Example Question - vertical shift
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.
Understanding Function Transformations through Graphs
The image presents a math problem with two graphs. The graph of a function f(x) is shown in gray, and the graph of another function g(x) is shown in pink. We are informed that g(x) has the same shape as f(x), which implies that g(x) is a transformation of f(x). The transformation appears to involve both a horizontal shift to the right and a vertical shift downward. Specifically, the point (0,1) on f(x) has been mapped to the point (3,0) on g(x). This indicates a horizontal shift of 3 units to the right and a vertical shift of 1 unit down. Now, let's examine the provided options to determine the correct equation for g(x): A. \( g(x) = f(x − 3)^2 − 1 \): Incorrect because this suggests f(x) is squared after a shift, which is inconsistent with uniform scaling. B. \( g(x) = f(x − 1)^2 − 3 \): Incorrect because the inner subtraction should reflect a horizontal shift of 3 units, not 1. C. \( g(x) = f(x − 3) − 3 \): Almost correct, but it shows an extra vertical shift down by 3 units instead of 1 unit. D. \( g(x) = f(x − 3) − 1 \): Correct because this represents a rightward shift of 3 units (x − 3) and a downward shift of 1 unit (− 1). Therefore, the correct answer is: D. \( g(x) = f(x − 3) − 1 \)
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