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Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = x^2 + cx $

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 6

Graphing with Calculus and Calculators

Derivatives

Differentiation

Volume

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We want to describe how the craft of F of X is equal to X squared plus C. That's Berries as C Berries. And we're going to then graft several members of the family to illustrate trends that we discovered. And in particular, we want to investigate how our maximum minimum and points of inflection move as sea changes. We also want to identify any transitional values of C at which the basics shape of the curve will change. So here you can see I have the functions listed for seizing with a negative one. Negative too negative. 1012 But before we look at these graphs more, let's just do a little bit of stuff with the general function. So the first thing you might know this is we could go ahead and factor this because we can pull out a X so we get X is equal to, or this is equal to X Times X plus C. And if we were to go ahead and set this equal zero, that's going to imply that either X is equal zero or X is equal to negative. Seen this using the zero product property, so we know that no matter what, we will always have an X intercept, that zero. And for whatever value we choose for C, the negative of that will be our other ex Anderson. All right, so that gives us a little bit of information. Now let's go ahead and look at the derivative of this so f prime of X. Because remember to find maximum or minimum points, we can look at critical values, which is where our directive is equal to Israel or undefined. So taking the derivative of X squared plus C x we would use power rule so X squared becomes to X and see X will see is a constant. So the derivative of X is just one. So we just end up with two explosive now to find our critical values. Remember, the thing we want to do said this equals zero, and that's going to imply that if we have any, Max's airmen's is going to occur at negative. See over. And you might recall that this year is just the Vertex because we're working with a quant traffic, so we could have did the whole negative be over two. A plug goes in to get the same point. So we know how are in this case is going to be a minimum, since we know our leading coefficient is larger than zero positive, we know how are minimum is going to change because it's just going to shift to the left or to the right based off this value are released, the position or long be why access the X axis is going to shit. I should say that. And now for our last part, let's go ahead and look a second derivative and see if we can find any points of inflection. So F Double Prime of X is going to be just to cause a derivative of two x will driven of X is one. So we're just up with two. And the derivative of See Sense is a constant would be zero. And we know that this here is strictly greater than zero, which is going to imply that our function is always con cave up. So there should be no points of inflection. And if we just go through and look at these values, we could kind of see what we expect to occur from this little analysis does happen so starting at C is equal to zero, which is this middle graph here. We could see when c 0 to 0. We get that our ex intercepts are only at zero. And if we were to go ahead and plug in zero into negative c over to we get X zero. Sorber, Texas. Also there. Now, when C is negative, one Arbor, Tex. Shifts down and celebrate as well as we get extra X intercept hat, which is exactly what we would expect again when c zero to negative, too. It shifts halfway in between those two values for seeing for Vertex. So it's now at one and we get our X intercept over here two and looking at season one, Exeter said. Get shifted to the left one. And our Vertex is right in the middle. Those two and sees it too. Same thing are other ex intercepted ship the negative, too. And our burr Tex is half that distance

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