Question

given an existing function: f(x)=0.5(x-2)2+3, what transformstiins would have to be made to result in g(x)=-2(x+3)2 -1?

Answer 1

Answer:

• vertical scaling by a factor of -4
• horizontal translation 5 units left
• vertical translation 11 units up

Step-by-step explanation:

We notice that the multiplier of the squared term in f(x) is 0.5; in g(x), it is -2, so is a factor of -4 times that in f(x).

If we scale f(x) by a factor of -4, we get ...

  -4f(x) = -2(x -2)² -12

In order for the squared quantity to be x+3, we have to add 5 to the value that is squared in f(x). That is, x -2 must become x +3. We have to replace x with (x+5) to do that, so ...

  (x+5) -2 = x +3

The replacement of x with x+5 amounts to a translation of 5 units to the left.

We note that the added constant after our scaling changes from +3 to -12. Instead, we want it to be -1, so we must add 11 to the scaled function. That translates it upward by 11 units.

The attached graph shows the scaled and translated function g(x):

  g(x) = -4f(x +5) +11