Math Problem Statement
give me the vertex equation of this graph show all work
Solution
To find the equation of the parabola and its vertex from the provided graph, I will follow these steps:
Observing the graph:
- The graph is a parabola opening upwards, indicating a positive quadratic term.
- The vertex appears to be at the point , based on the symmetry and minimum value.
General Vertex Form of a Parabola:
The equation of a parabola in vertex form is:
Where:
- is the vertex of the parabola.
- determines the width and direction of the parabola.
From the graph, we can observe:
- The vertex is , so and .
Thus, the equation simplifies to:
Finding :
To determine the value of , we need another point on the parabola. From the graph, one easily identifiable point is , as it seems to lie on the curve.
Substituting and into the equation:
Solving for :
Final Equation:
Thus, the equation of the parabola is:
This is the equation of the parabola shown in the graph.
Would you like further details on the steps? Here are some related questions:
- What is the general form of a quadratic equation, and how does it relate to the vertex form?
- How do you determine whether a parabola opens upwards or downwards?
- How can you determine the vertex from a graph without the equation?
- What effect does changing the value of have on the graph of the parabola?
- Can this equation be converted into standard form, and how?
Tip: The coefficient affects the "width" or "narrowness" of the parabola—higher absolute values make it narrower, while lower values make it wider.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Parabolas
Graphing
Formulas
Vertex form of a quadratic equation: y = a(x - h)^2 + k
Theorems
Parabola Vertex Theorem
Suitable Grade Level
Grades 9-12
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