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Two quadratic functions are shown.
Function 1: 
quad Function 2:

f(x)=3x^(2)+6x+7





x

g(x)


-2
13


-1
7


0
3


1
7




Which function has the least minimum value and what are its coordinates? (5 points)

Two quadratic functions are shown.\newlineFunction 11: \quad Function 22:\newlinef(x)=3x2+6x+7 f(x)=3 x^{2}+6 x+7 \newline\begin{tabular}{|l|l|}\newline\hline x \mathbf{x} & g(x) \mathbf{g}(\mathbf{x}) \\\newline\hline2-2 & 1313 \\\newline\hline1-1 & 77 \\\newline\hline 00 & 33 \\\newline\hline 11 & 77 \\\newline\hline\newline\end{tabular}\newlineWhich function has the least minimum value and what are its coordinates? (55 points)

Full solution

Q. Two quadratic functions are shown.\newlineFunction 11: \quad Function 22:\newlinef(x)=3x2+6x+7 f(x)=3 x^{2}+6 x+7 \newline\begin{tabular}{|l|l|}\newline\hline x \mathbf{x} & g(x) \mathbf{g}(\mathbf{x}) \\\newline\hline2-2 & 1313 \\\newline\hline1-1 & 77 \\\newline\hline 00 & 33 \\\newline\hline 11 & 77 \\\newline\hline\newline\end{tabular}\newlineWhich function has the least minimum value and what are its coordinates? (55 points)
  1. Calculate Vertex: Calculate the vertex of f(x) using the formula for the vertex of a quadratic function, x=b2ax = -\frac{b}{2a}. For f(x) = 33x^22 + 66x + 77, a = 33 and b = 66.
  2. Substitute xx: Substitute x=1x = -1 into f(x)f(x) to find the yy-coordinate of the vertex.
  3. Analyze Values: Analyze the values of g(x)g(x) given in the table to find the minimum value.
  4. Compare Minimum Values: Compare the minimum values of f(x)f(x) and g(x)g(x).

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