Mathematics, 18.12.2020 21:10 gavincoscarat760

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.

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Answer from: marieminnerly1628

a) The equation of the parabola in the vertex form is y = $\frac{1}{6}$ (x + 5)² - $\frac{9}{2}$

b) The equation of the parabola in the vertex form is y = $-\frac{1}{20}$ (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 of the parabola is (h, k)The focus is (h, k + p)The directrix is at y = k - p

The vertex form is y = a(x - h)² + k, where a = $\frac{1}{4p}$

∵ 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 = $-\frac{9}{2}$

→ Substitute it in equation (1) to find p

∵ $-\frac{9}{2}$ + p = -3

→ Add $\frac{9}{2}$ to both sides

∴ $-\frac{9}{2}$ + $\frac{9}{2}$ + p = -3 +

∴ p = $\frac{3}{2}$

→ Find the value of a using the rule above

∵ a = $\frac{1}{4(\frac{3}{2})}$

∴ a= $\frac{1}{6}$

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

∵ y = $\frac{1}{6}$ (x - -5)² + $-\frac{9}{2}$

∴ y = $\frac{1}{6}$ (x + 5)² - $\frac{9}{2}$

∴ The equation of the parabola in the vertex form is y = $\frac{1}{6}$ (x + 5)² - $\frac{9}{2}$

∵ 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 = $\frac{1}{4(-5)}$

∴ a = $-\frac{1}{20}$

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

∵ y = $-\frac{1}{20}$ (x - 10)² + 1

∴ y = $-\frac{1}{20}$ (x - 10)² + 1

∴ The equation of the parabola in the vertex form is y = $-\frac{1}{20}$ (x - 10)² + 1

Answer from: Quest

you forgot an image

step-by-step explanation:

Answer from: Quest

At a constant it will take one hour to travel 32 mi.at that speed it will take 1/2 hr to travel the remaining 16 mi.it will take 1+1/2 hr to travel the 48mi @ 32mph.so 1 hr = 60 min& 1/2 hr = 30 min1+1/2 hr = 90 min.there are 60s in each 90 × 60 = 3600 seconds.the real question seriously is he why are they driving so slow? ! lol.

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