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Fiad the roots of $f(x)=x^{4}-4 x^{2}$ and sketch its graph, Oan which intervals is $f(x)$ decreasing?...

Question

Fiad the roots of $f(x)=x^{4}-4 x^{2}$ and sketch its graph, Oan which intervals is $f(x)$ decreasing?

Fiad the roots of $f(x)=x^{4}-4 x^{2}$ and sketch its graph, Oan which intervals is $f(x)$ decreasing?



Answers

find the zeros of $f(x)$ and sketch its graph by plotting points. Use symmetry and increase/decrease information where appropriate.

$$
f(x)=x^{2}-4
$$

Okay. First, let's calculate this year's of dysfunction. So we're gonna take two weeks. Square minus four equals zero. Now we can move for to the other side. We have two x squared. It goes to four. We can divide by two both sides and we get X squared is equal to two Now, from there we can take the square root both sides and where we take the square root, we get plaster minors and a class A miner's square adult too, so the serious are minor, escorted to an escort of two. Now let's check the symmetry of dysfunction. So this function is a polynomial where all the terms have even degree in That means that the function is gonna be even. But listen, Jake. Anyway, if of negative eggs, you see well to to times negative X square minus for negative X squared is gonna be the same that x square. So we have two x squared minus four. And that's if affects. So that means that the function is eat now this blood some numbers negative. Three negative three. Sport is nine times to ease 18 minus 4 14 to negative two square east four times two is eight minus for four negative. One square is one times two is two minors for minus two zero square. He see rope. Times two is zero minus for mindful. One square is one times two. My nose. Ah, for its negative too. And we see that since the function is even, we're gonna obtain the same values for 200 to 2 and for three and 83. And if you make the graphic, we can see that effectively the function is symmetric Respect off the Y X.

So first the phone to find a solution we have doesn't put pickle juice, huh? Now we do it. A factor was X X squared, minus phone. Could you sell? So there's a muse, ext. Good zero X squared minus for you Could, you know, doesn't oblige that X. It's quite difficult for X. We could you press the manuscripts far. And thanks to the courage of wrestle managed to so have three solutions and not this step. Do you find a signature? Who had you get F minus tax code? You minus x tree for times. Minutes. Next. So we should get minus expand entry plus for thanks. So you could your manners expect tree matters for ex? So you could You minus f Thanks. So means another function F thanks. He's got Angie means now who's sumit chicks. But back to you Hey reaching so we can make a table off Except by now. Uh, why? Yeah, we had this zero equals one. We go to my next one, Will winning sandwich again, huh? Minus three. And, uh, Joe, we got does there already and far on three. But when you're inside was get going. T o Jenny seven minus trail. So could you fit in? So we have enough information now, so I'm going to start now. So zero coaches row on. Then one will get nickel to ministry. So monetary side So worry somewhere here my mystery do we gonna snow And two You got a perfect in So somewhere here So is that isn't being 15 This will be minus three so we can go on the grapples it is. And because of racism it respected the urgent mission being about you gone under Grant here, Miss Mentor.

For the function F of X equals two x to the fourth. We're going to be drawing a sketch of this graph in order to find where this function is going to be increasing and decreasing. So let's start out with drawing a table of values so we'll have our X values and ever. Why values? So we'll do negative three and negative to negative 10 going to three. So we'll put these each through our function. So two times a negative three to the fourth is equal to two times negative. Three to the fourth is 81. That's going to be 162. We don't have that on our graph. It's going to be somewhere way out here, so let's move on to negative two. So two times negative to to the fourth is equal to two times negative to to the fourth of 16. So that's 32 so we can see that on our graph. It's going to be right there. Our next point is two times negative. One to the fourth, which is equal to two times one is equal to two, so it's going to be way down here. Zero is too. Time zero to the fourth is equal to two time zero or zero next for 12 times. One to the 4th 6 to 2 times one from to the fourth of one That's going to be too. And they seem to mirror each other, Which makes sense because this is a this is an even function. So it is going to be symmetrical across the Y axis. Now we have two times 16 which is equal to 32. So we're going to have this somewhat flat Carrabba low looking shape as our graph. And now we're going to need to know where this is going to be increasing and where this is going to be decreasing. So we're increasing starting from zero. This is our turning point all the way up to infinity. So we're increasing from zero to infinity and we're decreasing from negative infinity all the way down to zero. So are decreases negative. Infinity to zero. So these are our two intervals that we're looking for for this for rabble

For this problem, we're going to be using the function f of X equals two X to the fourth. We're going to be sketching a graph of this function and then using that graph to find where we're going to see our function increase and decrease. So because we have X to the fourth we're going to expect and no other extremes, we're going to expect this to look like f of X equals X to the fourth, which is a sort of, ah, wider parabola. And it's going to be amplified by two, so it's going to just be taller than X to the fourth would normally be. So let's start out by having a table of X and Y values. We're going to go from negative three negative to negative 10123 That should be enough to see our general shape of this curved. So we're going to plug each of these into our function. So two times negative three to the fourth is equal to two times 81 which is equal to 162 two times negative to to the fourth is equal to two times 32 which is equal to 64 two times negative one to the fourth is equal to two times one, which is equal to two two time. Zero to the fourth is to time zero, which is, of course, equal to zero. Two times. One to the fourth is equal to two times one equals two, two times two to the fourth is equal to two times 32 which is equal to 64. And you'll probably notice no, that were mirroring what we originally had, because this is an even exponents, so it's going to be the same on both sides of the Y. X is just mirroring itself so that negative three we have a point at 162 which is too high for us to graft a negative two. We have appointed 64 so it's going to be since 2 64 out there, a negative one. We have a point at two, so that's going to be there at zero. We have a point at zero, so at the origin at one, we have a point at 12 right there at two. We have another point at 64 so we can connect our dots and see that we have this tall parable, a shape and we're going to now see where a our function is increasing and be where function is decreasing. So we're going to be increasing starting at this this turning point at the origin all the way up to infinity. So that's going to go from zero. Let's make that a bracket rather than a parenthesis zero to infinity. And when we're decreasing, we're going to go from negative infinity all the way to zero. So as long as so, as long as we're within negative infinity and zero, we're going to be decreasing, and as long as we're from zero to infinity on the X axis, we're going to be increasing.


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