Question

Consider the quadratic function y equals 2 x squared plus 12 x plus 14. Rewrite the function in vertex format.

Answer 1

The quadratic function
`\( y = 2x^2 + 12x + 14 \)` can be rewritten in vertex form as
`\( y = 2(x + 3)^2 + 5 \).`

Step-by-step explanation:

To rewrite the quadratic function \( y = 2x^2 + 12x + 14 \) in vertex form, we can complete the square. The vertex form of a quadratic function is given by
`\( y = a(x - h)^2 + k \), where \((h, k)\)` is the vertex of the parabola.

First, factor out the leading coefficient from the
`\(x^2\) and \(x\) terms: \( y = 2(x^2 + 6x) + 14 \)`. To complete the square, add and subtract
`\((6/2)^2 = 9\) inside the parentheses: \( y = 2(x^2 + 6x + 9 - 9) + 14 \).` Now, factor the perfect square trinomial and simplify:
`\( y = 2((x + 3)^2 - 9) + 14 \). Distribute the \(2\)` and combine like terms:
`\( y = 2(x + 3)^2 - 18 + 14 \).` Finally, simplify further to obtain the vertex form:
`\( y = 2(x + 3)^2 + 5 \).`

In the vertex form
`\( y = 2(x + 3)^2 + 5 \), the vertex \((-3, 5)\)` is easily identified, and the transformation from the original form to the vertex form is evident. The
`\(x + 3\)` inside the square reflects the horizontal shift of the parabola, while the
`\(+5\)` outside the square indicates the vertical shift. The vertex form provides a clearer understanding of the essential characteristics of the quadratic function.