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

Question

Accepted Answer

Hello everybody. Everything. 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 our function is F of X is equal to seven X cubed plus 20 X squared minus three X. Now, when looking at a question like this might be a little hesitant to go ahead and solve it because we do have some powers that go up a bit high and some numbers are maybe complicated to solve. But if we just break it down, little by little, we will get to an answer. So let's start dissecting this. And the first thing I'll do is factor out an X from our function. So we have X now multiplying an open parentheses, seven X squared plus 20 X minus three and close that parentheses. And then I can go ahead and solve this and, or factor this even more where I'll have the X outside of the parentheses, multiplying open parentheses. And inside of that first parentheses, we have seven X minus one and then we'll multiply another parentheses which will have X plus three inside of that parentheses. And now we've turned our function into a more simple factored form. And from here, we can go ahead and solve four zeros of our function. So to do that, we just set each X X term and inside it out, outside of the parent, out of the parentheses equal to zero. So we'll have the X, outside of the parentheses equal zero. Then we have seven X minus one equals zero. And then we have X plus three equals zero. So for the seven X minus one, we move the minus one over from the left to the right and we'll have seven, X is equal to seven, X is equal to positive one. And then we divide both sides by seven. So I'll have that X is equal to one divided by seven. And then for X plus three, we just move the three over from the left to the right. And we'll have that X is equal to negative three or we'll have points at zero comma zero for X equal zero, one divided by seven comma zero for X equals one divided by seven or negative three or, and negative three comma zero for X equals negative three. So those will be the zeros for our graph or our X intercepts. Now for the Y intercepts, I just have to set R X equal to zero. So we'll have F from zero is equal to zero. Multiplying an open parentheses seven times zero minus one closed parentheses. And then multiplying open parentheses, zero plus three. Now when we have an equation like this, whatever happens inside of the parentheses won't matter because we are at the end multiplying it by a zero, which is outside of the parentheses and anything multiplied by 00. So at the end, this would just be equal to zero or a Y intercept of zero comma zero. So now I can go ahead and look at the multiplicities of my function. Now we have that X equals zero, X equals one divided by seven and X equals negative three. I have a multiplicity of one. And how do we determine this? Well, we look at the exponent associated with each of those XES. So the X equals zero was the X outside of the parentheses which had no exponent. So that's an exponent of one which is so a multiplicity of one for that, the X equals one divided by seven was associated with parentheses seven X minus one. And that parentheses is just raised to the power of one, no exponent. So that also has a multiplicity of one and the X minus X equals negative three is associated with the parentheses that was X plus three, which also is not raised to any other power other than one. So they all have a multiplicity of one. And what does a multiplicity of one mean? Well, they'll have a multiplicity of one and therefore they will cross the X axis at each of those points. So that's what a multiplicity of one means, it just means that at each point that has a multiple set one or any odd number for that matter, you will cross the X axis. So now finally, before we start drawing, we want to look at what the ends of our graph will look like. So our dominant term, meaning the term with the highest exponent is seven X Q. Therefore graph. And well being the left hand side will be downwards will be going downwards and the right hand side will be going upwards. So now we have all the information we need to draw general graph of our function. We have the different X intercepts the, the Y intercept. And then we have how each end will look like and how in between those ends, the graph will move. So Mora and uh Y axis and an X axis and we'll plot the different points. So we have a point at zero comma zero. Then we had another point at one divided by seven comma zero. And we also had a point at negative three comma zero. So let's look at how our graph will look like we know the ends. So from negative three common zero onward or to the left, our graph will be going downwards into the third quadrant and below the X axis as for the other end. So from the point at one divided by seven comma zero onward or to the right and to the first quadrant above the X axis, a gravel were moving upwards. So now we're missing the area between, so let's look at the point between negative three and zero comma zero. So we said that each one has a multiplicity of one. That means that a negative three, we cross the X axis and we go from below the X axis to above the X axis. And now to be able to cross at zero because zero also has a multiple C of one. So we have to cross the X axis again. And instead of being above the X axis, now go under it, we have to change direction in between those two points go downwards intersect zero, comma zero and go below the X axis. And now at the next point of one divided by seven comma zero, that also has a multiplicity of one. So we have to go above the excess again. But since we're going down below, now we have to change direction again between zero comma zero and one divided by zero, comma one divided by seven comma zero, we have to change direction again and go upwards to cross the X axis at that point. So in the end, we'll have a graph that's going from the third quadrant intersects a point at negative three comma zero goes above the X axis into the second quadrant changes direction. In the second quadrant downward intersects the point at zero comma zero goes below the X axis again into the fourth quadrant and changes direction quickly to intersect the point at one divided by seven comma zero and cross above the excesses again into the first quadrant. So that is a general idea of what a graph should look like a pointless function. So I hope that was helpful. And until next time.

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

Key Concepts

Polynomial Functions

Degree and Leading Coefficient

Graphing Techniques

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