Question

The standard form for a parabola with vertex (h,k) and an axis of symmetry of x=h is:(x-h)^2=4p(y-k)The equation below is for a parabola. Write it in standard form. When answering the questions type coordinates with parentheses and separated by a comma like this (x,y). If a value is a non-integer then type is a decimal rounded to the nearest hundredth. 5x^2-50x-4y+113=0 The value for p is: AnswerThe vertex is the point: AnswerThe focus is the point: AnswerThe directrix is the line y=Answer

Answer 1

Given the equation of a parabola:

`5x^2-50x-4y+113=0`

• You can rewrite it in Standard Form by following these steps:

1. Add the y-term to both sides of the equation:

`\begin{gathered} 5x^2-50x-4y+113+(4y)=0+(4y) \\ \\ 5x^2-50x+113=4y \end{gathered}`

2. Subtract the Constant Term from both sides of the equation:

`\begin{gathered} 5x^2-50x+113-(113)=4y-(113) \\ \\ 5x^2-50x=4y-113 \end{gathered}`

3. Divide both sides of the equation by 5 (the leading coefficient)

`x^2-10x=(4)/(5)y-(113)/(5)`

4. The coefficient of the x-term is:

`b=-10`

Then, you need to add this value to both sides:

`((-10)/(2))^2=5^2`

Therefore.

`\begin{gathered} x^2-10x+5^2=(4)/(5)y-(113)/(5)+5^2 \\ \\ x^2-10x+5^2=(4)/(5)y+(12)/(5) \end{gathered}`

5. Rewrite the equation as follows:

`\begin{gathered} (x-5)^2=(4)/(5)(y+3) \\ \\ (x-5)^2=0.8(y+3) \end{gathered}`

• Having the equation written in Standard Form:

`(x-h)^2=4p(y-k)`

You can identify that:

`4p=(4)/(5)`

Solving for "p", you get:

`p=(4)/(5\cdot4)=(1)/(5)=0.2`

• You can identify that:

`\begin{gathered} h=5 \\ k=-3 \end{gathered}`

Therefore, the Vertex is:

`(5,-3)`

• By definition, the Focus of a parabola that opens upward is given by:

`(h,k+p)`

Then, in this case, this is:

`(5,-3+0.2)=(5,-2.8)`

• By definition, the Directrix for a parabola that opens upward is given by:

`y=k-p`

Then, in this case, you get:

`\begin{gathered} y=-3-0.2 \\ \\ y=-3.2 \end{gathered}`

Hence, the answers are:

• Standard Form:

`(x-5)^2=0.8(y+3)`

• Value for "p":

`p=0.2`

• Vertex:

`(5,-3)`

• Focus:

`(5,-2.8)`

• Directrix:

`y=-3.2`