Example Question - graph translation
Here are examples of questions we've helped users solve.
Transformation of Quadratic Function
The function g(x) is defined in terms of the function f(x) as follows: g(x) = 4[f(x) - 4]. Let's break down what this means for the graph of g(x) relative to the graph of f(x). The function f(x) = x^2 is the original quadratic function. Here's what happens step-by-step: 1. f(x) - 4: This operation translates the graph of f(x) vertically downward by 4 units. This moves every point on the graph of y = x^2 down 4 units on the y-axis. 2. 4[f(x) - 4]: This operation scales the graph of f(x) vertically by a factor of 4 after it has been shifted down by 4 units. Every y-coordinate of the points on the translated graph is multiplied by 4, stretching the graph away from the x-axis if y is positive and towards the x-axis if y is negative. Putting these two transformations together, we see that the original graph of f(x) is first shifted down by 4 units and then stretched vertically by a factor of 4. The correct answer is: D. The graph of g(x) is the graph of f(x) shifted down 4 units.transforms
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.
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