Which of the following statements correctly describes the transformations required to obtain the graph of [math: y=13x+6−2] from [math: y=1x]?

Question

BlackTom AI · Accepted Answer

Rewrite the expression inside the denominator: 3x + 6 = 3(x + 2), so
y = 1/(3x + 6) − 2 = 1/[3(x + 2)] − 2 = (1/3)·[1/(x + 2)] − 2.

1. Relevant knowledge points and concepts
- For f(x) = 1/x, replacing x by (x + a) produces a horizontal shift: f(x + a) shifts the graph left by a (if a > 0).
- Replacing x by b·x (i.e. f(bx)) produces a horizontal dilation (scale) by factor 1/b from the y-axis.
- Multiplying the whole function by a constant c (c·f(x)) is a vertical dilation by factor c from the x-axis.
- Adding/subtracting outside the function translates the graph up/down.

2. Step-by-step reasoning
Start with f(x) = 1/x.
- The factor (x + 2) inside gives f(x + 2) = 1/(x + 2), which is the graph of 1/x shifted 2 units left.
- The factor 3 multiplying (x + 2) gives f(3(x + 2)) = 1/[3(x + 2)]. In terms of horizontal dilation, f(3x) is a horizontal dilation by factor 1/3 from the y-axis (points move to x_new = x_old/3). Thus the 3 produces a horizontal compression of the original 1/x graph by factor 1/3 about the y-axis.
- Finally the −2 outside shifts the entire graph down by 2 units.

3. Intermediate algebraic steps shown
3x + 6 = 3(x + 2)
y = 1/(3x + 6) − 2 = 1/[3(x + 2)] − 2 = (1/3)·[1/(x + 2)] − 2

4. Why the correct answer is correct
The decomposition shows a horizontal dilation by factor 1/3 (because of the multiplier 3 inside the argument) measured from the y-axis, a horizontal translation 2 units left (x → x + 2), and a vertical translation 2 units down (−2). These three combined give the final graph.

5. Why other options are incorrect (briefly)
- Options that claim dilation from the x-axis confuse vertical versus horizontal scaling; the 3 is inside the input so it produces a horizontal effect (y-axis dilation), not a vertical dilation from the x-axis.
- Options stating translations to the right are wrong because x + 2 shifts left, not right.
- Options that use factor 3 instead of 1/3 describe the opposite (an expansion rather than the correct compression).

6. Conclusion
Therefore the correct description is option A: Dilation of factor 1/3 from the y-axis, translation 2 units left and 2 units down — which matches the provided answer A.

Which of the following statements correctly describes the transformations required to obtain the graph of [math: y=13x+6−2] from [math: y=1x]?单项选择题

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