Example Question - vertical translation
Here are examples of questions we've helped users solve.
Understanding Transformation of Functions
The function f(x) = x^2 is given, and the function g(x) is defined as g(x) = 4f(x) - 4 = 4(x^2) - 4. The transformation from f(x) to g(x) involves two steps: 1. The term 4f(x) indicates that the function f(x) is multiplied by 4, which stretches or scales the graph vertically by a factor of 4. This means that for any x, the value of g(x) will be four times the value of f(x), but the shape of the graph remains the same (it will just be stretched). 2. The term -4 at the end of the expression indicates that the graph is then shifted down by 4 units. It is a vertical translation downwards. Putting these together, we get: A vertical stretch of the graph of f(x) by a factor of 4, followed by a downward shift of the graph by 4 units. The correct answer to the question in the image is D: The graph of g(x) is the graph of f(x) shifted down 4 units. The option about stretching is incorrect because the vertical multiplication by 4 does not stretch the graph of f(x) by a factor of 1/4; it stretches it by a factor of 4. The combination of these transformations makes D the best choice among the provided options.
Logarithmic Function Transformations
The given logarithmic function f(x) = ln x is transformed to g(x) = ln(x - 2) + 5. Comparing the function g(x) with f(x): 1. The "+ 5" at the end of g(x) indicates a vertical translation upward by 5 units. 2. The "(x - 2)" inside the logarithm in g(x) suggests a horizontal translation to the right by 2 units since the effect of subtracting inside the function argument is to shift the graph to the right. Given the options provided: A. f(x) is translated up 5 units. (This is correct for the vertical translation.) B. f(x) is translated down 2 units. (This is incorrect; the function is not translated downward.) C. f(x) is translated left 2 units. (This is incorrect; the function is actually translated right.) D. f(x) is translated right 5 units. (This is incorrect; the translation is 2 units to the right, not 5.) Therefore, the correct answer is A. f(x) is translated up 5 units.
Vertical Translation of Quadratic Function
The question is asking which function translates the graph of the parent function f(x) = x^2 vertically up by 6 units. When you need to translate a graph vertically, you add or subtract a constant to the entire function. If you add a positive constant, the graph moves up; if you subtract a positive constant, it moves down. Since we need to move the graph up by 6 units, we will add 6 to the function. Thus, the correct function that represents this transformation is: g(x) = x^2 + 6 The correct answer is A: g(x) = x^2 + 6.
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