The equation of a parabola is given. y=−1/4x2+4x−19 What are the coordinates of the vertex of the parabola?

Guest Jun 4, 2017

#2**+1 **

We can get the equation in " vertex " form, y - k = a(x - h)^{2} , where (h,k) is the vertex.

y = -\(\frac14\)x^{2} + 4x - 19 Multiply through by -4.

-4y = x^{2} - 16x + 76 Subtract 76 from both sides of the equation.

-4y - 76 = x^{2} - 16x Add (16/2)^{2} , that is, 64 , to both sides of the equation.

-4y - 76 + 64 = x^{2} - 16x + 64

Now the right side of the equation is a perfect square trinomial and can be factored like this...

-4y - 76 + 64 = (x - 8)(x - 8)

-4y - 12 = (x - 8)^{2} Multiply both sides of the equation by \(-\frac14\) .

y + 3 = -\(\frac14\)(x - 8)^{2}

Now that the equation is in this form, we can see that the vertex is the point (8, -3) .

hectictar
Jun 4, 2017

#3**+1 **

I'm posting this solution as an alternate method to finding the vertex of a quadratic equation. Either method, presented by hecticlar or me, are acceptable methods.

Finding the vertex of a parabola is actually simple, or, at least, I think so. First, find the line of symmetry by using this formula:

\(\frac{-b}{2a}\)

However, we must identify what *a *and *b* stand for. Let's look at our quadratic function and analyze it. Here it is:

\(y=-\frac{1}{4}x^2+4x-19\)

As a review, *a *is the coefficient of the quadratic term, and *b *is the coefficient of the linear term. Let's plug it into the formula above, \(\frac{-b}{2a}\).

\(\frac{-4}{2(-\frac{1}{4})}\) | Let's solve this expression by simplifying the denominator first. |

\(\frac{-4}{-\frac{1}{2}}\) | I'll use a fraction rule that states that \(\frac{a}{\frac{b}{c}}=\frac{ac}{b}\). Let's apply it! |

\(\frac{-4*2}{-1}\) | Simplify this |

\(8\) | |

This is not our answer. The vertex is the point where either the minimum or maximum is on a parabola. The point we have found is the line that divides the parabola in half. To find the corresponding y-coordinate, substitute \(8\) into the function. Let's do that:

\(y=-\frac{1}{4}x^2+4x-19\) | Anywhere you see an x substitute in an 8 in its place. |

\(y=-\frac{1}{4}(8)^2+4(8)-19\) | According to order of operations do the exponent operations first |

\(y=-\frac{1}{4}*64+4*8-19\) | Continue to simplify until you get the y-coordinate. |

\(y=-16+4*8-19\) | Do 4*8 next, of course. |

\(y=-16+32-19\) | |

\(y=-3\) | |

After all of this, we have determined that the vertex is \((8,-3)\). This is your answer.

TheXSquaredFactor
Jun 5, 2017