Which of the following correctly identifies the transformation undergone from f(x)= x^3 to f(-5x+1)=(-5+1)^3

Question

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

Answer 1

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.