I’m trying to wrap my head around simple function transformations, and I keep mixing up what the “inside” vs “outside” does. I set f(x) = x^2 and defined g(x) = -2 f((1/3)x + 4) – 5. I love the idea that inside messes with x and outside messes with y, but my brain keeps flipping signs and scales.
Here’s my (probably wrong) attempt: I figured “+4” inside means shift right 4, the (1/3) means a horizontal shrink by 1/3, the “-2” means reflect across the y-axis, and the “-5” means move up 5. So I even convinced myself the vertex would be at (0, -37). That can’t be right, can it?
What’s the correct sequence of transformations here, and in what order should I apply them so I don’t keep tripping up? Do I need to factor something inside to see the horizontal shift properly? If I track a specific point like (1, 1) from y = x^2 through each step, where should it land on g? Any help appreciated!
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3 Responses
Oof, inside–outside is a mischievous duo! I always want to say “+4 means go right 4,” but then the algebra taps me on the shoulder and says, “Factor me first!” For g(x) = -2 f((1/3)x + 4) – 5 with f(x) = x^2, the key is to rewrite the inside as (1/3)(x + 12). That shows the horizontal story clearly: x + 12 means shift left 12, and the 1/3 means a horizontal stretch by a factor of 3 (not a shrink). Outside, the -2 flips across the x-axis and scales vertically by 2, and the -5 pushes everything down 5. If you expand it, g(x) = -2[(1/3)(x + 12)]^2 – 5 = -(2/9)(x + 12)^2 – 5, so the vertex lives at (-12, -5). I’m 93% sure the parabola opens downward and looks wide, because of that 2/9 in front.
A handy “don’t trip” trick: inside changes act on x in the opposite-feeling way, but factoring reveals the true shift. If you like to track a point, take (1, 1) on y = x^2. First match the inside: solve (1/3)x + 4 = 1 to find where that input comes from, giving x = -9. That places you at (-9, 1) before the outside stuff. Then apply the outside: multiply y by -2 to get -2, and shift down 5 to land at y = -7. So (1, 1) on f becomes (-9, -7) on g. As a bonus check, the vertex (0, 0) of f comes from solving (1/3)x + 4 = 0, i.e., x = -12, and then y goes 0 → -5, confirming the vertex (-12, -5).
Factor the inside: (1/3)x + 4 = (1/3)(x + 12), so do inside moves in this order-horizontal stretch by 3, then shift left 12-then do the outside normally: multiply by -2 (flip over x-axis, vertical stretch 2) and shift down 5, giving the vertex at (-12, -5). Example: track (1,1) by solving (1/3)x+4=1 → x=-9, then y=-2·1-5=-7, so (1,1) lands at (-9, -7).
Quick trick: factor the inside-(1/3)x + 4 = (x + 12)/3-so from y = x^2 you go horizontal stretch by 3, then shift left 12, then reflect across the x-axis and scale by 2, then shift down 5, giving vertex at (-12, -5); tracking (1,1): (1,1) → (3,1) → (-9,1) → (-9,-2) → (-9,-7).