# What is the axis of symmetry of the graph of y=7(x+1)(x-3)?

May 26, 2015

Given $y = 7 \left(x + 1\right) \left(x - 3\right)$

Note that this is a parabola in standard position (vertical axis of symmetry).

The axis of symmetry passes through the vertex.

One method of determining the vertex is by noting that the derivative of the function is equal to zero at the vertex

$y = 7 \left(x + 1\right) \left(x - 3\right)$
$= 7 {x}^{2} - 14 x - 21$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 14 x - 14$

If $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$\rightarrow x = 1$

(we could now calculate the value of $y$ at the vertex, but we don't really need it since we are looking for the vertical line passing through $x = 1$

The axis of symmetry is
$x = 1$

An other way:
In a parabola of this kind you can also find the midpoint between the two points where the curve crosses the $x$-axis.
As you will see $y = 0 \to x = - 1 \mathmr{and} x = + 3$.
$x = 1$ is halfway.
Same answer, less work, but this method is not always usable.