\log_2x\) is \(x = 1\) and the vertical asymptote is \(x = 0\).

Step 1: Graph the Base Function

The base function here is \(f(x) = \log_2x\). This is a basic logarithmic function in the form \(f(x) = log_bx\), where the base \(b = 2\). This function passes through the point \((1,0)\) because \(\log_b1 = 0\) for any base \(b\). It also passes through the point \((2,1)\) because \(\log_22 = 1\). Thus, sketch the graph of \(f(x) = log_2x\) passing through \((1,0)\) and \((2,1)\).

Step 2: Transform the Base Function

The function \(g(x) = -2\log_2x\) is a transformation of the base function \(f(x) = \log_2x\). The negative sign in front of the 2 indicates a reflection over the x-axis. The '2' is a vertical stretch by a factor of 2. Reflect and stretch the original graph to obtain the graph of \(g(x) = -2\log_2x\).

Step 3: Find the x-intercept and Vertical Asymptote

The x-intercept is the value of \(x\) where the function intersects the x-axis. This can be found by setting \(g(x) = -2\log_2x = 0\), and solving for \(x\), which gives \(x = 1\). A logarithmic function has a vertical asymptote at \(x = 0\) because the log of zero is undefined. So, the vertical asymptote is \(x = 0\).

Key Concepts

Logarithm Transformations

Graphing logarithmic functions often starts with understanding the base function, such as \(f(x) = \log_2 x\), and applying various transformations to it. Each transformation alters the graph in a specific way.

• Reflection: If a negative sign is placed in front of the function, for example \(y = -\log_2x\), the graph will be reflected over the x-axis. This changes the direction of the curve.
• Vertical Stretch or Compression: When a coefficient other than 1 is placed in front of the logarithm, such as \(y = k\log_2x\) where \(k\) is any constant, the graph will be vertically stretched (if \(k > 1\)) or compressed (if \(0 < k < 1\)). A vertical stretch makes the graph steeper, and a vertical compression makes it less steep.
• Horizontal Shift: Adding or subtracting a number inside the logarithm, like \(\log_2(x - h)\), shifts the graph horizontally. If \(h > 0\), the graph moves to the right. If \(h < 0\), it shifts to the left.
• Vertical Shift: Adding or subtracting a number outside the logarithm, such as \(\log_2x + k\), shifts the graph vertically. Upward for positive \(k\) and downward for negative \(k\).

Understanding how these transformations work helps students to quickly sketch graphs of more complex logarithmic functions without needing to plot points laboriously.

X-intercept of Logarithmic Graph

The x-intercept of a logarithmic graph is crucial because it provides information about when the function's output will be zero. To find the x-intercept, you simply set the function equal to zero and solve for \(x\).
For instance, take the given function \(g(x) = -2\log_2 x\). To find the x-intercept:

1. Set \(g(x)\) to zero: \(0 = -2\log_2 x\).
2. Solve for \(x\): Multiplying both sides by \(\frac{-1}{2}\), we get \(0 = \log_2 x\).
3. By the property that \(\log_b a = 0 \Rightarrow a = 1\), we find \(x = 1\).

This tells us that the point \( (1, 0) \) is the x-intercept of the graph. It's where the function crosses the x-axis. This point is the same for all logarithmic functions in the form \(\log_b(x)\), regardless of the base \(b\), because \(\log_b(1)\) is always zero.

Vertical Asymptote of Logarithm

Logarithmic functions have a characteristic feature called a vertical asymptote. This is a vertical line that the graph approaches but never actually touches or crosses.
To determine the vertical asymptote of a logarithmic function, you look at the domain of the function, which is the set of permissible \(x\)-values that make the function defined. For any logarithm function \(\log_b(x)\), where \(b > 0\), \(x\) must be positive as you cannot take the logarithm of a nonpositive number.
Thus, the vertical asymptote is always located at \(x = 0\), because as \(x\) approaches zero from the right (the positive side), the logarithm goes to negative infinity. No matter what transformations are applied to the function, this vertical asymptote does not change. It is a fundamental characteristic of logarithmic functions and helps define their distinctive shape.