Math Problem Statement
Solution
The problem asks you to rewrite the given function in vertex form and use it to determine the vertex of the parabola. Let's solve this step-by-step:
1. Rewrite in vertex form using completing the square.
The vertex form of a quadratic function is: where is the vertex.
Step 1. Factor out (if necessary, but here already):
Step 2. Complete the square:
To complete the square, take half of the coefficient of (which is ), square it, and add/subtract it inside the equation:
Add and subtract 9 inside the equation:
Simplify:
Final vertex form:
2. Determine the vertex:
From the vertex form , the vertex is:
3. Match this to the answer choices:
- The correct function in vertex form is .
- The vertex coordinates are .
4. Final Answer:
- Select .
- The vertex is .
Would you like further clarification or additional examples? Let me know!
Relative Questions:
- What does it mean for a quadratic function to be in vertex form?
- How do you complete the square for any quadratic equation?
- Why is the vertex useful in understanding the graph of a parabola?
- What is the significance of the and in the vertex form equation?
- How do you determine whether a parabola opens upward or downward?
Tip:
The coefficient of in the vertex form () determines the "width" and direction of the parabola. If , the parabola opens upwards, and if , it opens downwards!
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Completing the Square
Vertex Form of a Parabola
Formulas
Vertex form: f(x) = a(x - h)^2 + k
Completing the square: Add and subtract (b/2)^2 inside the equation
Theorems
Vertex Formula: The vertex of f(x) = a(x - h)^2 + k is (h, k).
Suitable Grade Level
Grades 8-10
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