ANALYZE THE EQUATION OF A PARABOLA

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