Question textThe circular function [math: f(x)=sin(x)] has been dilated in the [math: y] axis direction by a factor [math: 6]\displaystyle {6}, translated [math: 2π3]\displaystyle {\frac{2\,\pi}{3}} units in the positive direction of the [math: x] axis and translated [math: 6]\displaystyle {6} units in the negative direction of the [math: y] axis. The resulted function is [math: f(x)]=[input] Check Question 1多项填空题
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Question

Question textThe circular function [math: f(x)=sin(x)] has been dilated in the [math: y] axis direction by a factor [math: 6]\displaystyle {6}, translated [math: 2π3]\displaystyle {\frac{2\,\pi}{3}} units in the positive direction of the [math: x] axis and translated [math: 6]\displaystyle {6} units in the negative direction of the [math: y] axis. The resulted function is [math: f(x)]=[input] Check Question 1多项填空题

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BlackTom AI · Accepted Answer

The problem describes a sequence of transformations applied to the base function f(x) = sin(x).
First, a dilation in the y-axis by a factor of 6 scales the amplitude, turning sin(x) into 6·sin(x).
Next, a translation to the right by 2π/3 units changes the argument by subtracting 2π/3, giving sin(x − 2π/3) inside the amplitude-scaled form, so we have 6·sin(x − 2π/3).
Finally, a translation downward by 6 units shifts the whole graph down by 6, resulting in 6·sin(x − 2π/3) − 6.
Therefore, the resultant function is f(x) = 6 sin(x − 2π/3) − 6, which matches the provided answer exactly.
Note on the provided options: there is only one option listed, which is "6sin(x - 2π/3) - 6". Since no other choices are given, there is no comparison to be made among multiple incorrect forms in this case.