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 y=x2+10x+15y = x^2 + 10x + 15 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: y=x2+10x+15y = x^2 + 10x + 15

Step 2: Completing the square

We take the coefficient of xx, which is 10, divide it by 2, and then square it. This helps in creating a perfect square trinomial. (102)2=25\left( \frac{10}{2} \right)^2 = 25

Now, add and subtract 25 inside the equation: y=(x2+10x+25)25+15y = (x^2 + 10x + 25) - 25 + 15 y=(x+5)210y = (x + 5)^2 - 10

Step 3: Vertex form

The equation is now in vertex form: y=(x+5)210y = (x + 5)^2 - 10 So, the correct vertex form is y=(x+5)210y = (x + 5)^2 - 10.

Step 4: Determine the vertex

The vertex form of a quadratic function is y=a(xh)2+ky = a(x - h)^2 + k, where the vertex is at the point (h,k)(h, k). In this case, the vertex is (5,10)(-5, -10).

Step 5: Maximum or Minimum

Since the coefficient of x2x^2 (which is 1) is positive, the parabola opens upwards, meaning the vertex represents a minimum value at (5,10)(-5, -10).

Correct Answers:

  • C: The vertex form is y=(x+5)210y = (x + 5)^2 - 10.
  • E: The vertex is a minimum value at point (5,10)(-5, -10).

Would you like more details or have any questions?

Here are some related questions for further practice:

  1. How do you determine if the vertex is a maximum or minimum based on the coefficient of x2x^2?
  2. What would the vertex form be if the quadratic was y=x210x+15y = x^2 - 10x + 15?
  3. Can you explain why adding and subtracting the same value inside the equation helps in completing the square?
  4. How does the sign of the number added in the perfect square trinomial affect the vertex form?
  5. What is the general procedure for completing the square for any quadratic function?

Tip: Always remember that the coefficient of x2x^2 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