Complete the square to identify the vertex of y = x^2 - 16x + 14.
A. (8, 30)
B. (8, 14)
C. (8, 78)
D. (8, -50)
To complete the square, we want to rewrite the equation in the form y = a(x - h)^2 + k, where (h, k) is the vertex.
To do this, let's focus on the x^2 and x terms. We can rewrite x^2 - 16x as (x - 8)^2 - 64. (This is found by taking half of the coefficient of the x-term, squaring it, and then subtracting that result.)
We can then substitute this into our original equation to get:
y = (x - 8)^2 - 64 + 14
Simplifying, we get:
y = (x - 8)^2 - 50
Now we can see that the vertex is (8, -50).
So the answer is D. (8, -50)
To complete the square and identify the vertex of the quadratic equation y = x^2 - 16x + 14, follow these steps:
Step 1: Separate the quadratic term from the linear term.
y = (x^2 - 16x) + 14
Step 2: Take half of the coefficient of the x-term and square it.
-16/2 = -8 (-8)^2 = 64
Step 3: Add and subtract the value obtained in Step 2 inside the parentheses.
y = (x^2 - 16x + 64 - 64) + 14
Step 4: Rearrange to group the square term and the linear term together.
y = (x^2 - 16x + 64) - 64 + 14
y = (x - 8)^2 - 50
Step 5: The vertex form of the equation is y = (x - h)^2 + k, where (h, k) is the vertex.
Comparing the equation y = (x - 8)^2 - 50 with y = (x - h)^2 + k, we can see that the vertex is (h, k) = (8, -50).
Therefore, the answer is option D: (8, -50).
To complete the square and identify the vertex of the quadratic equation y = x^2 - 16x + 14, follow these steps:
Step 1: Take half of the coefficient of the x term and square it.
Half of -16 is -8, so (-8)^2 equals 64.
Step 2: Add the result from step 1 to both sides of the equation.
y + 64 = x^2 - 16x + 64 + 14
Simplifying the right side, we get:
y + 64 = (x - 8)^2 + 78
Step 3: Rewrite the equation with the square as a binomial squared.
y + 64 = (x - 8)^2 + 78
Step 4: Rearrange the equation to isolate y.
y = (x - 8)^2 + 78 - 64
Simplifying further, we have:
y = (x - 8)^2 + 14
Comparing this equation to the standard form of a quadratic equation:
y = a(x - h)^2 + k
We can identify that the vertex of the quadratic equation is (h, k).
The vertex of y = x^2 - 16x + 14 is (8, 14).
Therefore, the correct answer is B. (8, 14).