First, notice that the variable raised to the power of 2 is y. Therefore, the axis of symmetry will be a horizontal line. We will start by writing the given equation in the form x=a(y-k)^2+h. To do so, we will first multiply both sides of the equation by 2 to make its quadratic coefficient equal 1.
x=1/2y^2-4y+3 [0.4em] ⇕ [0.4em] 2x=y^2-8y+6
Now we can 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 ⇒ (b/2)^2=(8/2)^2=4^2
Let's do it!
Now, let's identify the values of a, h, and k.
x=1/2(y-4)^2-5
⇕
x= 1/2(y- 4)^2+( - 5)
We can see that a= 12, h= - 5, and that k= 4. Next, we can use this information to highlight some important characteristics of the parabola.
x=a(y-k)^2+h
x=1/2(y-4)^2+(- 5)
Direction of Opening
Right if a>0, Left if a<0
Right ( 1/2>0)
Vertex
( h, k)
( - 5, 4)
Axis of Symmetry
y= k
y= 4
Focus
( h+1/4 a, k)
( - 5+1/4( 12), 4) ⇕ (- 4.5,4)
Directrix
x= h-1/4 a
x= - 5-1/4( 12) ⇕ x=- 5.5
Length of Latus Rectum
|1/a| units
|1/12| units ⇕ 2 units
We can use the above information to draw the graph.
