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2.6 Rational Functions & Their Graphs

2.6 Rational Functions & Their Graphs. Objectives Find domain of rational functions. Use arrow notation. Identify vertical asymptotes. Identify horizontal asymptotes. Use transformations to graph rational functions. Graph rational functions. Identify slant (oblique) asymptotes.

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2.6 Rational Functions & Their Graphs

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  1. 2.6 Rational Functions & Their Graphs • Objectives • Find domain of rational functions. • Use arrow notation. • Identify vertical asymptotes. • Identify horizontal asymptotes. • Use transformations to graph rational functions. • Graph rational functions. • Identify slant (oblique) asymptotes. • Solve applied problems with rational functions.

  2. Vertical asymptotes • Look for domain restrictions. If there are values of x which result in a zero denominator, these values would create EITHER a hole in the graph or a vertical asymptote. Which? If the factor that creates a zero denominator cancels with a factor in the numerator, there is a hole. If you cannot cancel the factor from the denominator, a vertical asymptote exists. • If you evaluate f(x) at values that get very, very close to the x-value that creates a zero denominator, you notice f(x) gets very, very, very large! (approaching pos. or neg. infinity as you get closer and closer to x)

  3. Example • f(x) is undefined at x = 2 • As • Therefore, a vertical asymptote exists at x=2. The graph extends down as you approach 2 from the left, and it extends up as you approach 2 from the right.

  4. What is the end behavior of this rational function? • If you are interested in the end behavior, you are concerned with very, very large values of x. • As x gets very, very large, the highest degree term becomes the only term of interest. (The other terms become negligible in comparison.) • SO, only examine the ratio of the highest degree term in the numerator over the highest degree term of the denominator (ignore all others!) • As x gets large, becomes • THEREFORE, a horizontal asymptote exists, y=3

  5. What if end behavior follows a line that is NOT horizontal? • Using only highest-degree terms, we are left with • This indicates we don’t have a horizontal asymptote. Rather, the function follows a slanted line with a slope = 4. (becomes y=4x as we head towards infinity!) • The exact equation for the oblique asymptote may be found by long division! • NOTE: f(x) also has a vertical asymptote at x=1.

  6. Graph of this rational function

  7. What is the equation of the oblique asymptote? • y = 4x – 3 • y = 2x – 5/2 • y = 2x – ½ • y = 4x + 1

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