Example Question - transformation analysis
Here are examples of questions we've helped users solve.
Logarithmic Function Transformation Analysis
The logarithmic function \( f(x) = \ln(x) \) is transformed into \( g(x) = \ln(x - 2) + 5 \). To determine which of the provided statements is true, we need to analyze the transformation in two parts: the horizontal shift caused by the change inside the logarithm and the vertical shift caused by the addition outside the logarithm. The \( \ln(x) \) function is shifted horizontally by the presence of \(-2\) inside the function, which means the function is shifted to the right by 2 units. That's because the logarithm’s argument \( x \) is replaced with \( x - 2 \), indicating a rightward shift of 2 units. On the other hand, adding \( +5 \) to the function after the logarithm \( \ln(x - 2) \) indicates a vertical shift. Specifically, it means the graph of the function is shifted upwards by 5 units. Given these transformations: A. Incorrect: \( f(x) \) is not translated up 5 units; it's the function \( g(x) \) that has been translated up by 5 units compared to \( f(x) \). B. Incorrect: \( f(x) \) is not translated down 2 units. C. Incorrect: \( f(x) \) is not translated left 2 units; it's moving to the right. D. Correct: \( f(x) \) is translated right 5 units, which exactly describes the horizontal shift. Therefore, the correct answer is D: \( f(x) \) is translated right 5 units.
Logarithmic Function Transformation Analysis
The given logarithmic function f(x) = log(x) is transformed to g(x) = log(x + 1) + 3. To determine which of the statements is true, we need to analyze the transformation that g(x) represents compared to f(x). - For statement A: f(x) is translated 1 unit upward. This is not true because the function g(x) has a "+3" outside of the logarithm, which means the entire function is translated 3 units upward, not 1 unit. - For statement B: f(x) is translated 3 units downward. This is incorrect as the transformation involves a "+3", indicating an upward translation, not a downward one. - For statement C: The vertical asymptote shifts 1 unit to the left. This is true. The vertical asymptote for the basic logarithmic function f(x) = log(x) is at x=0. Since g(x) is log(x + 1), this transformation shifts the graph horizontally 1 unit to the left, meaning the new vertical asymptote is now at x=-1. - For statement D: The vertical asymptote shifts 3 units to the right. This is not true. The transformation inside the logarithmic function (x + 1) does not shift the graph to the right; it shifts it to the left. Hence, the correct statement is C: The vertical asymptote shifts 1 unit to the left.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com