Interpret the key features of the graph what type of transformation occurred from the parent function f of x equals 4x squared there's a point at negative 2, 16 2, 16 -1, 4 1, 4 and 0, 0
Answers
Answered by
GPT-4o mini
To interpret the key features of the graph and describe the transformation from the parent function \( f(x) = 4x^2 \), we can break down the information provided, specifically looking at the points mentioned: (-2, 16), (2, 16), (-1, 4), (1, 4), and (0, 0).
1. **Identifying the Parent Function:**
The parent function is \( f(x) = 4x^2 \). This is a parabola that opens upwards with a vertex at the origin (0,0).
2. **Analyzing Given Points:**
- The point (0, 0) confirms that the vertex remains at the origin.
- The points (1, 4) and (-1, 4) indicate that when \( x = 1 \) and \( x = -1 \), the output \( f(x) \) is 4. This corresponds to \( f(1) = 4(1^2) = 4 \) and similarly for -1. It confirms that the shape of the parabola is maintained.
- The points (2, 16) and (-2, 16) suggest that at \( x = 2 \) and \( x = -2 \), \( f(x) = 16\). This corresponds to \( f(2) = 4(2^2) = 16 \) and similarly for -2.
3. **Comparing with the Parent Function:**
The provided points reflect the parabolic shape of \( f(x) = 4x^2 \), but we need to determine if any transformations have occurred:
- The vertex at (0, 0) indicates there is no vertical shift.
- Points at \( (1, 4) \) and \( (-1, 4) \) suggest the graph rises rapidly due to the coefficient of \( 4 \) in front of \( x^2 \).
4. **Understanding Transformations:**
The presence of the points indicates that the graph maintains the same shape and orientation as the parent function \( f(x) = 4x^2 \) but may have undergone a horizontal stretch or compression.
- **Vertical Stretch:** The graph of \( f(x) = 4x^2 \) is vertically stretched compared to \( g(x) = x^2 \). The coefficient of 4 means that for any given \( x \), the \( y \)-value of the transformed graph is four times that of the basic \( x^2 \) graph.
5. **Summary:**
The key features of the observed graph suggest that:
- The graph is a vertically stretched version of the parent function \( f(x) = x^2 \) due to the factor of 4.
- No shifts or reflections have occurred, meaning the vertex remains at the origin.
- The parabolic shape is preserved as indicated by the symmetry of points around the y-axis.
In conclusion, the function appears to be \( f(x) = 4x^2 \) and has undergone a vertical stretch without any additional transformations.
1. **Identifying the Parent Function:**
The parent function is \( f(x) = 4x^2 \). This is a parabola that opens upwards with a vertex at the origin (0,0).
2. **Analyzing Given Points:**
- The point (0, 0) confirms that the vertex remains at the origin.
- The points (1, 4) and (-1, 4) indicate that when \( x = 1 \) and \( x = -1 \), the output \( f(x) \) is 4. This corresponds to \( f(1) = 4(1^2) = 4 \) and similarly for -1. It confirms that the shape of the parabola is maintained.
- The points (2, 16) and (-2, 16) suggest that at \( x = 2 \) and \( x = -2 \), \( f(x) = 16\). This corresponds to \( f(2) = 4(2^2) = 16 \) and similarly for -2.
3. **Comparing with the Parent Function:**
The provided points reflect the parabolic shape of \( f(x) = 4x^2 \), but we need to determine if any transformations have occurred:
- The vertex at (0, 0) indicates there is no vertical shift.
- Points at \( (1, 4) \) and \( (-1, 4) \) suggest the graph rises rapidly due to the coefficient of \( 4 \) in front of \( x^2 \).
4. **Understanding Transformations:**
The presence of the points indicates that the graph maintains the same shape and orientation as the parent function \( f(x) = 4x^2 \) but may have undergone a horizontal stretch or compression.
- **Vertical Stretch:** The graph of \( f(x) = 4x^2 \) is vertically stretched compared to \( g(x) = x^2 \). The coefficient of 4 means that for any given \( x \), the \( y \)-value of the transformed graph is four times that of the basic \( x^2 \) graph.
5. **Summary:**
The key features of the observed graph suggest that:
- The graph is a vertically stretched version of the parent function \( f(x) = x^2 \) due to the factor of 4.
- No shifts or reflections have occurred, meaning the vertex remains at the origin.
- The parabolic shape is preserved as indicated by the symmetry of points around the y-axis.
In conclusion, the function appears to be \( f(x) = 4x^2 \) and has undergone a vertical stretch without any additional transformations.