The equation of a parabola is given. y=−1/4x2+4x−19 What are the coordinates of the vertex of the parabola?
+9466 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) . 
+2441 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.
