Question
Analyze the graph of cubic function f(x)=x3
𝑓
(
𝑥
)
=
𝑥
3
and the transformed function g(x)
𝑔
(
𝑥
)
; then determine by how much the graph of f(x)
𝑓
(
𝑥
)
shifts horizontally to the right.
(1 point)
The function f(x)
𝑓
(
𝑥
)
shifts horizontally units to the right.
𝑓
(
𝑥
)
=
𝑥
3
and the transformed function g(x)
𝑔
(
𝑥
)
; then determine by how much the graph of f(x)
𝑓
(
𝑥
)
shifts horizontally to the right.
(1 point)
The function f(x)
𝑓
(
𝑥
)
shifts horizontally units to the right.
Answers
GPT-4o mini
To analyze the cubic function \( f(x) = x^3 \) and a transformed function \( g(x) \), we need to understand how transformations affect the graph of the function.
### Analyzing \( f(x) = x^3 \):
1. **Shape**: The graph of \( f(x) = x^3 \) is symmetric about the origin and has a characteristic "S" shape.
2. **Intercepts**:
- The function intersects the origin (0,0) since \( f(0) = 0^3 = 0 \).
3. **Behavior**:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to \infty \), \( f(x) \to \infty \).
4. **Increasing/Decreasing**: The function is increasing for all \( x \).
### Horizontal Shifts:
A horizontal shift occurs in the function when it's written in the form:
\[ g(x) = f(x - h) = (x - h)^3 \]
In this case, the graph shifts to the right by \( h \) units if \( h > 0 \) and shifts to the left if \( h < 0 \).
### Conclusion:
To determine how much \( f(x) \) shifts horizontally in the transformed function \( g(x) \), we would specifically need the expression for \( g(x) \).
Assuming the transformation is given or provided as \( g(x) = (x - h)^3\):
- The graph of \( f(x) = x^3 \) shifts to the right by \( h \) units.
In absence of a specific expression for \( g(x) \), if \( h > 0 \), then:
**The function \( f(x) \) shifts horizontally \( h \) units to the right.**
You can specify a particular \( h \) based on the provided transformation if available (for example, \( g(x) = (x - 2)^3 \) would imply a shift of 2 units to the right).
### Analyzing \( f(x) = x^3 \):
1. **Shape**: The graph of \( f(x) = x^3 \) is symmetric about the origin and has a characteristic "S" shape.
2. **Intercepts**:
- The function intersects the origin (0,0) since \( f(0) = 0^3 = 0 \).
3. **Behavior**:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to \infty \), \( f(x) \to \infty \).
4. **Increasing/Decreasing**: The function is increasing for all \( x \).
### Horizontal Shifts:
A horizontal shift occurs in the function when it's written in the form:
\[ g(x) = f(x - h) = (x - h)^3 \]
In this case, the graph shifts to the right by \( h \) units if \( h > 0 \) and shifts to the left if \( h < 0 \).
### Conclusion:
To determine how much \( f(x) \) shifts horizontally in the transformed function \( g(x) \), we would specifically need the expression for \( g(x) \).
Assuming the transformation is given or provided as \( g(x) = (x - h)^3\):
- The graph of \( f(x) = x^3 \) shifts to the right by \( h \) units.
In absence of a specific expression for \( g(x) \), if \( h > 0 \), then:
**The function \( f(x) \) shifts horizontally \( h \) units to the right.**
You can specify a particular \( h \) based on the provided transformation if available (for example, \( g(x) = (x - 2)^3 \) would imply a shift of 2 units to the right).