Question 1080182: Write a function whose graph represents the indicated transformation of the graph of f. f(x)= x^2-3; reflection in the x axis, followed by a translation 2 units left and 5 units up.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the original function is y = x^2 - 3
to reflect that function about the x-axis, the point (x,y) must become the point (x,-y)
that happens when y = f(x) becomes y = -f(x).
therefore, y = x^2 - 3 becomes y = -(x^2 - 3) which becomes y = -x^2 + 3.
to shift y = f(x) 2 units to the left, then the function becomes y = f(x+2)
therefore, y = -x^2 + 3 becomes y = -(x+2)^2 + 3.
your original function is y = x^2 - 3
your converted function that is reflected about the x-axis and shifted to the left 2 unit is y = -(x+2)^2 + 3
the first graph shows the reflection.
the second graph shows the reflection and the shift.
you can see on the first graph that -3,6 and -3,-6 are both reflections about the x-axis when x = -3, and you can see that 3,6 and 3,-6 are both reflections about the x-axis when x = 3.
you can see on the second graph that the same reflections are true, except that -3,6 on the original graph has its reflection at -5,-6 on the shifted graph, and 3,6 on the original graph has its reflection at 1,-6 on the shifted graph.
here's a reference on transformations that you might find helpful.
https://www.mathsisfun.com/sets/function-transformations.html
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