GRAPH OF A RATIONAL FUNCTION

Transcription

1 GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find the horizontal asmptote, if an, and draw it. A horizontal asmptote ma be found using the exponents and coefficients of the lead terms in the numerator and denominator. Please see Horizontal Asmptotes and Lead Coefficients below. Pick two or three x-values to the left and right of each vertical asmptote. If there are no vertical asmptotes, then just pick positive, negative, and zero. Put these values into the function f(x) and plot the points. This will give ou an idea of the shape of the curve. You will quickl see whether the graph goes above or below the horizontal asmptote. Find the -intercept b letting x 0. Put 0 into the function, do the arithmetic and get the -value. Plot the point ( 0, ). This will show ou where the curve crosses the -axis. Find an x-intercepts b letting 0. Replace f(x) b 0, then solve for x. Plot these points as (x, 0 ). This will show ou where the graph crosses the x-axis. B the wa, when ou set 0 it is possible that ou will get imaginar answers for x. Since we can not graph a + bi on the real plane, this means there are no x-intercepts. Finall, connect all of the points (x, ) with a smooth curve rather than chopp, straight line segments.

2 Horizontal Asmptotes and Lead Coefficients RATIONAL FUNCTION: cx + c x + c x cn dx + d x + d x d a a 1 a 3 b b 1 b 3 n Focus on the lead (first) term in the numerator and the lead term in the denominator. cx dx a b Look at the exponent in the top (this is a ) and the exponent in the bottom (this is b ). c 1, The equation of the HORIZONTAL ASYMPTOTE will be d if the top exponent is the same as the bottom exponent, i.e. when a b use the lead coefficients.. The equation of the HORIZONTAL ASYMPTOTE will be 0 if the top exponent is smaller than the bottom exponent, i.e. when a< b the degree of the denominator will be larger than the numerator, so the bottom becomes much larger than the top as x, 3, There will be NO HORIZONTAL ASYMPTOTE when the top exponent is larger than the bottom one. In other words, the degree of the numerator will be larger than the denominator, so there will alwas be some degree of x in the top. Thus when x gets ver large, will also get ver large. In a sense, as positive x gets ver large, the curve approaches + or depending on the signs of the lead coefficients. If the signs are the same, the curve goes to + when x also goes to +. Remember negative divided b negative is positive, and positive divided b positive is also positive. If the signs of the lead coefficients are opposite and positive x gets ver large, the curve goes to because negative divided b positive is negative and vice-versa.

3 Example 1: Graph b hand: 3x ( x Find vertical asmptotes and draw them. Vertical asmptotes occur where the denominator becomes zero. There is a vertical asmptote at x 6. Find the horizontal asmptote, if an, and draw it. A horizontal asmptote ma be found using the exponents and coefficients of the lead terms in the numerator and denominator. 3x In the exponents in the top and bottom are both 1. The coefficient of the ( x top is 3 and the coefficient of the bottom is 1, so there is a horizontal asmptote at 3 3 1

4 Pick two or three x-values to the left and right of each vertical asmptote. You will quickl see whether the graph goes above or below the horizontal asmptote. There is a vertical asmptote at x 6, so to the left pick x 4 and x 5 because the are close to the asmptote. When x 4, 3(4) f ( x ) (4 will be 6. When x 5, 3(5) f ( x ) will be -15. (5 Tr x 7 and x 8 to the right. The points produced will be (7,1) and (8,1) Find the -intercept b letting x 0. When x 0, 3(0) f ( x ) 0. (0 So the curves passes through the origin (0,0). Find an x-intercepts b letting 0 3x When 0 we also get x 0. ( x So the x-intercept and -intercept are the same. Finall, connect all of the points (x, ) with a smooth curve rather than chopp, straight line segments

5 Example : Graph b hand: x x ( 5) Find vertical asmptotes and draw them. Vertical asmptotes occur where the denominator becomes zero. There are vertical asmptote at x 5 and 5. Find the horizontal asmptote, if an, and draw it. A horizontal asmptote ma be found using the exponents and coefficients of the lead terms in the numerator and denominator. x In the exponents in the top and bottom are both. The coefficient of the ( x 5) top is 1 and the coefficient of the bottom is 1, so there is a horizontal asmptote at 1 1 1

6 Pick two or three x-values to the left and right of each vertical asmptote. You will quickl see whether the graph goes above or below the horizontal asmptote. There is a vertical asmptote at x 5. To the left of it pick x 7 and x 6 because the are close to the asmptote. To the right, tr x 4 and x 3. To the left of the other asmptote, x 5, pick x 3 and x 4. To the right of x 5 pick x 6 and x 7. This will produce the points: ( 7,.04), ( 6,3.7),( 4, 1.78),( 3,.5,(3,.5,(4, 1.78),(6,3.7),(7,.04) Find the -intercept b letting x 0. When x 0, f ( x ) (0) (0 5) 0. So the curves passes through the origin (0,0). Find an x-intercepts b letting 0 x When 0 we also get x 0. ( x 5) So the x-intercept and -intercept are the same. Finall, connect all of the points (x, ) with a smooth curve rather than chopp, straight line segments