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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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#1
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the 2 is an exponent

Guest Jun 4, 2017
#2
+4775
+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) .

hectictar  Jun 4, 2017
#3
+1224
+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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