Graph each polynomial function. ƒ(x)=(x-2)^2(x+3)

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Accepted Answer

Hello, everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states consider the given polynomial function and plot its graph where F of X or our function is equal to open parentheses, X minus nine, closed parentheses, squared, multiplying and open parentheses, X plus 13, closed parentheses. Now, when looking at a question like this, I see that we have it already in factor form. So we have two parentheses, multiplying each other and with one of them being squared, but they each are multiplying each other with an X inside of them. So this is kind of already in a effective form. So we can try to solve for the zeros of our function. And to do that, all we have to do is set, whatever is inside of each pricy is equal to zero. So in this case, we'll have X minus nine equals zero and we'll also have X plus 13 equal zero. So for X minus nine, we just move the nine over from the left to the right by adding and we'll have that X is equal to nine. And for extras 13, we move it from the left to the right by subtracting the 13. So we'll have that X is equal to negative 13, which means we'll have points at nine comma zero and negative 13 comma zero. No, I can look at those are my X intercepts so I can look at my Y intercept now and to do that, I'll just put zero as my X. So I'll have F M zero is equal to zero minus nine inside of the first parentheses. And that'll be squared multiplying the second parentheses with X plus inside of it. And once I distribute all that and do all that math, the answer will be or a Y intercept of zero comma 1053. Now, I can look at the multiplicities of my zeros to see how that's gonna change the shape of my graph. So X is equal to nine has a multiplicity of two. And to figure out the multiplicity, you just look at the exponent associated with this X. So for example, my X equals nine is associated with the parentheses of X minus nine. And that parentheses was raised to the second power. So it's a multiplicity of two. So we have a multiplicity of two and therefore, we will have a tangent at that point and change direction. So that is what will happen at X equals nine. Now, for X equals negative 13 that has a multiplicity of one because it wasn't raised to any other power. So that's the most of one. Therefore, we will cross the X axis at that point. And finally, I want to see how each end of my graph will look like. So I have to look at the dominant term or the dominant exponent in this case, um to do that, we just look at our function. We see that our first parentheses is X minus nine, but it's squared so that X will be squared. So now we have a power of two, however, that X squared will be multiplied by another X in the X was 13. And you, when you have X squared multiplied by an X, you have X cubed. So our dominant term here will be X Q. Therefore grap ends will be the left side of the graph will be going downwards and the right side of a graph will be going upwards. And now with all this general information, we can draw a general idea of what our graph will look like. So we draw a Y axis and an X axis and plug in some of the points that we know of. So we know we have a point at X equals nine. So we have a, a zero, X equals nine or our, our X intercept of nine comma zero. We also had one at X equals negative 13. So at negative negative 13 or the point negative 13 common zero, that was another intercept our Y intercept was way high into our Y axis. So we extend that upwards and just imagine we're very high up in 1053. And now we look at our graph moves. So the end of our graph on the left will be going downwards. So from negative 13 on into the fourth quadrant, we go down and the other end will be going upwards. So from nine comma zero, we go up into the first quadrant. Now we have to see what happens in between all those points. So A X equals negative 13, we cross the X axis and we go into the second quadrant and we go up up up until we do the Y intercept. And we know at some point that graph is gonna have to change directions and go back down because we have to meet the 0.0.9 comma zero. And at nine comma zero, we change direction. So we don't cross that X axis again, we only touch the nine comma zero, change direction and go up again into the first quadrant. So we start in from left to right. We start on the third quadrant. The graph is moving upwards until the point negative 13 comma zero. You intersect that point and continue to go up into the second quadrant. You have a Y intercept of 1053. And then at some point, the graph is going to have to change direction down once you go into the first quadrant, you change direction downward up to the 0.9 comma zero. And at that point, you don't cross the X axis, you simply touch that point and change directions up once again into the first quadrant. So that is a general idea of what the graph or function she look like. So I hope that was helpful. And until next time.

Graph each polynomial function. ƒ(x)=(x-2)^2(x+3)

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Techniques

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