# ANALYZE THE EQUATION OF A PARABOLA

The parabola will be in four different forms,

 (y - k)2 = 4a(x - h) Opening right (y - k)2 = -4a(x - h) Opening left (x - h)2 = 4a(y - k) Opening up (x - h)2 = -4a(y - k) Opening down

Write the following in standard form. Identify the

• Vertex
• Focus
• Axis of symmetry
• Direction of opening of parabola.
• Equation latus rectum and directrix
• Draw the graph

Problem 1 :

y = 3x2 + 24x + 50

Solution :

y = 3x2 + 24x + 50

y = 3[x2 + 8x] + 50

= 3[x2 + 2x(4) + 42 - 42] + 50

= 3[(x + 4)2 - 42] + 50

= 3[(x + 4)2 - 16] + 50

= 3(x + 4)2 - 48 + 50

y = 3(x + 4)2 + 2

y - 2 = 3(x + 4)2

Comparing with

(y - k) = 4a(x - h)2

The parabola is symmetric about y-axis and open upward.

4a = 3

a = 3/4

 Vertex (h, k) ==> (-4, 2) Focus (h, k + a) k + a = 2 + (3/4)= 11/4(-4, 11/4) Equation of latus rectum y = k + ay = 11/4 Equation of directrix y = k - ay = 2 - (3/4)y = 5/4 Axis of symmetry x = -4 Equation of directrix 4a = 3 units

Problem 2 :

-6y = x2

Solution :

x2 = -6y

The parabola is symmetric about y-axis and open downward.

4a = 6

a = 6/4

a = 3/2

 Vertex (h, k) ==> (0, 0) Focus (0, -a) (0,-3/2) Equation of latus rectum y = -ay = -3/2 Equation of directrix y = ay = 3/2 Axis of symmetry x = 0 Equation of directrix 4a = 6 units

Problem 3 :

3(y - 3) = (x - 6)2

Solution :

(x - 6)= 3(y - 3)

(x - h)= 4a(y - k)

The parabola is symmetric about y-axis and open upward.

4a = 3

a = 3/4

 Vertex (h, k) ==> (6, 3) Focus (h, k + a) k + a = 3 + (3/4)= 15/4(6, 15/4) Equation of latus rectum y = k + ay = 15/4 Equation of directrix y = k - ay = 3 - (3/4)y = 9/4 Axis of symmetry x = 6 Equation of directrix 4a = 6 units

Problem 4 :

-2(y - 4) = (x - 1)2

Solution :

(x - 1)= -2(y - 4)

(x - h)= -4a(y - k)

The parabola is symmetric about y-axis and open downward.

4a = 2

a = 2/4

a = 1/2

 Vertex (h, k) ==> (1, 4) Focus (h, k - a) k - a = 4 - (1/2)= 7/2(1, 7/2) Equation of latus rectum y = k - ay = 7/2 Equation of directrix y = k + ay = 4 + (1/2)y = 9/2 Axis of symmetry x = hx = 1 Equation of directrix 4a = 2 units

Problem 5 :

4(x - 2) = (y + 3)2

Solution :

4(x - 2) = (y + 3)2

(x - h)= 4a(y - k)

The parabola is symmetric about x-axis and open rightward.

4a = 1

a = 1/4

 Vertex (h, k) ==> (2, -3) Focus (h + a, k) h + a = 2 + (1/4)= 9/4(9/4, -3) Equation of latus rectum x = h + ax = 9/4 Equation of directrix x = h - ax = 2 - (1/4)x = 7/4 Axis of symmetry y = ky = -3 Equation of directrix 4a = 1 unit

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