# How do you graph the quadratic function and identify the vertex and axis of symmetry and x intercepts for y=(x-2)(x-6)?

Feb 17, 2018

#### Explanation:

To find the vertex (commonly known as the turning or stationary point), we can employ several approaches. I will employ calculus to do this.

First Approach:
Find the derivative of the function.

Let $f \left(x\right) = y = \left(x - 2\right) \left(x - 6\right)$
then, $f \left(x\right) = {x}^{2} - 8 x + 12$

the derivative of the function (using the power rule) is given as
$f ' \left(x\right) = 2 x - 8$
We know that the derivative is naught at the vertex. So,
$2 x - 8 = 0$
$2 x = 8$
$x = 4$
This gives us the x-value of the turning point or vertex. We will now substitute $x = 4$ into $f$ to obtain the corresponding y-value of the vertex.
that is, $f \left(4\right) = {\left(4\right)}^{2} - 8 \left(4\right) + 12$
$f \left(4\right) = - 4$
Hence the co-ordinates of the vertex are $\left(4 , - 4\right)$

Any quadratic function is symmetrical about the line running vertically through its vertex.. As such, we have found the axis of symmetry when we found the co-ordinates of the vertex.
That is, the axis of symmetry is $x = 4$.

To find x-intercepts: we know that the function intercepts the x-axis when $y = 0$. That is, to find the x-intercepts we have to let $y = 0$.
$0 = \left(x - 2\right) \left(x - 6\right)$
$x - 2 = 0 \mathmr{and} x - 6 = 0$
therefore, $x = 2 \mathmr{and} x = 6$
This tells us that the co-ordinates of the x-intercept are $\left(2 , 0\right)$ and $\left(6 , 0\right)$

To find the y-intercept, let $x = 0$
$y = \left(0 - 2\right) \left(0 - 6\right)$
$y = 12$
This tells us that the co-ordinate of the y-intercept is $0 , 12$
Now use the points we derived above to graph the function graph{x^2 - 8x +12 [-10, 10, -5, 5]}

Feb 17, 2018

$\text{see explanation}$

#### Explanation:

$\text{to find the intercepts}$

• " let x = 0, in the equation for y-intercept"

• " let y = 0, in the equation for x-intercepts"

$x = 0 \to y = \left(- 2\right) \left(- 6\right) = 12 \leftarrow \textcolor{red}{\text{y-intercept}}$

$y = 0 \to \left(x - 2\right) \left(x - 6\right) = 0$

$\text{equate each factor to zero and solve for x}$

$x - 2 = 0 \Rightarrow x = 2$

$x - 6 = 0 \Rightarrow x = 6$

$\Rightarrow x = 2 , x = 6 \leftarrow \textcolor{red}{\text{x-intercepts}}$

$\text{the axis of symmetry goes through the midpoint}$
$\text{of the x-intercepts}$

$x = \frac{2 + 6}{2} = 4 \Rightarrow x = 4 \leftarrow \textcolor{red}{\text{axis of symmetry}}$

$\text{the vertex lies on the axis of symmetry, thus has}$
$\text{x-coordinate of 4}$

$\text{to obtain y-coordinate substitute "x=4" into the}$
$\text{equation}$

$y = \left(2\right) \left(- 2\right) = - 4$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(4 , - 4\right)$

$\text{to determine if vertex is max/min consider the}$
$\text{value of the coefficient a of the "x^2" term}$

• " if "a>0" then minimum"

• " if "a<0" then maximum"

$y = \left(x - 2\right) \left(x - 6\right) = {x}^{2} - 8 x + 12$

$\text{here "a>0" hence minimum } \bigcup$

$\text{gathering the information above allows a sketch of }$
$\text{quadratic to be drawn}$
graph{(y-x^2+8x-12)(y-1000x+4000)=0 [-10, 10, -5, 5]}