We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website.
Please click on "Accept cookies" if you agree to the setting of cookies. Cookies that do not require consent remain unaffected by this, see
cookie policy and privacy policy.
DECLINE COOKIES

The equation of a parabola is given. y=1/8x^2+4x+20 What are the coordinates of the focus of the parabola?

Guest Jun 4, 2017

#1**0 **

The parabola focus is the point wherein the distance to a point on a parabola is equidistant to the distance to the directrix!

To find the focus, convert the quadratic to vertex form, \(y=a(x-h)^2+k\) where \((h,k+\frac{1}{4a})\) is the focus. Let's try and do this:

\(y=\frac{1}{8}x^2+4x+20\) | This is the original quadratic equation. In order ro convert the quadratic to the desired form above, we need to use a method called "completing the square." First, subtract 20 on both sides. |

\(y-20=\frac{1}{8}x^2+4x\) | Multiply by 8 on both sides to get rid of the pesky fraction |

\(8y-160=x^2+32x\) | This is where completing the square comes in handy. Do the linear x-term and half it. Take that quantity and square it. Add it to both sides. |

\(8y-160+(\frac{32}{2})^2=x^2+32x+(\frac{32}{2})^2\) | Simplify both sides of the equation |

\(8y+96=x^2+32x+256\) | What's the point of doing all this work? Well, the right hand side is a perfect square trinomial. |

\(8y+96=(x+16)^2\) | Subtract 96 on both sides of the equation |

\(8y=(x+16)^2-96\) | Divide by 8 on both sides |

\(y=\frac{1}{8}(x+16)^2-12\) | |

Our quadratic equation is finally in vertex form. Now, we can find the focus by using the formula I mentioned above, \((h,k+\frac{1}{4a})\). Let's plug those values into this quadratic equation. First, identify what *h, k, and a *are.

h=-16

k=-12

a=1/8

Let's plug these values in:

\((-16,-12+\frac{1}{4(\frac{1}{8})})\) | Do 4*1/8 first. |

\((-16,-12+\frac{1}{\frac{1}{2}})\) | I'll use a fraction rule that states that \(\frac{a}{\frac{b}{c}}=\frac{ac}{b}\) |

\((-16,-12+2)\) | Continue simplifying. |

\((-16,-10)\) | |

Now, you are finally done. The point of the focus is \((-16,-10)\).

TheXSquaredFactor Jun 5, 2017