1. Intro to Conics – Parabolas
I. Parabolas.
A) Quadratic form: y = ax2 + bx + c or x = ay2 + by + c.
B) Directions parabolas open.
1) y = ax2 + bx + c 2) x = ay2 + by + c
y = – ax2 + bx + c x = – ay2 + by + c

2. Intro to Conics – Parabolas
II. Parts of a Parabola.
A) Vertex point: the min / max point on a parabola.
1) Formula for finding the vertex of a parabola…
a) for y = ax2 + … is ( – b/2a , plug it in)
b) for x = ay2 + … is (plug it in , – b/2a)
B) Focus point: a special point located inside the parabola.
C) Axis of Symmetry: the line of symmetry that divides a
parabola into two equal halves.
1) Goes thru the vertex point. (vert: x = # or horiz: y = #)
D) Directrix line: a line that is equal distant from the focus to the parabola (at a right angle).

3. Intro to Conics – Parabolas
III. Finding the Focus and Directrix of a Parabola.
A) Focus point: (only works for ay = bx2 or ax = by2)
1) Isolate the squared term.
2) Divide the side that doesn’t have the exponent by 4
(or multiply by ¼ whichever is easier).
3) The # in front of the non-squared term is the focus.
(the squared letter gets a zero coordinate value).
(the non-squared term gets the focus number)
Examples: If you have already done the 3 steps above, then …
x2 = 5y y2 = – 5/7x
Focus = ( 0 , 5 ) Focus = ( – 5/7 , 0 )

4. Intro to Conics – Parabolas
III. Finding the Focus and Directrix of a Parabola.
B) The Directrix: you need to find the focus first.
1) The equation for the directrix line is found by …
a) Changing the sign of the focus.
b) Whichever letter had the focus is the letter that is
needed for the directrix equation.
Examples: Focus = ( 0 , 5 ) Focus = ( – 5/7 , 0 )
Directrix is y = – Directrix is x = 5/7

5. Intro to Conics – Parabolas
Examples: Find the focus point of the parabola.
1) – 2x2 = y 2) 1/25y2 = x 3) 4y = 6x2
(divide by – 2) (multiply by 25) (divide by 6)
x2 = – ½ y y2 = 25x /6y = 2/3y = x2
(multiply by ¼) (divide by 4) (multiply by ¼)
x2 = – 1/8y y2 = 25/4x /12y = 1/6y = x2
Focus (0 , –1/8) Focus (25/4, 0) Focus (0, 1/6)
Directrix y = 1/ Directrix x = -25/ Directrix y = -1/6

6. Intro to Conics – Parabolas
IV. Sketching Parabolas.
A) Find and graph the Vertex point.
(– b/2a , plug it in ) or ( plug it in , – b/2a )
B) Determine the direction the parabola opens.
1) y = ax2 + … opens up or down (+ is up, – is down).
2) x = ay2 + … opens left or right (+ is right, – is left).
C) Determine the “slope” of the parabola.
1) Taller ( a > 1 ) draw a narrow parabola.
2) Shorter ( 0 > a > 1 ) draw a wide parabola.