Question

Part 5: Use the information provided to write the vertex formula equation of each parabola. Please show the work.

1. x = y^2 + 2y - 3

2. y = x^2 + 18x + 76

3. x^2 + 16x + 64

4. x= y^2 + 20y + 90

Answer 1

Answer: 1. x = (y + 1)² - 4

2. y = (x + 9)² - 5

3. y = (x + 8)²

4. x = (y + 10)² - 10

Explanation:

Complete the square by dividing the x-value by 2, squaring it, and adding that value to both sides of the equation.

1) x = y² + 2y - 3

`x + 3 = y^2+2y\\\\x + 3 + \bigg((2)/(2)\bigg)^2=y^2+2y+\bigg((2)/(2)\bigg)^2\\\\x + 3+1=(y+1)^2\\\\\large\boxed{x=(y+1)^2-4}`

2) y = x² + 18x + 76

`y-76 = x^2+18x\\\\y-76 + \bigg((18)/(2)\bigg)^2=x^2+18x+\bigg((18)/(2)\bigg)^2\\\\y-76+81=(x+9)^2\\\\\large\boxed{y=(x+9)^2-5}`

3) y = x² + 16x + 64

`y-64 = x^2+16x\\\\y-64 + \bigg((16)/(2)\bigg)^2=x^2+16x+\bigg((16)/(2)\bigg)^2\\\\y-64+64=(x+8)^2\\\\\large\boxed{y=(x+8)^2}`

4) x = y² + 20y + 90

`x -90 = y^2+20y\\\\x -90 + \bigg((20)/(2)\bigg)^2=y^2+20y+\bigg((20)/(2)\bigg)^2\\\\x -90+100=(y+10)^2\\\\\large\boxed{x=(y+10)^2-10}`

Answer 2

4.
`\displaystyle x = [y + 10]^2 - 10`

3.
`\displaystyle y = [x + 8]^2`

2.
`\displaystyle y = [x + 9]^2 - 5`

1.
`\displaystyle x = [y + 1]^2 - 4`

Step-by-step explanation:

To do this, you must perform the complete the square method
`[((B)/(2))^2].`This formula will help us with our Vertex Equations when determining differences\sums of results in our C-values, which then get converted to finding our k-values in the Vertex Equation:

`\displaystyle x = a[y - h]^2 + k \\ y = a[x - h]^2 + k \\ \\ Standard\:Equation[s]: x = Ay^2 + By + C \\ y = Ax^2 + Bx + C \\ \\ \\ 4.\:x = [y^2 + 20y + 100] - 90 → x = [y + 10]^2 - 10\:(-10 = k; '-10' + 100 = 90) \\ 3.\:y = [x + 8]^2\:(TWO\:EIGHTS\:sum\:up\:to\:16\:AND\:multiply\:to\:64) \\ 2.\:y = [x^2 + 18x + 81] - 76 → y = [x + 9]^2 - 5\:(-5 = k; '-5' + 81 = 76) \\ 1.\:x = [y^2 + 2y + 1] - 3 → x = [y + 1]^2 - 4\:(-4 = k; '-4' + 1 = -3)`

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