Question

What is the axis of symmetry for the function f(x)=-(x+9)(x-21)

Answer 1

Answer: The axis of symmetry is the line x = 6

The roots of the function f(x) are x = -9 and x = 21. They are found by setting f(x) equal to zero and solving for x. Each factor is set equal to zero by the zero product property and you solve each sub-equation

x+9 = 0 leads to x = -9

x-21 = 0 leads to x = 21

Average these two roots to get the midpoint. Add them up and divide by 2.

So we add the roots to get -9+21 = 12

Then we divide by two: 12/2 = 6

On a number line, if we had point A at -9 and point B at 21, then point C is the midpoint at 6.

The axis of symmetry is the vertical line that passes through the vertex of the parabola. It is the vertical mirror line of symmetry.

Answer 2

`\bf \stackrel{f(x)}{0}=-(x+9)(x-21)\implies \begin{cases} 0=-x-9\implies &x=-9\\ 0=x-21\implies &21=x \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \boxed{-9}\rule[0.35em]{8em}{0.25pt}0\rule[0.35em]{5em}{0.25pt}\stackrel{\downarrow }{6}\rule[0.35em]{10em}{0.25pt}\boxed{21}`

now, this parabolic graph, has two zeros/solutions/x-intercepts, at -9 and 21.

for a quadratic equation, the vertex will be right in between the x-intercepts, namely in this case between -9 and 21, right in the middle, namely at 6, and is where the axis of symmetry is at, x = 6.