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What is the vertex form of the equation?
y=−x2+6x−7
(3 points)
To find the vertex form of the equation, we need to complete the square.
Step 1: Factor out the coefficient of x^2, which is -1:
y = -1(x^2 - 6x + 7)
Step 2: Take half of the coefficient of x (-6) and square it. This gives us (-6/2)^2 = 9:
y = -1(x^2 - 6x + 9 + 7 - 9)
Step 3: Rearrange the expression inside the parentheses:
y = -1((x^2 - 6x + 9) + 7 - 9)
Step 4: Factor the perfect square trinomial inside the parentheses:
y = -1((x - 3)^2 - 2)
Step 5: Distribute the -1 to the terms inside the parentheses:
y = -1(x - 3)^2 + 2
The vertex form of the equation is y = -1(x - 3)^2 + 2.
To find the vertex form of the equation, we need to complete the square.
Step 1: Identify the coefficients of x^2 and x
In the given equation y = -x^2 + 6x - 7, the coefficient of x^2 is -1, and the coefficient of x is 6.
Step 2: Divide the coefficient of x by 2 and square the result
Take half of 6, which is 3, and square it: 3^2 = 9.
Step 3: Add the square from step 2 to both sides of the equation
Add 9 to both sides of the equation:
y - 7 = -x^2 + 6x + 9
Step 4: Rearrange the equation
Rearrange the equation by switching the order of the terms:
y - 7 = -(x^2 - 6x + 9)
Step 5: Factor the trinomial inside the parentheses and evaluate it
Factor the trinomial x^2 - 6x + 9:
y - 7 = -(x - 3)^2
Step 6: Move the constant term to the other side of the equation
Add 7 to both sides of the equation:
y = -(x - 3)^2 + 7
Therefore, the vertex form of the equation is y = -(x - 3)^2 + 7.
To find the vertex form of the equation, we need to complete the square.
Step 1: Rewrite the equation in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
y = -x^2 + 6x - 7
Step 2: Group the x terms together.
y = -(x^2 - 6x) - 7
Step 3: Complete the square by taking half of the coefficient of x (-6/2 = -3) and squaring it (-3^2 = 9). Add this value inside the parentheses and subtract the same value outside the parentheses to maintain the balance of the equation.
y = -(x^2 - 6x + 9 - 9) - 7
Step 4: Simplify inside the parentheses.
y = -(x - 3)^2 + 9 - 7
Step 5: Combine like terms.
y = -(x - 3)^2 + 2
So, the vertex form of the equation y = -x^2 + 6x - 7 is y = -(x - 3)^2 + 2.