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Download Article A simple guide to find and grapth vertical asymptotes. Explore this Article parts. Related Articles. Part 1. Factor the denominator of the function. To simplify the function, you need to break the denominator into its factors as much as possible. For the purpose of finding asymptotes, you can mostly ignore the numerator. Recognize that some denominator functions may not be able to be factored. For this first step, you will just have to leave it in that form.

Find values for which the denominator equals 0. Still disregarding the numerator of the function, set the factored denominator equal to 0 and solve for x. Remember that factors are terms that multiply, and to get a final value of 0, setting any one factor equal to 0 will solve the problem. Depending on the number of factors that exist, you may find one or more solutions.

The solutions will be any values of x that make this true. Understand the meaning of the solutions. The work you have done to this point identifies values of x for which the denominator of the function equals 0. Recognize that a rational function is really a large division problem, with the value of the numerator divided by the value of the denominator.

Because dividing by 0 is undefined, any value for x for which the denominator will equal 0 represents a vertical asymptote for the full function. Part 2. Review the meaning of a graph. A graph of a function is a visual representation of the values of x and y that are solutions to a given equation. The graph may consist of individual points, a straight line, a curved line, or even some closed figures like a circle or an ellipse.

Any point that lies on the line could be a solution to the equation. Written in pairs of x,y , some possible solutions are 1,2 , 2,4 , 3,6 , or any pair of numbers in which the second number is double the first. Plotting these points on the x,y coordinate plane will show a continuous straight line that appears as a diagonal that goes upward from left to right. To see more samples of this type of graph, you may want to review Graph Linear Equations.

Some sample solutions are -1,-2 , 0,-1 , 1,1 , 2,7. If you plot these points, and others, you will find the graph of a parabola, which is a u-shaped curve. To review this type of graph, you can look at Graph a Quadratic Equation. If you need more help reviewing how to graph functions, read Graph a Function or Graph a Rational Function.

Recognize asymptotes. Limits and velocity Two young mathematicians discuss limits and instantaneous velocity. Instantaneous velocity We use limits to compute instantaneous velocity. Definition of the derivative. The definition of the derivative We compute the instantaneous growth rate by computing the limit of average growth rates.

Derivatives as functions. Wait for the right moment Two young mathematicians discuss derivatives as functions. The derivative as a function Here we study the derivative of a function, as a function, in its own right. Differentiability implies continuity We see that if a function is differentiable at a point, then it must be continuous at that point.

Basic rules of differentiation We derive the constant rule, power rule, and sum rule. The derivative of the natural exponential function We derive the derivative of the natural exponential function. The derivative of sine We derive the derivative of sine. Product rule and quotient rule.

Derivatives of products are tricky Two young mathematicians discuss derivatives of products and products of derivatives. The Product rule and quotient rule Here we compute derivatives of products and quotients of functions. Chain rule. An unnoticed composition Two young mathematicians discuss the chain rule. The chain rule Here we compute derivatives of compositions of functions. Derivatives of trigonometric functions We use the chain rule to unleash the derivatives of the trigonometric functions.

Higher order derivatives and graphs. Rates of rates Two young mathematicians look at graph of a function, its first derivative, and its second derivative. Higher order derivatives and graphs Here we make a connection between a graph of a function and its derivative and higher order derivatives. Concavity Here we examine what the second derivative tells us about the geometry of functions. Position, velocity, and acceleration Here we discuss how position, velocity, and acceleration relate to higher derivatives.

Standard form Two young mathematicians discuss the standard form of a line. Implicit differentiation In this section we differentiate equations that contain more than one variable on one side. Derivatives of inverse exponential functions We derive the derivatives of inverse exponential functions using implicit differentiation. Logarithmic differentiation. Multiplication to addition Two young mathematicians think about derivatives and logarithms. Logarithmic differentiation We use logarithms to help us differentiate.

Derivatives of inverse functions. We can figure it out Two young mathematicians discuss the derivative of inverse functions. Derivatives of inverse trigonometric functions We derive the derivatives of inverse trigonometric functions using implicit differentiation. The Inverse Function Theorem We see the theoretical underpinning of finding the derivative of an inverse function at a point.

More than one rate. A changing circle Two young mathematicians discuss a circle that is changing. More than one rate Here we work abstract related rates problems. Applied related rates. Pizza and calculus, so cheesy Two young mathematicians discuss tossing pizza dough. Applied related rates We solve related rates problems in context. Maximums and minimums. More coffee Two young mathematicians witness the perils of drinking too much coffee. Maximums and minimums We use derivatives to help locate extrema.

Concepts of graphing functions. Two young mathematicians discuss how to sketch the graphs of functions. Concepts of graphing functions We use the language of calculus to describe graphs of functions. Computations for graphing functions. Wanted: graphing procedure Two young mathematicians discuss how to sketch the graphs of functions. Computations for graphing functions We will give some general guidelines for sketching the plot of a function. Mean Value Theorem. The Extreme Value Theorem We examine a fact about continuous functions.

The Mean Value Theorem Here we see a key theorem of calculus. Replacing curves with lines Two young mathematicians discuss linear approximation. Explanation of the product and chain rules We give explanation for the product rule and chain rule. A mysterious formula Two young mathematicians discuss optimization from an abstract point of view. Basic optimization Now we put our optimization skills to work. Applied optimization. Volumes of aluminum cans Two young mathematicians discuss optimizing aluminum cans.

Applied optimization Now we put our optimization skills to work. A limitless dialogue Two young mathematicians consider a way to compute limits using derivatives. Basic antiderivatives We introduce antiderivatives. Falling objects We study a special type of differential equation. Approximating the area under a curve. What is area? Two young mathematicians discuss the idea of area.

Introduction to sigma notation We introduce sigma notation. Approximating area with rectangles We introduce the basic idea of using rectangles to approximate the area under a curve. Definite integrals. Computing areas Two young mathematicians discuss cutting up areas. The definite integral Definite integrals compute net area.

Antiderivatives and area. Meaning of multiplication A dialogue where students discuss multiplication. Relating velocity, displacement, antiderivatives and areas We give an alternative interpretation of the definite integral and make a connection between areas and antiderivatives. First Fundamental Theorem of Calculus.

This is a good idea (for someone who knows derivatives), but a potential problem in general is that the derivative may go to ±∞ at points where. firehousehouston.com › Maths › Math Article. The method opted to find the horizontal asymptote changes involves comparing the degrees of the polynomials in the numerator and denominator of the function.