For the quadratic function $f (x) = a x^{2} + b x + c,$ the $y -$ coordinate of the vertex is the maximum 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} - 6 x + 5 ⇕ g (x) = - 3 x^{2} + (- 6) x + 5

We can see above that $a = - 3,$ $b = - 6,$ and $c = 5 .$ We will now use these values to find the desired information.

Since

a = - 3

is less than

0,

the parabola will open downwards. This means it will have a maximum 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 = - 6

in

- \frac{b}{2 a} .

- \frac{b}{2 a}

▼

Substitute values and evaluate

$a = - 3$ , $b = - 6$

- \frac{- 6}{2 ( - 3 )}

$a (- b) = - a \cdot b$

- \frac{- 6}{- 6}

$- \frac{a}{- b} = \frac{a}{b}$

\frac{- 6}{6}

Calculate quotient

- 1

Now we have to calculate

g (- 1) .

To do so, we will substitute

- 1

for

x

in the given function.

g (x) = - 3 x^{2} - 6 x + 5

$x = - 1$

g (- 1) = - 3 (- 1)^{2} - 6 (- 1) + 5

▼

Simplify right-hand side

Calculate power

g (- 1) = - 3 (1) - 6 (- 1) + 5

g (- 1) = - 3 + 6 + 5

Add and subtract terms

g (- 1) = 8

This tells us that the maximum value of the function is

8 .

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 less than

0,

the range is all values less than or equal to the maximum value,

8 .

Domain : Range : All real numbers y \leq 8

Since

a = - 3

is less than

0,

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

- 1 .

Increasing Interval : Decreasing Interval : To the left of - 1 To the right of - 1

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