Math Problem Statement
Solution
We are given the quadratic function:
This can be rewritten in the form:
To do this, we will complete the square to transform the equation into the vertex form.
Step 1: Factor out the coefficient of
The equation is . First, we factor out from the terms involving :
Step 2: Complete the square
To complete the square inside the parentheses, take half of the coefficient of , square it, and add it inside the parentheses, while compensating by subtracting the same value outside the parentheses. Half of is , and .
We add and subtract 4 inside the parentheses:
Simplify:
Step 3: Simplify
Now, simplify the constants:
Step 4: Identify , , and
From the vertex form , we can see that:
Step 5: Determine the vertex
The vertex of the quadratic function occurs at . Therefore, the vertex is at:
Final Answer:
- The vertex is at
Let me know if you would like more details, or if you have any questions.
Here are 5 related questions for you to consider:
- How do you complete the square for a general quadratic function?
- What is the significance of the vertex in a quadratic function?
- How does the value of affect the direction of the parabola's opening?
- Can a quadratic function have more than one vertex?
- How can the vertex form of a quadratic be used to determine the axis of symmetry?
Tip: The vertex form of a quadratic is particularly useful when you need to quickly identify the vertex and the axis of symmetry!
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex Form
Completing the Square
Formulas
Standard form of a quadratic: ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Theorems
Quadratic Function Transformation
Completing the Square
Suitable Grade Level
Grades 9-12
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