Convert each of the following quadratic functions to vertex form by completing the square y=x^2+x+5
To convert a quadratic function to vertex form by completing the square, follow these steps:
1. Start with the quadratic function in the standard form: y = ax^2 + bx + c.
In this case, the given quadratic function is y = x^2 + x + 5.
2. Make sure the coefficient of the x^2 term (a) is 1.
In our given function, the coefficient of x^2 is already 1.
3. Group the x and constant terms together:
y = (x^2 + x) + 5.
4. To complete the square, take half the coefficient of x, square it, and add it inside the parentheses. Add and subtract this value to keep the equation balanced.
The coefficient of x is 1/2, so half of it is 1/4. Square of 1/4 is 1/16. Adding 1/16 inside the parentheses:
y = (x^2 + x + 1/16 - 1/16) + 5.
5. Combine like terms inside the parentheses:
y = (x^2 + x + 1/16) - 1/16 + 5.
6. Simplify:
y = (x + 1/4)^2 - 1/16 + 5.
7. Combine constants:
y = (x + 1/4)^2 + 79/16.
8. The quadratic function is now in vertex form: y = a(x - h)^2 + k, where (h, k) represents the vertex.
In this case, the vertex is (-1/4, 79/16).
Therefore, the given quadratic function y = x^2 + x + 5 can be converted to vertex form as y = (x + 1/4)^2 + 79/16.