Math Problem Statement
Solution
To solve this problem, let's analyze each function and identify its graph by observing the transformations of the standard quadratic function . The transformations involve vertical shifts and reflections. Here's the approach:
-
:
- This function represents an upward-opening parabola (positive coefficient of ) shifted up by 2 units.
- Look for a graph with a vertex at and opening upwards.
-
:
- This function represents a downward-opening parabola (negative coefficient of ) shifted up by 2 units.
- Look for a graph with a vertex at and opening downwards.
-
:
- This is an upward-opening parabola shifted down by 2 units.
- Look for a graph with a vertex at and opening upwards.
-
:
- This is a downward-opening parabola shifted down by 2 units.
- Look for a graph with a vertex at and opening downwards.
Now, let's match each function to the graphs (A, B, C, D) based on these observations:
- Graph A: Upward-opening parabola with a vertex at , matching .
- Graph B: Upward-opening parabola with a vertex at , matching .
- Graph C: Downward-opening parabola with a vertex at , matching .
- Graph D: Downward-opening parabola with a vertex at , matching .
Final Matching:
- matches Graph B
- matches Graph C
- matches Graph A
- matches Graph D
Would you like further details on any specific step?
Here are five questions to deepen understanding:
- How would you determine the vertex of a quadratic function in general?
- What happens to the graph of if you add or subtract a constant?
- How does the sign of the coefficient of affect the direction in which a parabola opens?
- Can you describe the effect of horizontal shifts on the graph of a parabola?
- How would you graphically identify a reflection in the x-axis?
Tip: Remember, a positive coefficient of means the parabola opens upward, while a negative coefficient means it opens downward.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Graphing Parabolas
Transformations
Formulas
f(x) = ax^2 + bx + c for a quadratic function
Vertex form: f(x) = a(x - h)^2 + k where (h, k) is the vertex
Theorems
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Suitable Grade Level
Grades 9-12
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