Graphing factored form

 

In factored form....

 

y=a(x-r) (x-s)

 

• The "r" and "s" inform us about the two x-intercepts (points where the parabola crosses the x-axis) of the parabola.

 

How? 

The zeros (x-intercepts) are found by setting each "factor" equal to zero.

Therefore, x-r=0 and x-s=0.

x-r+r=0+r, so x=r

x-s+s=0+s, so x=s

Therefore, "r" and "s" are our x-intercepts (zeros).

 

In order to find the vertex of the parabola, we need to find the axis of symmetry, meaning the "x" value of the vertex.

 

• Axis of symmetry is the midpoint of the two x-intercepts.

• Therefore, we add the two x-intercepts and divide them by two, in order to find the axis of symmetry. 

• x=(r+s)/2

 

Once we find that, we have the "x" value of the vertex, however, vertex is a point, so we need the "y" value. 

• For finding the "y" value we can plug-in the "x" value in the equation, and solve for "y"

 

Let's try an example to see if you understood!

 

y=(x-3) (x+2) 

 

Let's find the x-intercepts!

• The x-intercepts are given, and they are (3,0) and (-2,0) 

 Really important rule!!

Make sure to switch the signs when writing the x-intercepts, so positive becomes negative, and negative becomes positive.

 

Time to find the axis of symmetry!

• Now we have to add the two intercepts, and divide them by two, so we can get the midpoint. 

• x=(3+(-2))/2

• x=(3-2)/2

• x=1/2

• x=0.5

 

The axis of symmetry is 0.5!

 

Now we have to solve for the vertex.

We have found the "x" value, so it will be (0.5,y).

• To fnd the "y" value we need to substitute the "x" value.

• Therefore, it will be y=(0.5-3) (0.5+2)

• y=(-2.5) (2.5)

• y=(-6.25)

 

The vertex is (0.5, -6.25).

 

Now, we can graph the equation, since we have the vertex, and two other points (x-intercepts).

 

Let's try another example and graph it!

y=(x-2) (x+4)

• x-intercepts would be (2,0) and (-4,0)

Axis of symmetry?

• x=(2+(-4))/2

• x=(2-4)/2

• x=(-2)/2

• x=(-1)

 

Vertex?

• Substitute the "x" value in the equation.

• y=(-1-2) (-1+4)

• y=(-3) (3)

• y=(-9)

 

Vertex is (-1,-9) 

 

How would it look after graphing the vertex, and the x-intercepts?

It would look like....

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Video time!!