Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
