Which of the following correctly identifies the transformations undergone from f(x)=x

Question

𝑓
(
𝑥
)
=
𝑥
to f(3x)−13 =3x−13
𝑓
(
3
𝑥
)
−
1
3

=
3
𝑥
−
1
3
(1 point)
Responses

compressed horizontally and shifted down
compressed horizontally and shifted down

reflected over the y-axis and shifted up
reflected over the y -axis and shifted up

stretched horizontally and shifted down
stretched horizontally and shifted down

reflected over the x-axis and shifted up

Answer 1

To analyze the transformation from \( f(x) = x \) to \( f(3x) - 13 = 3x - 13 \), we can break it down step by step.

1. Start with \( f(x) = x \):

   • This is simply the identity function.

2. Transform to \( f(3x) \):

   • The expression \( f(3x) \) means we are compressing the function horizontally by a factor of 3. This is because for every value of \( x \), we are now evaluating the function at \( 3x \) instead of \( x \). Consequently, this transformation results in the line \( y = 3x \).

3. Transform to \( f(3x) - 13 \):

   • Finally, subtracting 13 from the function translates the graph vertically downward by 13 units.

Putting these transformations together, we observe that the function is compressed horizontally (due to the \( 3x \)) and translated down (due to the \(-13\)).

Therefore, the correct identification of the transformations is:

compressed horizontally and shifted down.