Identify the parent function and describe its transformations. Then find the equation of the transform function represented by the graph below 1 2 3 C. Equation: -10 List of Transformations (-4,-3) (0.1)
The Correct Answer and Explanation is :
The graph you’ve provided seems to be a representation of a transformed quadratic function. Let’s analyze it step by step to identify the parent function, the transformations, and the equation of the transformed function.
Parent Function
The parent function for a quadratic equation is ( f(x) = x^2 ). This function produces a parabola that opens upwards with its vertex at the origin (0,0).
Identifying Transformations
Based on the graph:
- Vertical Stretch/Compression and Reflection: The parabola opens downwards, indicating a vertical reflection of the parent function. Additionally, the vertex is not at (0,1), suggesting a vertical stretch or compression.
- Vertical Translation: The vertex of the parabola is not at the origin, which means the entire graph has been shifted vertically and horizontally.
- Horizontal Translation: The vertex of the parabola is shifted horizontally from the origin.
Coordinates of Vertex
The vertex of the parabola in the provided graph seems to be at ((-4, -3)). This gives us the horizontal and vertical translations.
Equation of the Transformed Function
Given the vertex form of a quadratic function is:
[ f(x) = a(x-h)^2 + k ]
where ((h, k)) is the vertex of the parabola and (a) determines the vertical stretch/compression and the reflection (if (a) is negative, the parabola opens downwards).
- ( h = -4 )
- ( k = -3 )
- Since the parabola opens downwards and appears relatively steep, let’s consider ( a = -10 ) as indicated.
Thus, the equation of the transformed function based on the vertex form is:
[ f(x) = -10(x + 4)^2 – 3 ]
Explanation
The transformation involves:
- Reflection: The negative coefficient (-10) in front of the quadratic term reflects the graph across the x-axis, causing the parabola to open downwards.
- Vertical Stretch: The coefficient’s magnitude (10) stretches the parabola, making it narrower than the parent parabola.
- Horizontal Translation: Adding 4 inside the squared term ((x + 4)) shifts the entire parabola 4 units to the left of the origin.
- Vertical Translation: Subtracting 3 from the entire function lowers the vertex by 3 units below the x-axis.
These transformations collectively convert the standard parabola into the one depicted in the graph, affecting its shape, orientation, and position on the coordinate plane.
Since I cannot directly access or visualize the image URL you provided, this analysis is based on typical transformations applied to quadratic functions. If there’s a specific aspect of the graph not covered by this description, additional details might be needed to refine the analysis.