Math Problem Statement
Graph the following function using transformations. y equals 4 left parenthesis x minus 1 right parenthesis squared minus 3
Solution
To graph the function using transformations, we can break it down step by step. This function is a transformed version of the basic quadratic function .
Step 1: Start with the parent function
The parent function is: This is a basic parabola that opens upward with its vertex at .
Step 2: Apply horizontal shift
The term indicates a horizontal shift of the graph to the right by 1 unit. Now, the vertex moves from to .
Step 3: Apply vertical stretch
The coefficient 4 in front of means there is a vertical stretch by a factor of 4. This makes the parabola narrower compared to the standard .
Step 4: Apply vertical shift
The term at the end of the function represents a vertical shift. The graph is shifted downward by 3 units. Now, the vertex moves from to .
Summary of Transformations:
- Horizontal shift: Right 1 unit.
- Vertical stretch: Stretched by a factor of 4.
- Vertical shift: Down 3 units.
The final graph is a parabola with the vertex at , opening upwards, and narrower than the standard parabola due to the vertical stretch.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Graph Transformations
Formulas
y = a(x - h)^2 + k
Theorems
Transformations of quadratic functions
Suitable Grade Level
Grades 9-11
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