Question

Suppose that the function LaTeX: f(x) f ( x ) is shifted horizontally to the right by LaTeX: a a , reflected across the LaTeX: x x -axis, and shifted vertically down by LaTeX: b b to become the function LaTeX: g(x) g ( x ) . Write the function LaTeX: g(x) g ( x ) in terms of LaTeX: f(x) f ( x ) and explain each of the shifts

Answer 1

Answer:

$g(x) = (x-a,-y - b)$

Step-by-step explanation:

Given

Represent f(x) as follows:

$f(x) = (x,y)$

Transformations:

Horizontally shifted right by a

Reflected across x axis

Vertically shifted down by b

Taking the transformations one after the other.

Horizontally shifted right by a

When a function is shifted right, the resulting function is:

$f' = (x-a,y)$

Reflected across x axis

Here, the x axis remains unaltered while the y axis is negated

$f' = (x-a,y)$

becomes

$f" = (x-a,-y)$

Vertically shifted down by b

When a function is shifted down by b, the resulting function is:

$f"' = f" - b$

i.e, subtract b from the function (f(x) or y, as the case may be)

So, we have:

$f"' = (x-a,-y - b)$

Represent f"' with g(x)

$g(x) = (x-a,-y - b)$