Math Problem Statement
Solution
The given function is , and the problem asks for the equation of the graphed function shown in the image. The options provided are:
A.
B.
C.
D.
Let's analyze the transformations step-by-step:
-
Horizontal Shifts:
A shift left or right is represented by . If the graph shifts left by units, it becomes , and if it shifts right by units, it becomes . -
Vertical Shifts:
A vertical shift is represented by adding or subtracting a constant outside the function: .
Analysis of the Graph:
-
Horizontal Shift:
The graph of is typically centered at the origin . The graph in the image appears to have been shifted right by 2 units. This corresponds to the transformation . -
Vertical Shift:
The vertex of the parabola normally starts at . In the graph, it has moved up by 3 units. This corresponds to adding 3 to the function.
Combining Transformations:
Combining the horizontal and vertical transformations, the equation of the transformed function is:
This corresponds to option D: .
Would you like more details or have any questions?
Related Questions:
- How do vertical transformations affect the graph of a quadratic function?
- How can you identify horizontal shifts in polynomial functions?
- What are the effects of reflecting a function over the x-axis or y-axis?
- How do you find the vertex of a transformed quadratic function?
- What is the general form of a transformed quadratic function?
Tip: Always check both the horizontal and vertical shifts independently to accurately determine the function's transformation.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Function Transformations
Formulas
f(x) = (x-h)^2 + k
Theorems
Quadratic Transformation Theorem
Suitable Grade Level
Grades 9-11
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