Finding Horizontal Asymptote We know that if f(x) is a function and if the line ax + by + c = 0 meets the curve only at infinity, then the line ax + by + c = 0 is called an asymptote. Horizontal asymptote: It is an asymptote which is horizontal. The asymptote is thus parallel to x axis, so is of the form y = a, for all real values of a. Thus if f(x) = y is a function, then y =a, becomes undefined for the function, then the line y = a, is called the asymptote for the function y = f(x). To find out horizontal asymptote, we can visualise the curve and get when y=a, for any a, so that the curve becomes undefined. A simple example is the equation y= . This we can write as x = . The curve becomes undefined when y=0 and therefore the line y = 0 or x axis is a horizontal asymptote for the curve x= . The methods of finding out horizontal asymptotes for a curve, if it is a rational function: Step 1: Find out the degree of numerator and denominator. Know More About Unit Conversion Worksheet
Step 2: There are three possibilities. i. degree of numerator > degree of denominator. Then we can leave this as this type of curve has no horizontal asymptotes. Suppose numerator is more than the denominator then it may turn out to be a linear or quadratic of some function of degree n-m the difference. ii. degree of numerator =degree of denominator. Then find the fraction made by the leading coefficients and equate to y. This is the horizontal asymptote. Eg: If numerator has 3x続 and denominator has 2x as leading coefficient then y = is horizontal asymptote. iii.degree of numerator < degree of denominator: then the horizontal asymptote is x axis, or y=0. Eg: numerator has 4x4 and denominator as x then the asymptote is y=0. So let us assume if the degree of numerator as n and degree of denominator as m. Learn More Completing the Square Worksheet
Properties of Similar Triangles Similar Triangles are two triangles that are having congruent corresponding angles and the ratios of the corresponding sides are in proportion. This proportion is also called as similarity ratio. The similar triangles are also called as equiangular triangle. This is because in equilateral triangles, both the triangles have equal angles. The similar triangles have common shape but different sizes. In the above figure, the two triangles are of same shape but different sizes. And so, the above two triangles are called as similar triangles. Properties of Similar Triangles Given below are some of the properties of similar triangles: Corresponding Angles of both the triangles are equal. <P=<A, <Q=<B, <R=<C
The corresponding sides of the triangles are having same ratio. AB/PQ= BC/QR=AC/PR = X, where X is called as the similarity ratio Rules of Similar Triangles There are some rules to test the similarity of triangles. They are as follows: Angle Angle Angle (AAA) Side Side Side (SSS) Side Angle Side (SAS) AAA Similarity: When three angles of the triangles are equal, we can say that the two triangles are similar triangles. That is, the corresponding angles are having equal measurement. SSS Similarity: When three corresponding sides of the triangles are equal, we can say that the triangles are similar triangles. SAS Similarity: When two sides in one triangle are in the same proportion to the corresponding sides of the other and the included angles are equal, we can say that both are similar triangle. Read More About Congruent Triangles Worksheet
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