In Exercises 1–4, the graph of a quadratic function is given. Wri...

Question

In Exercises 1–4, the graph of a quadratic function is given. Write the function's equation, selecting from the following options.

Accepted Answer

Hey everyone in this problem, we have to find the equation of the quadratic function using the given graph. And we're showing this graph here and we're told two particular points on this graph we have the vertex at one half and two we have the y intercept at 03. Okay, now, if we look at this possible answers were given, we see that they look like they're kind of in vertex form and were given information about the vertex in the graph. So let's go ahead and try to write this in vertex form. Now we have a quadratic function. When we write the equation in vertex form it's going to be y is equal to a X minus H squared plus K. I don't recall that the point given by H K is the vertex of this parabola here of this quadratic function. So in our case the vertex given in our graph is one half to So this gives us our agent K values H is going to be one half K is going to be two. And we can substitute this into the general equation. And we get that. Why is equal to a X minus a half squared plus two? No, that gives us a lot of our function. But what we don't know is a. Now if we look at our graph, we see that this parabola opens upwards. Okay, so since it opens upwards, we expect that the value of a will be greater than zero. Okay, that leading coefficient is going to be positive because this opens up. So a is going to be bigger than zero. How can we find it? Well in this equation the things that we have that aren't constant or that we don't know. Ry A and X. Okay, so if we choose a particular point and we sub in the X and Y values the only thing left that's unknown is going to be a we're going to be able to solve for a. Okay, so let's choose a particular point. Now we're given the point where the y intercept is 03. So let's use that point. Okay so we have the y intercept and 03. Let's go ahead and substitute that into the equation we have going okay so we have three, the y value is equal to a times X which is zero minus a half squared plus two. This gives us a three is equal to minus a half square. That's gonna be 1/4. We get 1/4 A plus two. So we have 1/4. A is equal to If we move the two over to the other side, That's equal to 1 3 -2. And then we multiply both sides by four and we get that a is equal to four. Okay, four is bigger than zero. This is positive like we expected. Okay so we have a value of four. No, if we rewrite our function now with this a value we're gonna have that Y is equal to four times X minus one half all squared plus two. So if we look at our answer traces, we see that none of these match what we have so far. What can we do? Well none of these have that constant out in front of this term that squared. So let's try to move this constant into the square and see where that takes us. So we know that we can write for as two squared Times X - squared plus two. Okay, these have the same exponents so we can bring it into the brackets. So we're gonna have two times x minus one, half all of this squared plus two. And then we can expand inside of that square two times X gives us two X. And then we have two times negative a half gives us minus one. So we get two x minus one squared plus two. And that is going to be the equation for the parabola for the quadratic function that we were given in that graph. And if we go back up to our answer choices and now that we've brought that four into the brackets, we see that this corresponds with answer choice B. So we have y is equal to two, X minus one squared plus two. Thanks everyone for watching. I hope this video helped see you in the next one

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In Exercises 1–4, the graph of a quadratic function is given. Write the function's equation, selecting from the following options.

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