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3 Responses
A quick, reliable way is: treat the inside as an input filter and the outside as an output filter. For the inside, set u = 3(x−1) and solve for x to see what happens to the graph’s x-coordinates: x = u/3 + 1. That tells you the actual order on the graph: first a horizontal compression by a factor of 3 (distances from the y-axis divided by 3), then a shift right by 1. For the outside, read in the obvious order: y → −2y + 4 means vertical stretch by 2, reflect across the x-axis, then shift up 4.
So for y = −2·f(3(x−1)) + 4:
– Horizontal: compress by 3, then shift right 1.
– Vertical: stretch by 2, reflect in the x-axis, then shift up 4.
Small analogy: think of a two-stage machine-first the “input knob” squeezes the x-values and slides them right, then the “output lever” flips and scales the heights and lifts the whole graph up. As a quick check, if f has a point (3, 2), it lands at x = 3/3 + 1 = 2, and y = −2·2 + 4 = 0, so (3,2) → (2,0), which matches “compress by 3, right 1; then reflect, stretch, up 4.”
A quick, reliable way is: changes inside f(·) act on x (horizontal), and changes outside act on y (vertical). For y = -2·f(3(x−1)) + 4, read it inside-out: x→x−1 gives a shift right by 1; the 3·(x−1) compresses horizontally by a factor of 3 (equivalently, scale factor 1/3); multiplying the whole f by −2 reflects across the x-axis and stretches vertically by 2; finally, +4 shifts up 4. A neat check is to solve 3(x−1)=u, so x=u/3+1: a point (u, f(u)) on the original becomes (u/3+1, −2f(u)+4) on the new graph, which keeps horizontal vs vertical straight. Example: if f(x)=x^2, then y=−2·f(3(x−1))+4 = −2·[3(x−1)]^2+4 = −18(x−1)^2+4, so the vertex moves to (1,4), the parabola is 3 times narrower, flipped upside down, and stretched by 2 vertically; the point (u,y)=(3,9) on y=x^2 maps to (3/3+1, −2·9+4)=(2, −14). In practice you can think “right 1, then horizontal compress by 3” for the inside; the order of these two horizontal steps doesn’t really matter for the final picture, as long as you keep the amounts consistent. For more examples and a systematic overview, see Khan Academy’s transformations of functions: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:shift-functions/v/shifting-functions-introduction
Quick decode: y = -2·f(3(x−1)) + 4 means shift right 1, compress horizontally by factor 3 (about x=1), reflect across the x-axis, stretch vertically by 2, then shift up 4-inside is x-stuff, outside is y-stuff. Does it click more if you think “x → 3(x−1)” or if you map points via u=3(x−1) so x=u/3+1? (Nice refresher: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:functions-transformations/a/transformations-of-functions)