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Answer the questions below about the quadratic function.

g(x)=2x^(2)-16 x+35
Does the function have a minimum or maximum value?
Minimum
Maximum
What is the function's minimum or maximum value?

◻
Where does the minimum or maximum value occur?

x=

Answer the questions below about the quadratic function. $\newline$ $g(x)=2 x^{2}-16 x+35$ $\newline$ Does the function have a minimum or maximum value? $\newline$ Minimum $\newline$ Maximum $\newline$ What is the function's minimum or maximum value? $\newline$ $\square$ $\newline$ Where does the minimum or maximum value occur? $\newline$ $x=$

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Write a quadratic function in vertex form

Full solution

Q. Answer the questions below about the quadratic function.

\newline

g(x)=2 x^{2}-16 x+35

\newline

Does the function have a minimum or maximum value?

\newline

Minimum

\newline

Maximum

\newline

What is the function's minimum or maximum value?

\newline

\square

\newline

Where does the minimum or maximum value occur?

\newline

x=

Identify Function Type: Identify the type of function and determine if it has a minimum or maximum value. $\newline$ Since the coefficient of $x^2$ in $g(x) = 2x^2 - 16x + 35$ is positive ( $2$ ), the parabola opens upwards, indicating a minimum value.
Convert to Vertex Form: Convert the quadratic equation to vertex form to find the minimum value and the x-coordinate of the vertex. $\newline$ Complete the square for the quadratic term: $\newline$ $1$ . Factor out the coefficient of $x^2$ from the first two terms: $g(x) = 2(x^2 - 8x) + 35$ . $\newline$ $2$ . Calculate $\left(\frac{-8}{2}\right)^2 = 16$ . $\newline$ $3$ . Add and subtract $16$ inside the bracket: $g(x) = 2(x^2 - 8x + 16 - 16) + 35$ . $\newline$ $4$ . Simplify inside the bracket: $g(x) = 2((x - 4)^2 - 16) + 35$ . $\newline$ $5$ . Distribute and combine like terms: $g(x) = 2(x - 4)^2 - 32 + 35 = 2(x - 4)^2 + 3$ .
Find Minimum Value: Identify the minimum value and the x-coordinate where it occurs. $\newline$ The vertex form of the equation is $g(x) = 2(x - 4)^2 + 3$ . The vertex is $(4, 3)$ . $\newline$ - The x-coordinate of the vertex (where the minimum occurs) is $x = 4$ . $\newline$ - The minimum value of the function is $3$ .

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