For each polynomial function, identify its graph from choices A–F...

Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)^2

Accepted Answer

Hello, everybody. I hope you're doing all right. Today, today we're going to get this question that states consider the given polynomial function and plot its graph where the function is F of X is equal to negative. And then parentheses, X minus 17 closed parentheses squared, multiplying another parentheses with X minus 29 inside of that parentheses. And that parentheses also squared. No, we look at this question. I think, you know, it, it looks kind of complicated. There's two squares, there's a lot of negatives and such, but we can work through it little by little and sure enough, we'll get to a, an answer at the end. So we see that we have two parentheses squared, which we can also think about that. They're giving us a function in a factored form which means we can solve for the zeros of this function. So to do that, we just set each parentheses equal to zero. So we have X minus 17, which is inside of the first parenthesis equal to zero. And we'll also have X minus which is inside the second parentheses equal to zero. So we have X is equal to 17. As one of the zeros and X is equal to 29 as another zero. Now remember that this means zeros, those mean that you are intercepting the X axis at these points. So at X equals 17 and at X equals 29 we will be intercepting, intersecting the X axis. Now, we can try to solve for A Y intercept. So to do that, we do F of zero, so we plug in zero for our X and see where it intersects the Y. So we'll have F of zero is equal to negative and then open parentheses, zero minus 17, closed parentheses squared, multiplying in open parentheses zero minus closed parentheses squared. Now, once we do that math, we will get to an answer of negative or negative 243,049 which means our Y intercept will be at zero comma negative two negative 49. So now I'm going to see if there's any clues on what our graph will look like. Now we have two squares multiplying each other, which means that realistically we can, once you multiply them each other, you will have and X to the fourth power because when you multiply exponents, you add them so are dominating value here will be negative X to the fourth power. After multiplying the two squares together, you will have a value of negative of fourth power. Which means graph will have an end behavior of down down. So because we have a negative X to the fourth power our graph on the on each end will be aiming downwards. Now we have to think about the zeros. So both or or the zeros and their multiplicity. So let's focus on the multiplicity. So we have X equals and X equals have a multiplicity of two. Therefore graph will change direction. So if you have a multiplicity and even multiplicity in our case, which is two, then the graph will change direction. So now we kind of have an idea of what the graph would look like and we can try to graph this. So like we said, we will intercept the X axis at X equals tw uh 17. So I 17 comma zero, we have a point and we'll have another point at 29 comma zero. Those are, are two zeros. Now our Y that was zero, comma negative 243,049 which will be very low in our graph. And not enough to, there's just not enough to graph that. So for the purpose of this graph, we won't intercept the Y but we will make sure that the graph looks like it is headed down there and intercepting down there. Now we said the end behavior of our graph will be down on both sides. So from the 17 to when we go left, we'll go downwards. And when we look at our zero X equals 29 to the right, we will also go downwards. And now when we talked about the multiplicity, we said that it will change direction. So at 17, it will change direction. And then we know that at some point, it'll have to change direction again to go and intersect the Y, the X axis at X equals 29. And then at X equal 29 that also has a multiplicity of two, which is even. So we change direction once again downwards. So at the end, our graph should look kind of like an M where we intersect the X axis at X equals 17 and X equals and we go infinitely downwards. So I hope that was helpful. And until next time.

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)^2

Key Concepts

Polynomial Functions

Factoring and Roots

Graph Behavior and Transformations

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