Problem 94 - Algebra 2

Question

Which function is a shrink of \(y=\log x\) by a factor of \(\frac{1}{2} ?\) $$ \begin{array}{llll}{\text { F. } \log \sqrt{x}} & {\text { G. } \log x^{2}} & {\text { H. } \log \left(x-\frac{1}{2}\right)} & {\text { I. } \log \left(x+\frac{1}{2}\right)}\end{array} $$

Step-by-Step Solution

Answer

The function \(y = \log \sqrt{x}\) is a shrink of \(y = \log x\) by a factor of \(\frac{1}{2}\). Therefore, the correct answer is F.

Step 1: Identify the shrinking factor in the function

The function \(y = \log x\) is said to be shrunk by a factor of \(\frac{1}{2}\). This means that the change we seek should be inside the argument of the logarithmic function because a horizontal shrink or stretch happens in the argument of the function. Therefore, the values of \(x\) are the ones that are halved, not the values of \(y\).

Step 2: Review each option

Now we can analyze each of the provided options: \n\n1. \(y = \log \sqrt{x}\)\nA factor of \(\frac{1}{2}\) can be thought of as a square root, so the argument inside the logarithmic function is correct. \n\n2. \(y = \log x^{2}\)\nThis function actually stretches the original function by a factor of \(2\), so it cannot be the answer. \n\n3. \(y = \log (x - \frac{1}{2})\)\nThis function shifts the original function to the right by \(\frac{1}{2}\), not a shrink. \n\n4. \(y = \log (x + \frac{1}{2})\)\nThis function shifts the original function to the left by \(\frac{1}{2}\), not a shrink.

Step 3: Choose the correct answer

From the above analysis, it becomes clear that the function which represents a shrink of \(y = \log x\) by a factor of \(\frac{1}{2}\) is the function \(y = \log \sqrt{x}\). Therefore, the correct answer is F.

Key Concepts

Function Shrink

When we talk about a "shrink" in functions, we're usually referring to a transformation that compresses the graph of the function in some way. For the logarithmic function \( y = \log x \), a shrink by a factor of \( \frac{1}{2} \) means that the rate at which the function increases is slowed down.
In mathematical terms, this is achieved by modifying the argument of the logarithm. Instead of \( \log x \), we consider \( \log (\sqrt{x}) \), because taking the square root of \( x \) is equivalent to raising \( x \) to the power of \( \frac{1}{2} \). This effectively compresses the function horizontally.

• A shrink compresses the function towards the y-axis.
• For \( y = \log \sqrt{x} \), notice how quickly it approaches the vertical line as compared to \( y = \log x \).

Understanding function shrink is crucial when solving problems that involve graph transformations because it helps visualize how original functions change shape.

Horizontal Transformations

Horizontal transformations impact how a function's graph shifts or reshapes along the x-axis. These transformations can involve shifts (translations), stretches, or shrinks.
In logarithmic functions like \( y = \log x \), a horizontal transformation may occur within the logarithmic argument \( x \). For instance, to achieve a horizontal shrink or stretch, you modify the argument of the log. Consider that if you have a factor in the function like \( \log(\sqrt{x}) \), this constitutes a horizontal shrink.

• Horizontal Shifts: Adding or subtracting a constant from \( x \) shifts the graph left or right (e.g., \( \log(x - \frac{1}{2}) \)).
• Horizontal Stretches/Shrinks: Multiplying \( x \) by a factor compresses or expands the graph.

Recognizing these transformations helps in understanding how to manipulate and interpret the graphs of logarithmic equations.

Logarithmic Equations

Logarithmic equations are equations involving logarithms, typically in the form \( y = \log_b(x) \), where \( b \) is the base and \( x \) is the argument. These equations are a powerful tool in solving problems involving exponential growth or decay.
To solve logarithmic equations, one often needs to rewrite them using the properties of logarithms, such as the power rule: \( \log_b(x^a) = a\log_b(x) \). For example, \( \log(x^2) = 2\log(x) \) suggests a horizontal stretch by a factor of 2, rather than a shrink.

• Understand how the base and the argument transform the graph.
• Complications often arise with negative arguments or when base \( b \) is less than 1.

Mastering these concepts requires practice in rewriting and solving equations and understanding their graphs.