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

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).

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 )

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 = – 5 Directrix is x = 5/7

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 4/6y = 2/3y = x2 (multiply by ¼) (divide by 4) (multiply by ¼) x2 = – 1/8y y2 = 25/4x 2/12y = 1/6y = x2 Focus (0 , –1/8) Focus (25/4, 0) Focus (0, 1/6) Directrix y = 1/8 Directrix x = -25/4 Directrix y = -1/6

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.