First, notice that the variable raised to the power of 2 is x. Therefore, the axis of symmetry will be a vertical line. We will start by writing the given equation in the form y=a(x-h)^2+k. To do so, we will first rearrange the equation so that the x-terms and the constant are on the right-hand side.
3y-x^2=8x-11 [0.3em] ⇕ [0.3em] 3y=x^2+8x-11
We can now simplify the right-hand side by completing the square. We have to add and subtract ( b2)^2. In this case, we have that the linearcoefficient b is 8.
b=8 ⇒ (8/2)^2=(8/2)^2=4^2
Let's do it!
Now, let's identify the values of a, h, and k.
y=1/3( x+4 )^2 - 9
⇕
y= 1/3( x-( - 4) )^2 + ( - 9)
We can see that a= 13, h= - 4, and that k= - 9. Next, we can use this information to highlight some important characteristics of the parabola.
y=a(x-h)^2+k
y=1/3(x-(- 4))^2+(- 9)
Direction of Opening
Up if a>0, Down if a<0
Up ( 2>0)
Vertex
( h, k)
( - 4, - 9)
Axis of Symmetry
x= h
x= - 4
Focus
( h, k+1/4 a)
( - 4, - 9+1/4( 13)) ⇕ (- 4,- 8.25)
Directrix
y= k-1/4 a
y= - 9-1/4( 13) ⇕ y=- 9.75
Length of Latus Rectum
|1/a| units
|1/13| units ⇕ 3 units
We can use the above information to draw the graph.
