To graph the quadratic function given in intercept form

f (x) = a (x - p) (x - q),

we must start by identifying the values of

a,

p,

and

q .

g (x) = - x (x + 6) ⇕ g (x) = - 1 (x - 0) (x - (- 6))

We can see that

a = - 1,

p = 0,

and

q = - 6 .

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

Identify the $x -$ intercepts.
Plot the $x -$ intercepts and find the axis of symmetry.
Calculate the vertex and plot it.
Connect the points with a parabola.
Identifying the $x -$ intercepts

Recall that the $x -$ intercepts of a function written in intercept form are the values of $p$ and $q .$ Thus, the points where our function intercepts the $x -$ axis are $(0, 0)$ and $(- 6, 0) .$

Finding the Axis of Symmetry
The axis of symmetry is halfway between the $x -$ intercepts, $(p, 0)$ and $(q, 0) .$ Therefore, it has the following equation.
$x = \frac{p + q}{2}$
Since we already know the values of $p$ and $q,$ we can substitute them into the formula.
$x = \frac{p + q}{2}$
SubstituteII
$p = 0$ , $q = - 6$
$x = \frac{0 + ( - 6 )}{2}$

▼

Simplify right-hand side

IdPropAdd
Identity Property of Addition
$x = \frac{- 6}{2}$
CalcQuot
Calculate quotient

$x = - 3$
The axis of symmetry of the parabola is the vertical line with equation $x = - 3 .$ Let's plot the $x -$ intercepts and the axis of symmetry on a coordinate plane.

Calculating the Vertex
The $x -$ coordinate of the vertex is the same as the formula for the axis of symmetry, $x = - 3 .$ To find the $y -$ coordinate, we need to substitute $- 3$ for $x$ in the given equation.
$g (x) = - x (x + 6)$
Substitute
$x = - 3$
$g (- 3) = - (- 3) (- 3 + 6)$

▼

Simplify right-hand side

SubTerms
Subtract terms
$g (- 3) = - (- 3) (3)$
Multiply
Multiply

$g (- 3) = 9$
We found the $y -$ coordinate, and now we know that the vertex is $(- 3, 9) .$ Let's plot it on our coordinate plane.

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.

Identifying the x-intercepts

Finding the Axis of Symmetry

Calculating the Vertex

Connecting the Points

Identifying the $x -$ intercepts

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