ANALYZE THE EQUATION OF A PARABOLA

The parabola will be in four different forms,

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

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

Problem 3 :

3(y - 3) = (x - 6)²

Solution :

(x - 6)² = 3(y - 3)

(x - h)² = 4a(y - k)

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

4a = 3

a = 3/4

Problem 4 :

-2(y - 4) = (x - 1)²

Solution :

(x - 1)² = -2(y - 4)

(x - h)² = -4a(y - k)

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

4a = 2

a = 2/4

a = 1/2

Problem 5 :

4(x - 2) = (y + 3)²

Solution :

4(x - 2) = (y + 3)²

(x - h)² = 4a(y - k)

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

4a = 1

a = 1/4

Problem 6 :

Sketch the graph of the given equation and fill in the blanks for the given information. 

a) Coordinate of vertex

b) Direction it opens

c)  Axis of symmetry

d) Coordinate of focus

e) Equation for directrix

i) (x + 1)² = -8(y - 4)

ii) (y + 2)² = -4(x - 2)

iii) (x - 2)² = -12(y - 5)

Solution :

i) (x + 1)² = -8(y - 4)

Comparing the given function with (x - h)² = -4a(y - k)

(x - (-1))² = -8(y - 4)

a) Vertex is at (h, k) ==> (-1, 4)

b) The parabola opens down.

c)  Axis of symmetry : x = -1

4a = 8

a = 2

d) Coordinate of focus is at (-1, 2)

e) Equation of directrix is y = 6

ii) (y + 2)² = -4(x - 2)

Comparing the given function with (y - k)² = -4a(x - h)

(y - (-2))² = -4(x - 2)

a) Vertex is at (h, k) ==> (-2, 2)

b) The parabola opens left.

c)  Axis of symmetry : y = -2

4a = 4

a = 1

d) Coordinate of focus is at (1, -2)

e) Equation of directrix is x = 3

iii) (x - 2)² = -12(y - 5)

Comparing the given function with (x - h)² = -4a(y - k)

(x - 2)² = -12(y - 5)

a) Vertex is at (h, k) ==> (2, 5)

b) The parabola opens down.

c)  Axis of symmetry : x = 2

4a = 12

a = 3

d) Coordinate of focus is at (2, 2)

e) Equation of directrix is y = 8