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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=(2 x+4)(4-x)$$

(a) $x=-2$(b) $x=4$(c) $y=-2$(d)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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(a) Find the $x$ -intercep…

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Graph the function$$f(…

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(a) Determine the $x$ -int…

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For the graph of $y=f(x)$<…

by finding our X intercept vertical ascent, oats and horizontal asuntos. Hopefully we're going to be able to sketch the graph of the function to expose four divided by four minus X. So starting with part A we're going to find the X intercept. And what we do here is really just set this entire function equal to zero and solve for X. Multiplying both sides by four minus X to get rid of that fraction on the left hand side, we're left with two. X plus four is still equal to zero. Let's subtract four from both sides. We have two. X is equal to negative four and divide both sides by two. We can see that our X intercept is going to occur at X is equal to negative two. That's going to sit right in here. We know our graph will pass through that point at some point Part B We want to find a vertical ascent to. And what's important here is our denominator. So four minus X. And really we can just set this equal to zero and solve for X. And that'll give us that a sum total. So we can see here is that X is going to be equal to four, so at X is equal to four out here, we're going to have this vertical ascent. Oh, meaning our graph isn't quite going to reach it, but it's probably going to get very close and for part C we want this horizontal asuntos So here we're going to take the dominant terms in our numerator and denominator. So it's going to be two X and a negative X. Simplify this. So these exes are going to cancel out which is going to leave us with two over a negative one. So we are going to have a horizontal as I'm told, that occurs when y is equal to a negative too. So when y is equal to negative two, we haven't asked him to down here. Let's go ahead and sketch this now. And although with the information we have here, we don't know exactly where the turning points of a graph are gonna be. So I like to sketch this or plug, there's no graphing calculators have a better idea and then I can check my work essentially. So if we look at this, you can see that we do have curves that sit down here following the information we gave ourselves as well as in here. So you can see that it does represent what we have found. We have our X intercept and our ascent oats, and we can at least give ourselves a rough sketch of this, given that information.

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