Question
Which of the following correctly identifies the transformation undergone from f(x)= x^3 to f(-5x+1)=(-5+1)^3
Reflected across the y-axis, compressed horizontally and translated horizontally to the left
Reflected across the y-axis and compressed horizontally to the left
Reflected across the y-axis and translated to the left horizontally
Reflected across the y-axis, compressed horizontally and translated horizontally to the right
Reflected across the y-axis, compressed horizontally and translated horizontally to the left
Reflected across the y-axis and compressed horizontally to the left
Reflected across the y-axis and translated to the left horizontally
Reflected across the y-axis, compressed horizontally and translated horizontally to the right
Answers
GPT-4o mini
To analyze the transformation from \( f(x) = x^3 \) to \( f(-5x + 1) \), we need to break down the transformation step by step.
1. **Starting Function**: \( f(x) = x^3 \)
2. **Transformation**: We want to evaluate \( f(-5x + 1) \).
- First, replace \( x \) with \( -5x \):
\[
f(-5x) = (-5x)^3 = -125x^3
\]
This represents a **reflection across the y-axis** (because of the negative sign) and a **horizontal compression** (because of the factor of 5).
- Next, replace \( x \) with \( -5x + 1 \):
\[
f(-5x + 1) = (-5x + 1)^3
\]
This represents a **translation horizontally to the left** by 1 unit.
Putting these transformations together:
- The reflection across the y-axis occurs because of the \( -5 \) multiplier.
- The compression occurs because of the factor \( 5 \).
- The translation to the left occurs because of the addition of 1 to the \( x \) value within the function.
Therefore, the correct identification of the transformation is:
**Reflected across the y-axis, compressed horizontally, and translated horizontally to the left**.
Thus, the correct choice from your provided options is:
**Reflected across the y-axis, compressed horizontally and translated horizontally to the left.**
1. **Starting Function**: \( f(x) = x^3 \)
2. **Transformation**: We want to evaluate \( f(-5x + 1) \).
- First, replace \( x \) with \( -5x \):
\[
f(-5x) = (-5x)^3 = -125x^3
\]
This represents a **reflection across the y-axis** (because of the negative sign) and a **horizontal compression** (because of the factor of 5).
- Next, replace \( x \) with \( -5x + 1 \):
\[
f(-5x + 1) = (-5x + 1)^3
\]
This represents a **translation horizontally to the left** by 1 unit.
Putting these transformations together:
- The reflection across the y-axis occurs because of the \( -5 \) multiplier.
- The compression occurs because of the factor \( 5 \).
- The translation to the left occurs because of the addition of 1 to the \( x \) value within the function.
Therefore, the correct identification of the transformation is:
**Reflected across the y-axis, compressed horizontally, and translated horizontally to the left**.
Thus, the correct choice from your provided options is:
**Reflected across the y-axis, compressed horizontally and translated horizontally to the left.**