Answer :
To understand what the graph of \( g(x) = f(4x) \) looks like, let us analyze the given functions step by step.
Given the function \( f(x) = x^2 \), we need to determine the form of the transformed function \( g(x) \).
### Step-by-Step Analysis
1. Original Function:
\( f(x) = x^2 \) is a standard quadratic function with its graph being a parabola that opens upwards, with the vertex at the origin \((0, 0)\).
2. Transformation:
We want to find \( g(x) = f(4x) \).
Substituting \( 4x \) into the function \( f \):
[tex]\[ g(x) = f(4x) = (4x)^2 \][/tex]
Simplifying, we get:
[tex]\[ g(x) = 16x^2 \][/tex]
### Graphical Interpretation
3. Effect on the Graph:
- The transformation from \( f(x) = x^2 \) to \( g(x) = 16x^2 \) affects the graph in a specific way: it compresses the graph horizontally by a factor of 4.
- In general, \( g(x) = f(ax) \) horizontally compresses the graph of \( f(x) \) by a factor of \( a \) if \( a > 1 \) (stretching if \( 0 < a < 1 \)).
4. Comparison of Graphs:
- The original graph \( f(x) = x^2 \):
- Vertex at (0, 0),
- Symmetric about the y-axis,
- Parabola opens upwards.
- The modified graph \( g(x) = 16x^2 \):
- Also a parabola that opens upwards,
- Still symmetric about the y-axis,
- The graph is narrower compared to \( f(x) = x^2 \), because every \( x \) value is effectively "multiplied by 4" before squaring.
### Detailed Graph Analysis
5. Critical Points:
- For a few key \( x \)-values, let's compare \( f(x) \) and \( g(x) \):
- At \( x = 1 \):
[tex]\[ f(1) = 1^2 = 1 \quad \text{and} \quad g(1) = 16(1)^2 = 16 \][/tex]
- At \( x = 2 \):
[tex]\[ f(2) = 2^2 = 4 \quad \text{and} \quad g(2) = 16(2)^2 = 64 \][/tex]
- The value of \( g(x) \) grows much faster than \( f(x) \) due to the coefficient 16.
### Conclusion
- The graph of \( g(x) = f(4x) = 16x^2 \) will be a parabola opening upwards, narrower than the original function \( f(x) = x^2 \).
### Graph Sketch
- Original graph \( f(x) = x^2 \):
[tex]\[ \text{Sketch:} \quad \cup \text{ (standard wide parabola)} \][/tex]
- Transformed graph \( g(x) = 16x^2 \):
[tex]\[ \text{Sketch:} \quad \cup \text{ (narrower parabola)} \][/tex]
This visualization guides us to the correct transformed graph, ensuring careful interpretation of horizontal compression by a factor of 4.
Given the function \( f(x) = x^2 \), we need to determine the form of the transformed function \( g(x) \).
### Step-by-Step Analysis
1. Original Function:
\( f(x) = x^2 \) is a standard quadratic function with its graph being a parabola that opens upwards, with the vertex at the origin \((0, 0)\).
2. Transformation:
We want to find \( g(x) = f(4x) \).
Substituting \( 4x \) into the function \( f \):
[tex]\[ g(x) = f(4x) = (4x)^2 \][/tex]
Simplifying, we get:
[tex]\[ g(x) = 16x^2 \][/tex]
### Graphical Interpretation
3. Effect on the Graph:
- The transformation from \( f(x) = x^2 \) to \( g(x) = 16x^2 \) affects the graph in a specific way: it compresses the graph horizontally by a factor of 4.
- In general, \( g(x) = f(ax) \) horizontally compresses the graph of \( f(x) \) by a factor of \( a \) if \( a > 1 \) (stretching if \( 0 < a < 1 \)).
4. Comparison of Graphs:
- The original graph \( f(x) = x^2 \):
- Vertex at (0, 0),
- Symmetric about the y-axis,
- Parabola opens upwards.
- The modified graph \( g(x) = 16x^2 \):
- Also a parabola that opens upwards,
- Still symmetric about the y-axis,
- The graph is narrower compared to \( f(x) = x^2 \), because every \( x \) value is effectively "multiplied by 4" before squaring.
### Detailed Graph Analysis
5. Critical Points:
- For a few key \( x \)-values, let's compare \( f(x) \) and \( g(x) \):
- At \( x = 1 \):
[tex]\[ f(1) = 1^2 = 1 \quad \text{and} \quad g(1) = 16(1)^2 = 16 \][/tex]
- At \( x = 2 \):
[tex]\[ f(2) = 2^2 = 4 \quad \text{and} \quad g(2) = 16(2)^2 = 64 \][/tex]
- The value of \( g(x) \) grows much faster than \( f(x) \) due to the coefficient 16.
### Conclusion
- The graph of \( g(x) = f(4x) = 16x^2 \) will be a parabola opening upwards, narrower than the original function \( f(x) = x^2 \).
### Graph Sketch
- Original graph \( f(x) = x^2 \):
[tex]\[ \text{Sketch:} \quad \cup \text{ (standard wide parabola)} \][/tex]
- Transformed graph \( g(x) = 16x^2 \):
[tex]\[ \text{Sketch:} \quad \cup \text{ (narrower parabola)} \][/tex]
This visualization guides us to the correct transformed graph, ensuring careful interpretation of horizontal compression by a factor of 4.
