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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
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3+0 Answers

 #1
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the 2 is an exponent 

Guest Jun 4, 2017
 #2
avatar+5576 
+1

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

 

y  =  -\(\frac14\)x2 + 4x - 19                      Multiply through by -4.

 

-4y  =  x2 - 16x + 76                     Subtract 76 from both sides of the equation.

 

-4y - 76  =  x2 - 16x                      Add  (16/2)2  , that is,  64  , to both sides of the equation.

 

-4y - 76 + 64  =  x2 - 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) .  smiley

hectictar  Jun 4, 2017
 #3
avatar+1493 
+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 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, is the coefficient of the quadratic term, and 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
edited by TheXSquaredFactor  Jun 5, 2017

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