Math Problem Statement
Use completing the square to rewrite the given function in vertex form. Determine whether the vertex is a maximum or minimum value. Select the TWO correct answers.
y = x^2 + 10x + 15 Responses A The vertex form is y =
- 10The vertex form is y = ( x + 5 ) 2 + 10 B The vertex form is y =
- 10.The vertex form is y = ( x − 5 ) 2 + 10. C The vertex form is y = − 10.The vertex form is y = ( x + 5 ) 2 − 10. D The vertex is a maximum value at point (5, 10)The vertex is a maximum value at point (5, 10) E The vertex is a minimum value at point (−5, −10)
Solution
To rewrite the given function in vertex form using the method of completing the square, we follow these steps:
Step 1: Start with the quadratic expression
The given equation is:
Step 2: Completing the square
We take the coefficient of , which is 10, divide it by 2, and then square it. This helps in creating a perfect square trinomial.
Now, add and subtract 25 inside the equation:
Step 3: Vertex form
The equation is now in vertex form: So, the correct vertex form is .
Step 4: Determine the vertex
The vertex form of a quadratic function is , where the vertex is at the point . In this case, the vertex is .
Step 5: Maximum or Minimum
Since the coefficient of (which is 1) is positive, the parabola opens upwards, meaning the vertex represents a minimum value at .
Correct Answers:
- C: The vertex form is .
- E: The vertex is a minimum value at point .
Would you like more details or have any questions?
Here are some related questions for further practice:
- How do you determine if the vertex is a maximum or minimum based on the coefficient of ?
- What would the vertex form be if the quadratic was ?
- Can you explain why adding and subtracting the same value inside the equation helps in completing the square?
- How does the sign of the number added in the perfect square trinomial affect the vertex form?
- What is the general procedure for completing the square for any quadratic function?
Tip: Always remember that the coefficient of tells you if the parabola opens upwards (minimum) or downwards (maximum).
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Completing the Square
Vertex Form
Formulas
y = ax^2 + bx + c
Vertex form: y = a(x - h)^2 + k
Theorems
Completing the Square
Properties of Parabolas
Suitable Grade Level
Grades 9-11
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