To practice writing the equations of vertical parabolas, write the equations of these parabolas in vertex form: focus at (-5, -3), and directrix y = -6
focus at (10, -4), and directrix y = 6.

a) The equation of the parabola in the vertex form is y = (x + 5)² -

b) The equation of the parabola in the vertex form is y =   (x - 10)² + 1

Step-by-step explanation:

The standard form of the equation of the parabola is (x - h)²  = 4p(y - k), where

The vertex form is y = a(x - h)² + k, where a =

a)

∵ The focus at (-5, -3)

∴ h = -5

∴ k + p = -3 ⇒ (1)

∵ The directrix at y = -6

∴ k - p = -6 ⇒ (2)

→ Add equations (1) and (2) to find k

∴ 2k = -9

→ Divide both sides by 2 to find k

∴ k =

→ Substitute it in equation (1) to find p

∵   + p = -3

→ Add   to both sides

∴   +   + p = -3 +  

∴ p =

→ Find the value of a using the rule above

∵ a =

∴ a=

→ Substitute a, h and k in the vertex form of the equation above

∵ y =   (x - -5)² +  

∴ y = (x + 5)² -

∴ The equation of the parabola in the vertex form is y = (x + 5)² -

b)

∵ The focus at (10, -4)

∴ h = 10

∴ k + p = -4 ⇒ (1)

∵ The directrix at y = 6

∴ k - p = 6 ⇒ (2)

→ Add equations (1) and (2) to find k

∴ 2k = 2

→ Divide both sides by 2 to find k

∴ k = 1

→ Substitute it in equation (1) to find p

∵  1 + p = -4

→ Subtract 1 from both sides

∴ 1 - 1 + p = -4 - 1

∴ p = -5

→ Find the value of a using the rule above

∵ a =

∴ a =

→ Substitute a, h and k in the vertex form of the equation above

∵ y =   (x - 10)² + 1

∴ y =   (x - 10)² + 1

∴ The equation of the parabola in the vertex form is y =   (x - 10)² + 1

the answer is a if my teacher taught me correct