In Exercises 57–62, use the vertex and the direction in which the...

Question

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function?

y = - x^2 + 4x - 3

Accepted Answer

everyone in this problem? We are given an equation of a parabola. Okay. We're asked to use its vertex and the direction in which it opens to solve for its domain and range. And to tell if it is a function or not. Alright, so the equation we're given is why is equal to negative x squared plus 10 X minus 21. Now let's recall if we want to find the vertex we can use the vertex form of a parabola. And the general equation for vertex of a parabola is given by the following. Y is equal to a times x minus H squared plus k. Okay. And if we have an equation in this form then our vertex it's just given by H. Okay. All right, let's go ahead. Now, in order to write our equation in this form we need to complete the square. Okay, so we have y is equal to negative X squared plus 10 X minus 21. Okay, in order to complete the square, we're gonna take the first two terms. Okay, we're gonna take the any terms with X. We're going to work with them. Okay, So we're gonna have negative X word plus X. Mhm. Now, in order to complete the square we're gonna want to have negative x minus five. Okay, why do we want X -5? Well, because when we square this we're going to get x squared minus 10 X. And then we have the negative. So we're gonna get negative X squared plus 10 X. Like we want. Okay, so we always want this number that you're subtracting or adding is going to be half Of the term that you have associated with your X -X - squared. Alright, so that deals with the X terms. But we know that when we square this we're gonna have a constant term to Okay, So if we square this we're gonna end up with negative X squared minus 10 X plus 25. Okay. And applying that negative all the way through, we get negative X lost minus 25. Okay. All right. So let's go ahead and see how we can use this with our original equation. Okay? We have why? And we can write negative X squared plus 10 X minus 25 as this negative X plus or x minus five squared. Ok, but what do we do with that minus 25 wow. Let's go ahead and write it up. We have negative X squared plus 10 X minus 21. That's our original function. In order to complete the square we need to subtract 25. So let's subtract and add 25. That's going to be equivalent to adding zero. Negative 25 plus 25. We're just left with our original equation. Okay, so we can rearrange this into negative X squared plus X minus 25. Okay? And then we have negative 21 plus 25. That's going to be plus four. We know those first three terms we just did the completing the square those first three terms are going to be equal to negative X -5. Sward. Okay. And then we have our plus four. All right now this looks good. This looks exactly like the vertex form we wanted. Okay, We have a is equal to -1. We have a church is equal to five and we have K. Is equal to four. Okay. And let me just highlight those because we're gonna come back to them a few times. Well this tells us that our vertex. Okay, so let's write our vertex our vertex again. Is that H K. H S five K is four. So our vertex is going to be at 54. Okay. And a is negative. It's less than zero which tells us that this problem is going to open downwards. Alright, so we know where our vertex is. We know which way this opens. So let's get going and find the domain and range. Like we're asked to let's give ourselves some more space and I'm going to draw a quick diagram. Okay, so we have access here um X. Y. And we have at 54. We have our vertex. Okay, so this is our vertex here and we know that the parable opens down and so it's gonna look something like this. Now we're thinking about our domain domain of a paralysis. A quadratic equation. We're thinking about the X values and in this case we can choose any X value. Okay, if we put any X value into this equation it will work and we can see that we can move along this X axis and find some point associated with any X value. Okay, so we're just going to have any X value and if we want to write this in different notation we get negative infinity to positive infinity. Okay? Any X value. When we're thinking about the range on the other hand, we see that there are values that Y does not reach. Okay, so the vertex is that y is equal to four? And the parable opens down so why will never be above four? Okay. So we're gonna have y less than or equal to four. And if we want to write this in different notation, well we're going to have negative infinity. Up to four. Okay. We're gonna have a square bracket there because we're including four. Alright, so we have our domain, we have a range. Last thing we want to do is figure out if this is a function. Now if this is a function we want it to pass our vertical line test. So we draw a vertical line and we see how many points it hits along our function. Okay. And for any vertical line we draw it's only going to hit one point. Okay, For every X value there's only one associated Y value. And so this is a function and I'm going to put in brackets that we know that because of our vertical line test, which we did in red on our graph. Alright, so let's go up and answer this question. We have that the domain is negative infinity to infinity. The range is negative, infinity. Up to and including four and that this is a function. And so we're going to have answer C. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 57–62, use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? y = - x^2 + 4x - 3

Key Concepts

Vertex of a Parabola

Direction of Opening

Domain and Range

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