what is the vertex form of the equation
y=-x^2+6x-7
show all the steps
To write the given equation in vertex form, which is in the form of "y = a(x - h)^2 + k", we need to complete the square.
Step 1: Rewrite the equation in the form of "y = ax^2 + bx + c" by moving the constant term to the right side:
y + 7 = -x^2 + 6x
Step 2: To complete the square, add the square of half the coefficient of the x-term to both sides of the equation:
y + 7 + (6/2)^2 = -x^2 + 6x + (6/2)^2
y + 7 + 9 = -x^2 + 6x + 9
y + 16 = -(x^2 - 6x + 9)
Step 3: Factor the quadratic on the right side:
y + 16 = -(x - 3)^2
Step 4: Move the constant term to the right side:
y = -(x - 3)^2 - 16
Therefore, the vertex form of the equation y = -x^2 + 6x - 7 is y = -(x - 3)^2 - 16.
To find the vertex form of the equation y = -x^2 + 6x - 7, we can complete the square. Here are the steps:
Step 1: Rearrange the equation to group the x terms together:
y = -x^2 + 6x - 7
Step 2: Take out any common factor from the x^2 and x terms, if possible. In this case, there are no common factors.
Step 3: To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of the x term is 6, so half of it is 3. The square of 3 is 9.
y = -(x^2 - 6x + 9 - 9) - 7
Step 4: Inside the parentheses, we have a perfect square trinomial: x^2 - 6x + 9. We can write it as the square of a binomial: (x - 3)^2.
y = -(x - 3)^2 + 9 - 7
Step 5: Simplify the equation:
y = -(x - 3)^2 + 2
So, the vertex form of the equation y = -x^2 + 6x - 7 is y = -(x - 3)^2 + 2.
To find the vertex form of the equation, we need to complete the square. Here are the steps to do that:
Step 1: Ensure that the coefficient of the x^2 term is 1. In this case, it already is, so we can proceed.
Step 2: Separate the x terms:
y = -(x^2 - 6x) - 7
Step 3: Find the value to complete the square. Take half of the coefficient of the x term, square it, and add/subtract it inside the parentheses. In this case:
Take half of (-6), which is -3. Square it: (-3)^2 = 9.
Add 9 and -9 inside the parentheses:
y = -(x^2 - 6x + 9 - 9) - 7
Step 4: Group the terms that can be factored:
y = -[(x^2 - 6x + 9) - 9] - 7
Step 5: Simplify the expression inside the parentheses:
y = -[(x - 3)^2 - 9] - 7
Step 6: Expand the negative sign outside the parentheses:
y = -(x - 3)^2 + 9 - 7
Step 7: Simplify further:
y = -(x - 3)^2 + 2
The equation in vertex form is y = -(x - 3)^2 + 2, where the coordinates of the vertex are (3, 2).