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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

The standard form for a parabola with vertex (h,k) and an axis of symmetry of x=h-example-1
User Tyshaun
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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

User Jota Pardo
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