Graph each polynomial function. Factor first if the polynomial is...

Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-2x(x-3)(x+2)

Accepted Answer

Hello. Today we're gonna be graphing the following polynomial function. So the polynomial function given to us is negative three, X times the quantity X minus four times the quantity X plus one. Now, in order to graph this polynomial function, we're first going to need to find the regions of increasing and decreasing on the graph. But in order to do that, we're going to need to first get the zeros of X. So in order to get the zeros of X, we're going to take our polynomial function and set it equal to zero, then we're gonna go ahead and solve for the values of X. Now notice that our polynomial function is already in the shape of a factored form. So what we can do is take each factor of X set equal to zero by using the property of zero. So that's going to give us negative three, X is equal to zero, X minus four is equal to zero and X plus one is equal to zero. And we just need to solve for each value of X. On the left hand side, if we divide both sides of this equation by negative three, we are going to be left with X is equal to zero in the middle. If we add four to both sides, we get X is equal to positive four. And on the right hand side, if we subtract both sides by negative one, we get X is equal to negative one. So now that we have the zeros of X, we're gonna go ahead and plot them on a number line. This number line is going to represent the domain of the function going from negative infinity to positive infinity. Next, we're going to plotter zeros which is negative 10 and positive four. And what we need to do now is we need to take the, the testing point from each of the regions in order to determine the regions of increasing and decreasing. But there is a shortcut that we can take in order to determine the behavior of the end points. See, notice that every factor of X in our equation has an exponent of one, we can go ahead and add all values of X together to give us X cubed and notice that the leading coefficient has a negative number. So the highest order term in this polynomial function is going to be negative three X cubed. Since the exponent is an odd number that tells us that the end behaviors of the graph are going to be facing in opposite directions. And because we have a negative exponent or a negative coefficient that tells us that as X approaches negative infinity, the graph is going to increase to positive infinity. And as X approaches positive infinity, the graph is going to decrease to negative infinity. So now we have the end behaviors for the regions of the graph. Now we just need to take a testing point between negative one and zero and zero and four and find the behaviors for those regions. So what's a testing point that we can take between negative one and zero? Well, that can be when X is equal to negative 0.5. If we plug this into the graph, what we get is three times negative 0.5. I'm sorry, negative three times negative 0.5 which is going to give us 1.5 times negative is 0.5 minus four, which is negative 4.5 times negative 0.5 plus one, which is going to give us positive 0.5 and multiplying these three values together is going to give us negative 3.375. Now notice that when we took the value of X is equal to negative 0.5 and plugged it into the equation, we got a negative number. So that tells us that the region between negative one and zero is going to be decreasing. Now we just need to take a test point between zero and four. That testing point can be when X is equal to one. And if we plug this into the uh into the original function, we get negative three times one, which is negative three times one minus four, which is negative three times one plus one, which is going to be two, a negative three times negative three times two is going to give us positive 18. Now, since we got a positive number that tells us that the region between zero and four is going to be increasing. Finally, there's one more piece of information that we need and that is going to be the Y intercept. In order to get the Y intercept, we're looking for the region of the graph that intercepts the Y axis. And that is going to be when X is equal to zero. So what we're going to do is we're going to take the value of X equal to zero and plug it into the original equation. But notice that when we do that we're going to have negative three times zero, which is zero times everything else, which is just going to be zero. So that tells us that the Y intercept is going to be located at zero comma zero. So now that we have the regions of increasing and decreasing and we have the white intercept, we can go ahead and use this information to make a rough sketch of the graph. So we're gonna start by plotting our zeros, we have negative one, we have zero comma zero, which is also going to be the white intercept. And we have positive form. Now we already know the behaviors of the end points. As X approaches negative infinity, the graph is going to increase to positive infinity. And as X approaches positive infinity, the graph is going to decrease to to negative infinity. Next, we know that the region between negative one and zero is decreasing. So we can represent that by drawing it upside down hill in that region. And finally, the region between zero and four is going to be increasing. And we can represent that by drawing it upward hill and connecting the dots. And this is going to be the rough sketch of the graph for the polynomial function. So now what we need to do is we just need to go ahead and select the option that accurately represents this graph. And that option in this case is going to be a just to make sure we have the zeros at negative one at zero and at positive four with zero comma zero being the Y intercept and the shape of the graph roughly matches the one that we drew below. So with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-2x(x-3)(x+2)

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