Question

compare $g(x)=3x^{2}+18x + 7$ to $f$, shown in the table. which function has a lesser minimum value? explain.\
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$x$	$(x,f(x))$	\
---	---	\
$1$	$(1,0)$	\
$3$	$(3, - 12)$	\
$5$	$(5, - 16)$	\
$7$	$(7, - 12)$	\
$9$	$(9,0)$	\

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choose the correct answer below.\
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\\(\bigcirc\\) a. $g(x)$; the $y$-coordinate of the vertex of $g(x)$ is less than the $y$-coordinate of the vertex of $f(x)$.\
\\(\bigcirc\\) b. $g(x)$; the $x$-coordinate of the vertex of $g(x)$ is less than the $x$-coordinate of the vertex of $f(x)$.\
\\(\bigcirc\\) c. $f(x)$; the $x$-coordinate of the vertex of $f(x)$ is less than the $x$-coordinate of the vertex of $g(x)$.\
\\(\bigcirc\\) d. $f(x)$; the $y$-coordinate of the vertex of $f(x)$ is less than the $y$-coordinate of the vertex of $g(x)$.\

Explanation:

Step1: Find vertex of \( g(x) \)

For a quadratic function \( ax^2 + bx + c \), vertex \( x \)-coordinate is \( -\frac{b}{2a} \). For \( g(x) = 3x^2 + 18x + 7 \), \( a = 3 \), \( b = 18 \).
\( x = -\frac{18}{2 \times 3} = -3 \).
Substitute \( x = -3 \) into \( g(x) \):
\( g(-3) = 3(-3)^2 + 18(-3) + 7 = 27 - 54 + 7 = -20 \).
So vertex of \( g(x) \) is \( (-3, -20) \), minimum \( y \)-coordinate is \( -20 \).

Step2: Analyze \( f(x) \) from table

The table for \( f(x) \) has symmetric points: \( (1,0) \) and \( (9,0) \), \( (3,-12) \) and \( (7,-12) \). The axis of symmetry (vertex \( x \)-coordinate) is midpoint of \( 1 \) and \( 9 \): \( \frac{1 + 9}{2} = 5 \). At \( x = 5 \), \( f(5) = -16 \). So minimum \( y \)-coordinate of \( f(x) \) is \( -16 \).

Step3: Compare minimum values

Minimum \( y \)-coordinate of \( g(x) \) is \( -20 \), of \( f(x) \) is \( -16 \). Since \( -20 < -16 \), \( g(x) \) has a lesser minimum (comparing \( y \)-coordinates of vertices).

Answer:

A. \( g(x) \); The \( y \)-coordinate of the vertex of \( g(x) \) is less than the \( y \)-coordinate of the vertex of \( f(x) \).