Question

The parent function f(x)=x^2 is translated such that the function g(x)=-x^2+6x-5 represented the new function. What is true about the transformation that was performed? Check all that apply.

Answer 1

Answer:

g(x) has an axis of symmetry at x = 3

g(x) is shifted right 3 units from the graph of f(x)

g(x) is shifted up 4 units from the graph of f(x)

Step-by-step explanation:

Answer 2

Answer:

• g(x) has an axis of symmetry at x = 3
• g(x) is shifted right 3 units from the graph of f(x)
• g(x) is shifted up 4 units from the graph of f(x)

Step-by-step explanation:

The vertex form of g(x) is ...

... g(x) = -(x -3)² +4

This is offset to the right by 3 and up by 4 from the parent function. (It is also first reflected across the x-axis.)

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Vertex form

You know the leading coefficient is -1 because that's what it is for x² in the given form. When you factor -1 from the first two terms, of the given form, you have ...

... g(x) = -1(x² -6x) -5

Half the x coefficient inside parentheses will be the constant in the squared binomial term, so that term is (x -3)². The constant in that square is +9, so adding that value inside and outside parentheses in g(x) gives ...

... g(x) = -1(x² -6x +9) -5 +9

... g(x) = -(x -3)² +4 . . . . . vertex form

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About transformations

g(x) = f(x -a) causes the graph of f(x) to be shifted "a" units to the right. For a function f(x) with an axis of symmetry at x=0, it moves the axis of symmetry to x=a.

g(x) = f(x) +a causes the graph of f(x) to be shifted "a" units up.

g(x) = -f(x) causes the graph of f(x) to be reflected across the x-axis.

Here, we have all three of these transformations. First is the reflection:

... f₁(x) = -f(x) = -x²

Then we have shifting to the right 3 units. (also moves the axis of symmetry)

... f₂(x) = f₁(x-3) = -(x -3)²

Finally, we have shifting up 4 units.

... g(x) = f₂(x) +4 = -(x -3)² +4