Put the quadratic into vertex form and state the coordinates of the vertex
y=x^2+4x+9
Vertex form:y=___________
Vertex:(,)
To put the quadratic equation into vertex form, we can complete the square.
Starting with the given equation:
y = x^2 + 4x + 9
Step 1: Take out the common factor from the first two terms (x^2 and 4x) to factor out the coefficient of x^2 (which is 1 in this case):
y = 1(x^2 + 4x) + 9
Step 2: To complete the square, take half of the coefficient of x (which is 4) and square it: (4/2)^2 = 4^2 = 16.
Then add and subtract this value inside the bracket:
y = 1(x^2 + 4x + 16 - 16) + 9
Step 3: Rearrange the terms inside the bracket and simplify:
y = 1((x + 2)^2 - 16) + 9
Step 4: Expand the bracket:
y = 1(x + 2)^2 - 1(16) + 9
Step 5: Simplify further:
y = (x + 2)^2 - 16 + 9
y = (x + 2)^2 - 7
So the equation in vertex form is:
y = (x + 2)^2 - 7
The coordinates of the vertex are obtained from the equation as (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the x-coordinate is -2 (from the equation: x + 2 = 0) and the y-coordinate is -7 (from the equation: y = -7).
Therefore, the coordinates of the vertex are (-2, -7).