For a quadratic function $f (x) = a x^{2} + b x + c,$ the $y -$ coordinate of the vertex is the minimum value of the function when $a > 0 .$

Let's identify the values of

a,

b,

and

c

in the given quadratic function.

g (x) = 3 x^{2} + 18 x - 5 ⇕ g (x) = 3 x^{2} + 18 x + (- 5)

We can see above that $a = 3,$ $b = 18,$ and $c = - 5 .$ We will now use these values to find the desired information.

Since

a = 3

is greater than

0,

the parabola will open upwards. This means it will have a minimum value, which is given by

g (- \frac{b}{2 a}) .

Before we find the value of the function at this point, we need to substitute

a = 3

and

b = 18

in

- \frac{b}{2 a} .

- \frac{b}{2 a}

▼

Substitute values and evaluate

$a = 3$ , $b = 18$

- \frac{1 8}{2 ( 3 )}

- \frac{1 8}{6}

Calculate quotient

- 3

Now we have to calculate

g (- 3) .

To do so, we will substitute

- 3

for

x

in the given function.

g (x) = 3 x^{2} + 18 x - 5

$x = - 3$

g (- 3) = 3 (- 3)^{2} + 18 (- 3) - 5

▼

Simplify right-hand side

Calculate power

g (- 3) = 3 (9) + 18 (- 3) - 5

g (- 3) = 27 - 54 - 5

Subtract terms

g (- 3) = - 32

This tells us that the minimum value of the function is

- 32 .

Unless there are any specified restrictions on the

x -

values, the domain of a quadratic function is all real numbers. Therefore, the domain of this function is all real numbers. Furthermore, since

a = 3

is greater than

0,

the range is all values greater than or equal to the minimum value,

- 32 .

Domain : Range : All real numbers y \geq - 32

Since

a = 3

is greater than

0,

the function decreases to the left of the minimum value and increases to the right of the minimum value, which we know occurs at

x = - 3 .

Decreasing Interval : Increasing Interval : To the left of - 3 To the right of - 3

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