Features of Graphs of Quadratic Equations Quick Check

Question

2 of 52 of 5 Items
Question
Complete the square to identify the vertex of y=x2−16x+14.(1 point)
Responses

(8,−50)
left parenthesis 8 comma negative 50 right parenthesis

(8,14)
left parenthesis 8 comma 14 right parenthesis

(8,78)
left parenthesis 8 comma 78 right parenthesis

(8,30)
left parenthesis 8 comma 30 right parenthesis
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Answer 1

To find the vertex of the quadratic equation \(y = x^2 - 16x + 14\) by completing the square, follow these steps:

1. Start with the original equation: \[ y = x^2 - 16x + 14 \]

2. Group the \(x\)-terms and rewrite the equation: \[ y = (x^2 - 16x) + 14 \]

3. Complete the square for the expression \(x^2 - 16x\): \[ \text{Take half of } -16, \text{ which is } -8, \text{ and square it: } (-8)^2 = 64. \]

4. Add and subtract 64 inside the parentheses: \[ y = (x^2 - 16x + 64 - 64) + 14 \] \[ y = (x - 8)^2 - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

5. Now the equation is in vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. Here, \(h = 8\) and \(k = -50\).

Therefore, the vertex of the quadratic equation \(y = x^2 - 16x + 14\) is: \[ \boxed{(8, -50)} \]