CONVERT THE EQUATION TO THE STANDARD FORM OF A PARABOLA

Write each of the following equations in the standard form for the equation of a parabola, where the standard form is represented by one of the following equations :

Problem 1 :

y² - 14y - 2x + 43 = 0

Solution :

Given, equation is y² - 14y - 2x + 43 = 0

y² - 14y = 2x - 43

y² - 14y + 49 = 2x - 43 + 49

(y - 7)² = 2x + 6

(y - 7)² = 2(x + 3)

So, the equation is in the standard form of  (y - 7)² = 2(x + 3).

Problem 2 :

x² + 10x - 12y = -61

Solution :

Given, equation is x² + 10x - 12y = -61

x²  + 10x = 12y - 61

x² + 10x  + 25 = 12y - 61  + 25

(x  +5)² = 12y - 36

(x  + 5)² = 12(y - 3)

So, the equation is in the standard form of  (x + 5)² = 12(y - 3).

Problem 3 :

-9y = x² - 8x  - 10

Solution :

Given equation is -9y = x² - 8x - 10

x² - 8x = 10 - 9y

x² - 8x + 16 = 10 + 16 - 9y

(x - 4)² = 26 - 9y

(x - 4)² = -9(y - 26/9)

So, the equation is in the standard form of

(x - 4)² = -9(y - 26/9)

Problem 4 :

-7x = y² - 10y  + 24

Solution :

y² - 10y = -24 - 7x

y² - 10y + 25 = -24 - 7x + 25

(y - 5)² = -7x + 1

(y - 5)² = -7(x - 1/7)

So, the equation is in the standard form of

(y - 5)² = -7(x - 1/7)

Problem 5 :

x = 3y² - 24y + 50

Solution :

Given equation is x = 3y² - 24y + 50

3y² - 24y =  x - 50

3(y² - 8y) = x - 50

3(y² - 8y + 16) = x - 50  + 16

(y - 4)² = x - 34

So, the equation is in the standard form of (y - 4)² = x - 34.

Problem 6 :

y = 2x² + 12x + 15

Solution :

Given equation is y = 2x² + 12x + 15

2x² + 12x = y - 15

2(x² + 6x) = y - 15

2(x² + 6x + 9) = y - 15 + 9

2(x + 3)² = y - 6

So, the equation is in the standard form of 2(x + 3)² = y - 6.

Problem 7 :

3x² - 3x - 5 - y = 0

Solution :

Given equation is 3x² - 3x - 5 - y = 0

3x²- 3x = y + 5

3(x² - x) = y + 5

3(x² - x + 1/4) = y + 5 + 1/4

3(x - 1/2)² = y + 21/4

So, the equation is in the standard form of 

3(x - 1/2)² = y + 21/4. 

Problem 8 :

5y² + 5y = x - 6

Solution :

5(y² + y) = x - 6

5(y² + y + 1/4) = x - 6 + 1/4

5(y + 1/2)² = x - 23/4

So, the equation is in the standard form of

 5(y + 1/2)² = x - 23/4.