Describe the translation of ƒ(x) = x³ to g(x)

Question

A vertical shift is a translation where the function moves up or down on the y-axis
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1 point
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g(x) = x³ - 4
g(x) = x³ + 2
g(x) = x³
g(x) = x³ + 4

Answer 1

To determine the translation of the function \( f(x) = x^3 \) to \( g(x) \) based on the options provided, we need to examine how vertical shifts affect the function.

1. A vertical shift of \( k \) units is represented by adding \( k \) to the function:
   • If \( k > 0 \), the function shifts upwards by \( k \) units.
   • If \( k < 0 \), the function shifts downwards by \( |k| \) units.

Analyzing the given options:

• \( g(x) = x^3 - 4 \): This represents a downward shift by 4 units.
• \( g(x) = x^3 + 2 \): This represents an upward shift by 2 units.
• \( g(x) = x^3 \): This represents no shift (the same function).
• \( g(x) = x^3 + 4 \): This represents an upward shift by 4 units.

So, depending on the specific context of the question (i.e., whether you're looking for an upward or downward shift), the translations could be identified accordingly.

Since you mentioned that a vertical shift generally relates to movement up or down, here are potential interpretations of the translations:

• If you want an upward shift, the candidates would be \( g(x) = x^3 + 2 \) or \( g(x) = x^3 + 4 \).
• If you want a downward shift, the candidate would be \( g(x) = x^3 - 4 \).

If asked which function represents a vertical shift upwards by 4 units, the answer would be \( g(x) = x^3 + 4 \). Conversely, if looking for a downward shift, it would be \( g(x) = x^3 - 4 \).

Please specify if you want details about a specific type of shift or context!