Question

1) Given the function f(x) = x^2. If the function is shifted left 4 units then shifted up by 7 units and reflected over the y-axis, what is the equation of the new function?

2) Given the function f(x) = √x. If the function is shifted right 4 units, then shifted down 5 units and reflected about the y−axis, what is the equation of the new function?

Answer 1

Answer:

1. g(x) = x^2 -8x +23
2. g(x) = √(-x -4) -5

Step-by-step explanation:

The transformations of interest are ...

  f(-x) . . . . reflects across the y-axis

  f(x -h) +k . . . . translates by (h, k)

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1)

The translation left 4 and up 7 is translation by (-4, 7), so the translated function is ...

  f(x +4) +7

Reflection over the y-axis replaces x with -x (and also reflects the left-translation). The reflected translated function is ...

  g(x) = f(-x +4) +7 = (-x +4)^2 +7 = x^2 -8x +16 +7

  g(x) = x^2 -8x +23

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2)

Translation right 4 and down 5 is translation by (4, -5), so the translated function is ...

  f(x -4) -5

Reflection over the y-axis replaces x with -x. The new equation is ...

  g(x) = f(-x -4) -5

  g(x) = √(-x -4) -5