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Find the distance between the vertices of the parabolas $y=-\frac{1}{2} x^{2}+4 x$ and $y=2 x^{2}-8 x-1$.

17.12 units

Algebra

Chapter 4

Polynomial and Rational Functions.Applications to Optimization

Section 2

Quadratic Functions

Linear Functions

Polynomials

Rational Functions

Missouri State University

Harvey Mudd College

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So we're gonna need to find the distance between our Vergis ease of over two equations I renamed her first equation toe. Why one which is equal to negative 1/2 X squared plus four x on our second equation. Just too wide to which equals two X squared minus eight X minus one. So first things first. Fine. The vertex of y one. No, but notice you could just put both of these into a graphing calculator and get our Vertex is from that. But assuming you don't have a graphing calculator, we're gonna do this by completing this foreigner. So why one factor at a 1/2 from everything was gonna be X squared? Minus eight acts which gives us thinking of 1/2 times X squared minus eight x plus 64 1964 are not 64. That's for eat. We're trying to find half of eight. I was just gonna be four squared are 16. So one is gonna be able to negative 1/2 X minus four. I swear. Then, by the 16 now, all we have left to do is distributed or 1/2 back in. So getting negative 1/2 x minus four squared Plus it waken See that our vertex what I want so called spare text one that for eight now, Same thing for why to if you just want to complete the square here, where If you just want to put this into a graphing calculator and it's perfectly fine, we're gonna assume that we don't have a graphing calculator. So our first step is gonna be the factor out to e X squared minus four x minus one had get two times X squared, minus four plus four minus four. Last 1/2 we continue. Okay, thanks. Minus two squared. And then we'll combine like terms in the end just to get minus nine. So what we have left to do is distributed or two and what you is equal Few times X minus two squared minus nine. Now we can see that our Vertex called for attacks to Is that to kind of night? A negative mind. So we have four comma, eight in two common negative nine. So our distance between the Vertex we'll use for a distance more real. She's just gonna be the square room important to my Why won't spread the cross next to mine aspects one square. So our vertex to in this case, has a Y value negative nine that are burnt. Text one has a why value eight x value. Our second veritext doesn't expel you too, that our first for Texas an X value before. So now we just saw and we get negative 17 squared, which is just to 89. And then we have tu minus four, which is negative. Two squared. That's just gonna be four. As is just gonna be equal to the square root of 293 which this fine how it is or approximately equal 17 11

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