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How many $x$ -intercepts can a function defined on an interval have if it is increasing on that interval? Explain....

Question

How many $x$ -intercepts can a function defined on an interval have if it is increasing on that interval? Explain.

How many $x$ -intercepts can a function defined on an interval have if it is increasing on that interval? Explain.



Answers

How many horizontal asymptotes can the graph of a given rational
function have? Give reasons for your answer.

In this problem, we want to know how can we tell if an absolute value function will have two X intercepts without graphing. Uh Now I do have the dez most graphing calculator open to help show you how to do this. Um But I'm going to show you real easy how you would know if an absolute value function uh would have two X intercepts without graphing first of all. Let's look at the basic absolute value function. Notice that the power of X here is one. This would be excellent first. So this is what the absolute value of X minus four. Or even basically the absolute value of X would look like it has a single X intercept right there where it touches the X axis. Now, if you look for functions that are going to have uh two X intercepts, you want to look for paralysis which basically are graphs of quadratic expressions or quadratic functions, A quadratic function is when the degree of the expression the highest power X is too uh noticing uh This expression here is not an absolute value function, but the function is x squared minus five X plus three. It is a quadratic expression because of the X squared term. If we grafted we would get a graph uh Parabola would be the graph And we have two x intercept shot This graph crosses the x axis in two places. So now we want to look at an absolute value function that is going to have uh two X intercepts. Uh So if you look at this third function that we're about to graph, it resembles this second quadratic function that we just grab the parabola. But this time we're going to grab the absolute value of X squared minus five X plus three. Okay, so here is the graph of the absolute value function. Uh The absolute value of x squared minus five X plus three. Notice it has uh two X intercepts. So here is how you can tell when an absolute value function will have two X intercepts when you are taking the absolute value of a quadratic expression. Okay, you need the absolute value and you need to have an x squared term. Basically this green function. This absolute value function is uh very related obviously to this uh regular quadratic function X squared minus five X plus three. The blue function. The regular quadratic function is basically this portion with this blue curve down here and then this portion all the absolute value does remember that absolute value turns uh negative answer is positive. So the only difference between the blue function and the green function, the absolute value function is the absolute value function. The green one uh took this negative uh blue portion. The part that was below the x axis. This negative portion and turned it positive. But since the original quadratic expression X squared minus five X plus three, that quadratic function had two X intercepts uh the absolute value of that quadratic function is also going to have to x intercepts

Can you tell if an absolute value functional have two X intercepts without graphing. Well, here here are two different ways. First, let's suppose the function looks like some number times X plus or minus some number plus some number. All right. If a is positive and see his negative, then it will have to two X intercepts. For example, FX equals X plus one minus three. Okay, I need to go once the left down three and then because it's got a positive one there upon upon. And then you could see that across this twice. Okay? Because this forces it the Vertex underneath the X axis. And because this is positive means it's gonna open up. So it has to hit twice. And then the other case, if a is negative and C is positive. So, for example, f of X equals negative X plus one plus three. All right, so we get this one shifted over one up three and then upside down, so it's gonna hit twice. Also, three other way is when you're finding the X intercept you're trying to find where the function has a wide value of zero, so you can set f of X equal to zero and then sulfur X

Problem. 86. Can a graph of polynomial function have no axe Intercept? Yes, you can. Off course you can't. So, like why you goes toe x square plus one So he doesn't have intercepts, right? We can claim that if that degrees are it must force to the left and rights to the right or or writers to the left and force the right. So there has to be a here. The access And there has to be a point between this graphs because it's continuous. Right? There has to be one, at least one. So if a paranormal function doesn't have, ah acts intercepts, it has to be even degree, right?

This question asks us to say without graphing how many ex intercepts this graph will have and we do that. We find that by using the discriminative and the discriminative is B squared minus four a. C and so B squared. So that will be nine minus four times a, which is a negative, too. Times C, which is a negative one. And so here I have a negative times negative, which is positive times and negative, which will be negative. So this will be nine minus and four times two is eight times one is still eight. So this will be one. And so when we have a positive, we're going to have to roots or two ex intercepts.


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