write a rule if f(x) = x^2-3x+1, vertically stretched by a factor of 3 and a reflection in the y-axis followed by a translation 2 units left.
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Step-by-step explanation:
We may rewrite f(x) = (x-1)^2
We may rewrite f(x) = (x-1)^2We stretch by having 3(x-1)^2
We may rewrite f(x) = (x-1)^2We stretch by having 3(x-1)^2We reflect about the y axis by changing x to -x: 3(-x-1)^2 = 3(x+1)^2
We may rewrite f(x) = (x-1)^2We stretch by having 3(x-1)^2We reflect about the y axis by changing x to -x: 3(-x-1)^2 = 3(x+1)^2We move to the left by adding 2 to x: 3(x+2+1)^2 = 3(x+3)^2
We may rewrite f(x) = (x-1)^2We stretch by having 3(x-1)^2We reflect about the y axis by changing x to -x: 3(-x-1)^2 = 3(x+1)^2We move to the left by adding 2 to x: 3(x+2+1)^2 = 3(x+3)^2A way to verify this: Consider y = x^2 to be the unit parabola. f(x) = (x-1)^2 moves the unit parabola one to the right, so it is symmetric about x = 1; We then vertically stretch it by a factor of 3; reflection about the y-axis moves the parabola to be symmetric about x = -1; moving to the left by 2 means the parabola is now symmetric about x = -3; thus, we have a stretched unit parabola symmetric about x = -3, so g(x) = 3(x+3)^2
We may rewrite f(x) = (x-1)^2We stretch by having 3(x-1)^2We reflect about the y axis by changing x to -x: 3(-x-1)^2 = 3(x+1)^2We move to the left by adding 2 to x: 3(x+2+1)^2 = 3(x+3)^2A way to verify this: Consider y = x^2 to be the unit parabola. f(x) = (x-1)^2 moves the unit parabola one to the right, so it is symmetric about x = 1; We then vertically stretch it by a factor of 3; reflection about the y-axis moves the parabola to be symmetric about x = -1; moving to the left by 2 means the parabola is now symmetric about x = -3; thus, we have a stretched unit parabola symmetric about x = -3, so g(x) = 3(x+3)^2We may write g(x) = 3x^2 + 18x + 27, also.
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