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Let f(x) = x^2 + 16x +57.What is the vertex and minimum value of f(x)?

Let f(x) = x^2 + 16x +57.What is the vertex and minimum value of f(x)?-example-1

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Given a quadratic function f(x):


f(x)=ax^2+bx+c

we can find the vertex of the parabolla that represents using the following expression:


Vertex\colon(-(b)/(2a),f(-(b)/(2a)))

then, in this case, we have the following values for a, b and c:


\begin{gathered} f(x)=x^2+16x+57 \\ \text{then:} \\ a=1 \\ b=16 \\ c=57 \end{gathered}

now lets find the x coordinate of the vertex:


-(b)/(2a)=-(16)/(2(1))=-(16)/(2)=-8

next, to find the y-coordinate of the vertex, we have to evaluate the x-coordinate on the function, that is, we have to find f(-8):


\begin{gathered} x=-8 \\ \Rightarrow f(-8)=(-8)^2+16(-8)+57 \\ =64-128+57=-7 \\ \Rightarrow f(-8)=-7 \end{gathered}

we have that f(-8) = -7, therefore, the vertex of the parabola (and minimum) is located at the point (-8,-7)

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