The ????- intercept is (0,1). The ????- intercept is (1,0). Degree is 3.End behavior: as ??????? , ????(????)?? , as ?????? , ????(????)???....

Use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or ?1. There may be more than one correct answer.

The $y$ -intercept is $(0,0) .$ The $x$ -intercepts are $(0,0),$ $(2,0) .$ Degree is $3 .$ End behavior: as $x \rightarrow-\infty,$ $f(x) \rightarrow-\infty,$ as $x \rightarrow \infty, f(x) \rightarrow \infty.$

Okay. We know the general form of this equation. Given a function of degree three and leading coefficient off either one or negative one. We know the general formula is gonna General form is gonna be plus or minus X cubed, plus a X squared plus bx pussy. You've probably seen this in the textbook, Lester. Okay? Not going to solve it. You know, the half of zero equals one, so this means that one equals negative zero cubed, plus a of zero squared, plus be of ax plus c. Therefore, we know. Sorry, we can plug in B of zero over here because we're playing in zero everywhere. So be of zero plus C. And we know this is equivalent to see. Okay, Now we know that half of zero f one is also zero, so we can do the same thing over here. Zero equals negative. One cubed plus a times one squared plus B of one plus one. This means that a equals negative be so after vax is gonna be aftereffects equals negative acts, cubed minus beat X squared plus B X plus one. And we know that being was one equals negative one. Therefore, our final equation is gonna be f of X equals negative, X cubed, minus X squared, plus acts plus one

It's question. He wants us to find the function, to write down the function, knowing that when X leads to negative infinity, why leads to positive impunity. And when X leads to positive infinity, why leads to positive infinity as well? That means that the coefficient has a positive living coefficients, so the functions thoughts with the positive leading coefficient. And now we know that the degrees for So it's positive one drink one I mean export for because at the beginning it was mentioned that the leading coefficient is either one or negative one. Now we know it's a positive export Ford, and also there is no X intercept. And the wind, sir said, is zero and one. So if we just add one year, I guess this function said's fight all the conditions. If you try to fact arise this expression, you cannot. It couldn't be fact arised. And if you try to solve for zero, you're gonna have export four equals negative one, and then X equals the fourth route off negative one, which is undefined. It's impossible. That's why it doesn't have any X intercept because there is no real value for X. So this will be our function which satisfy only given conditions

For this problem. We're told that this is a second degree function, so f of X must involve X squared of some sort. We also know that as X approaches negative infinity after the X will approach negative infinity as X approaches positive infinity of X will approach negative infinity. So on a graph that would look like this. So the parabola we know it's a problem that it's X squared must be upside down. So this must be f of X equals negative X squared of some sort and last two sets of criteria. The intercepts The Y intercept must be zero common nine. So can rewrite this as f of X equals negative X squared plus nine and the ex intercepts must be negative. Three comma 03 Common zero Well, we know that being a parabola and having the X intercept or the wide receptive nine well already guarantee that the ex intercepts are negative three comma zero and three comma zero. So our final answer for the function is f of X equals negative X squared plus night

Question. He wants us to write down the function. So by knowing that when X leads to positive infinity this way, why leads to positive infinity? And when explains to negative infinity, why leads to negative infinity? So it's obvious that that's gonna be a positive coefficient, that the leading coffee shit is positive. So we have positive a X Power three. Why? Because the degree is to the third degrees. The function is to the third degree. So now we have f of X equals a export three. Then we get a plus B X squared, plus see X, and we don't have a constant here. How did I know that? Because it is zero since the binder said zero and zero. So if I plug zero for all the exes here, then the Y intercept or the constant value will be zero. And now we know that the leading coefficient is one or negative one, and we said that it is a positive. So that's a positive export three. And also we can write dysfunction in the factor for so f of X equals. Knowing that one off the X intercepts is zero and zero, which means that exercise is a common factor here. And then we have a mother X intercept, which is two and zero. So you know that when we have one off the factors X minus two, then X equals positive, too. Because we do. X minus two equals zero. Right, So one of the two parentheses will be X minus two. And since its power three, So we need to square. I'm, um We need to square that in order to have a cubic function at the end. If you expand this print bracket and simply fine, you're gonna have f of X equals exports. Tree minus four x squared. Plus for ex dysfunction satisfy all the given condition. It satisfies the Y intercept. 00 It says Fire both ex intercepts. It says Father degrees three and positive leading coefficient.