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Published byCynthia Amie Fitzgerald Modified over 4 years ago

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EXAMPLE 1 Graph a rational function (m < n) Graph y =. State the domain and range. 6 x 2 + 1 SOLUTION The degree of the numerator, 0, is less than the degree of the denominator, 2. So, the line y = 0 (the x - axis) is a horizontal asymptote. The numerator has no zeros, so there is no x -intercept.The denominator has no real zeros, so there is no vertical asymptote.

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EXAMPLE 1 Graph a rational function (m < n) The graph passes through the points (–3, 0.6), (–1, 3), (0, 6), (1, 3), and (3, 0.6). The domain is all real numbers, and the range is 0 < y ≤ 6.

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EXAMPLE 2 Graph a rational function (m = n) Graph y =. 2x22x2 x 2 – 9 SOLUTION The zero of the numerator 2x 2 is 0, so 0 is an x - intercept. The zeros of the denominator x 2 –9 are +3, so x = 3 and x = –3 are vertical asymptotes. The numerator and denominator have the same degree, so the horizontal asymptote is y = amam bnbn = 2 1 = 2 Plot points between and beyond the vertical asymptotes.

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EXAMPLE 2 Graph a rational function (m = n)

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EXAMPLE 3 Graph a rational function (m > n) Graph y =. x 2 +3x – 4 x – 2 SOLUTION The numerator factors as (x + 4)(x –1), so the x - intercepts are – 4 and 1. The zero of the denominator x – 2 is 2, so x = 2 is a vertical asymptote. The degree of the numerator, 2, is greater than the degree of the denominator, 1,so the graph has no horizontal asymptote. The graph has the same end behavior as the graph of y = x 2–1 = x. Plot points on each side of the vertical asymptote.

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EXAMPLE 3 Graph a rational function (m > n)

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GUIDED PRACTICE for Example 1, 2 and 3 Graph the function. 1. y = 4 x 2 + 2 3x23x2 x 2 – 1 2. y =

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GUIDED PRACTICE for Example 1, 2 and 3 Graph the function. 3. f (x) = x 2 –5 x 2 +1 4. y = x 2 – 2x – 3 x – 4

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