To draw the graph of the given quadratic function, written in standard form, we must start by identifying the values of

a,

b,

and

c .

f (x) = - x^{2} - 6 x + 3 \Leftrightarrow f (x) = - 1 x^{2} + (- 6) x + 3

We can see that

a = - 1,

b = - 6,

and

c = 3 .

Now, we will follow four steps to graph the function.

Find the axis of symmetry.
Calculate the vertex.
Identify the $y -$ intercept and its reflection across the axis of symmetry.
Connect the points with a parabola.
Finding the Axis of Symmetry
The axis of symmetry is a vertical line with equation $x = - \frac{b}{2 a} .$ Since we already know the values of $a$ and $b,$ we can substitute them into the formula.
$x = - \frac{b}{2 a}$
SubstituteII
$a = - 1$ , $b = - 6$
$x = - \frac{- 6}{2 ( - 1 )}$

▼

Simplify right-hand side

Multiply
Multiply
$x = - \frac{- 6}{- 2}$
RemoveNegFracAndDenom
$- \frac{a}{- b} = \frac{a}{b}$
$x = \frac{- 6}{2}$
CalcQuot
Calculate quotient

$x = - 3$
The axis of symmetry of the parabola is the vertical line with equation $x = - 3 .$
Calculating the Vertex
To calculate the vertex, we need to think of $y$ as a function of $x,$ $y = f (x) .$ We can write the expression for the vertex by stating the $x -$ and $y -$ coordinates in terms of $a$ and $b .$
$Vertex : (- \frac{b}{2 a}, f (- \frac{b}{2 a}))$
Note that the formula for the $x -$ coordinate is the same as the formula for the axis of symmetry, which is $x = - 3 .$ Thus, the $x -$ coordinate of the vertex is also $- 3 .$ To find the $y -$ coordinate, we need to substitute $- 3$ for $x$ in the given equation.
$f (x) = - x^{2} - 6 x + 3$
Substitute
$x = - 3$
$f (- 3) = - (- 3)^{2} - 6 (- 3) + 3$

▼

Simplify right-hand side

CalcPow
Calculate power
$f (- 3) = - (9) - 6 (- 3) + 3$
Multiply
Multiply
$f (- 3) = - 9 + 18 + 3$
AddTerms
Add terms

$f (- 3) = 12$
We found the $y -$ coordinate, and now we know that the vertex is $(- 3, 12) .$
Identifying the $y -$ intercept and its Reflection

The $y -$ intercept of the graph of a quadratic function written in standard form is given by the value of $c .$ Thus, the point where our graph intercepts the $y -$ axis is $(0, 3) .$ Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since $a = - 1,$ which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.

Finding the Axis of Symmetry

Calculating the Vertex

Identifying the y-intercept and its Reflection

Connecting the Points

Identifying the $y -$ intercept and its Reflection

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