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?

x = - 4(y - 1)^2 + 3

Accepted Answer

hey everyone in this problem we're given the equation of a parabola and we're asked to use its vertex and direction of opening to solve for the domain and range and to figure out if this relation is a function or not. The equation we're given is x is equal to negative eight times y minus five squared plus nine. And what you'll notice is that we have X equals. Okay. And some white term squared, we see something like this, we want to think okay now we have a parabola that's opening either to the left or to the right. Okay. Instead of opening up and down like we're used to seeing when we have Y equals. Okay so our X and our wire a little bit swapped so we're gonna have a problem opening to the side. Alright, so let's recall our general vertex form and now when we have an X equals equation we can write this as X is equal to a times y minus k squared plus each. That's our general equation in the vertex can be given by H. Okay, alright comparing this to our equation, We see that RK is going to be negative five or sorry, positive five because we want minus K. Here we have minus five. So K is five and h is the term that gets added at the end which in this case is nine. Okay so the vertex of our parabola is going to be nine fight. Alright, so we found the vertex. Now we need to figure out what direction it opens and again because we have X equals and the Y term is squared. Our problems either opening to the right or its opening to the left. Okay, here we have a Okay, a is -8. So a is less than zero which tells us that our parabola is going to open to the left. All right. So negative A. Recall means our parable opens to the left. So if we do a quick little sketch here, we have X. Here, this will be nine and we have the Y axis here, this will be five. So our vertex is at 95 and our parabola is opening up to the left. So it looks something like this very roughly. All right. So quick little sketch Now we want to determine the domain, the range. And we want to figure out if this is a function. So let's look at the domain first. So we're looking at the domain, we're considering all of the X values. Now for considering the X values. Let's take a look at our diagram and let's think about where a vertex is. The X value of the vertex is nine. And this problem is opening left. Okay, so this function can have X values equal to nine at the vertex or anything less than nine. But this problem will never have X values to the right of nine. So our X values have to be less than or equal to nine if we want to write this in um different notation. Just in set notation or not? In set notation? Sorry? We have negative infinity. Okay. So any number all the way down to negative infinity or up to nine. Okay. And this includes nine. So we want to square bracket there because the vertex is right at X equals nine. Alright. And then we're looking at the range, we're thinking about possible Y values. Well we can plug any Y value into this equation and we'll get an X value out of that. Okay, so this problem opens up and we're going to have any possible Y value. So there's no restriction on why? And so this is going to be negative infinity to positive infinity. Alright. The last thing we want to do. So we found our domain, we found our range. The last thing we want to do is figure out if this is a function or not. Now we're called the vertical line test. If this function passes a vertical line test or Sorry if the expression the relation passes the vertical line test, it's a function. The vertical line test tells us if we draw a vertical line anywhere on our graph and it hits two or more points more than one point. This is not a function. Okay, so this one in this case when we draw a vertical line we hit two different parts of our function or our relations. So this is not a function. Okay. And in brackets I'm gonna put V L T vertical line test. Okay. Alright. So essentially what the vertical line test shows us is that when we choose any X value, there are two possible Y values. Okay. When we have that we don't have a function. All right. So looking at the possible answers, we have a domain going from negative eight up to up to including nine. We have a range negative infinity to positive infinity. And this is not a function. So we're gonna have answer. B. 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? x = - 4(y - 1)^2 + 3

Key Concepts

Vertex of a Parabola

Direction of Opening

Domain and Range

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