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ASYMPTOTES TUTORIAL

ASYMPTOTES TUTORIAL. Horizontal Vertical Slant and Holes. Do you remember fractions? . What does 1/100 mean? Is 1/100 or 1/1000 bigger? What is 1/(1/10)? What is 1 divided by a small positive number?.

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ASYMPTOTES TUTORIAL

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  1. ASYMPTOTES TUTORIAL Horizontal Vertical Slant and Holes

  2. Do you remember fractions? • What does 1/100 mean? • Is 1/100 or 1/1000 bigger? • What is 1/(1/10)? What is 1 divided by a small positive number?

  3. Suppose we have the expression 1/x-3. What happens as the values we plug in for x approach 3? • Consider the function: • What happens as x gets very large? How could we test this out?

  4. Definition of an asymptote • An asymptote is a straight line which acts as a boundary for the graph of a function. • When a function has an asymptote (and not all functions have them) the function gets closer and closer to the asymptote as the input value to the function approaches either a specific value a or positive or negative infinity. • The functions that we will study first that have asymptotes are rational functions. • Do you think it’s possible for a function to cross an asymptote?

  5. Vertical Asymptotes Vertical asymptotes occur when the following condition is met: The denominator of the simplified rational function is equal to 0. Why? What is anything divided by a small number? What does that look like on a graph? Why would we simplify? How is that related to fractions?

  6. Finding Vertical AsymptotesExample 1 Given the function The first step is to cancel any factors common to both numerator and denominator. Why would we do this? The second step is to see where the denominator of the simplified function equals 0. What happens when the denominator is close to zero? Why?

  7. Finding Vertical Asymptotes Example 1 Con’t. The vertical line x = -1 is the only vertical asymptote for the function. As the input value x to this function gets closer and closer to -1 the function itself looks and acts more and more like the vertical line x = -1.

  8. Graph of Example 1 The vertical dotted line at x = –1 is the vertical asymptote.

  9. Finding Vertical AsymptotesExample 2 If First simplify the function. Factor both numerator and denominator and cancel any common factors.

  10. Finding Vertical Asymptotes Example 2 Con’t. The asymptote(s) occur where the simplified denominator equals 0. The vertical line x=3 is the only vertical asymptote for this function. As the input value x to this function gets closer and closer to 3 the function itself looks more and more like the vertical line x=3.

  11. Graph of Example 2 The vertical dotted line at x = 3 is the vertical asymptote

  12. Finding Vertical AsymptotesExample 3 If Factor both the numerator and denominator and cancel any common factors. In this case there are no common factors to cancel.

  13. Finding Vertical AsymptotesExample 3 Con’t. The denominator equals zero whenever either or This function has two vertical asymptotes, one at x = -2 and the other at x = 3 How could we test this out? What happens at 2.9? 2.99? Why?

  14. Graph of Example 3 The two vertical dotted lines at x = -2 and x = 3 are the vertical asymptotes

  15. Horizontal Asymptotes? What happens to this function as x gets really large? Can you figure out anything using your brain and logic instead of a Calculator? Feel free to test with a calculator but imagine x getting bigger and bigger first. How about here? What happens as x gets large? Why? How could you reason through this without your calculator? What would happen if we changed the degree of the numerator to a bigger value?

  16. Horizontal Asymptotes When we ask the question “what happens as our inputs x get really large?” we are asking about horizontal asymptotes. There are three main cases: 1. Suppose the degree of the numerator is less than the degree of the denominator. What happens as input of x get really large? How about when the approach negative infinity? 2. The degree of the numerator is equal tothe degree of the denominator. In this case the asymptote is the horizontal line y = a/bwhere a is the leading coefficient in the numerator and b is the leading coefficient in the denominator. Why? How could we test this out? 3. When the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote. Why not? What happens as x gets large?

  17. Finding Horizontal AsymptotesExample 4 If then there is a horizontal asymptote at the line y=0 because the degree of the numerator (2) is lessthan the degree of the denominator (3). This means that as x gets larger and larger in both the positive and negative directions (x→ ∞ and x → -∞) the function itself looks more and more like the horizontal line y = 0

  18. Graph of Example 4 The horizontal line y = 0 is the horizontal asymptote.

  19. Finding Horizontal Asymptotes Example 5 If then because the degree of the numerator (2) is equal to the degree of the denominator (2) there is a horizontal asymptote at the line y=6/5.Note, 6 is the leading coefficient of the numerator and 5 is the leading coefficient of the denominator. As x→∞ and as x→-∞ g(x) looks more and more like the line y=6/5

  20. Graph of Example 5 The horizontal dotted line at y = 6/5is the horizontal asymptote.

  21. Finding Horizontal Asymptotes Example 6 If There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator. What happens as x keeps getting bigger?

  22. Graph of Example 6

  23. Slant Asymptotes • Slant asymptotes occur when the degree of the numerator is exactly one bigger than the degree of the denominator. In this case a slanted line (not horizontal and not vertical) is the function’s asymptote. • To find the equation of the asymptote we need to use long division – dividing the numerator by the denominator.

  24. Finding a Slant Asymptote Example 7 • If • There will be a slant asymptote because the degree of the numerator (3) is one bigger than the degree of the denominator (2). • Using long division, divide the numerator by the denominator.

  25. Finding a Slant AsymptoteExample 7 Con’t.

  26. Finding a Slant AsymptoteExample 7 Con’t. We can ignore the remainder The answer we are looking for is the quotient and the equation of the slant asymptote is

  27. Graph of Example 7 The slanted line y = x + 3 is the slant asymptote

  28. Holes • Holes occur in the graph of a rational function whenever the numerator and denominator have common factors. The holes occur at the x value(s) that make the common factors equal to 0. Why? What’s up with that? • The hole is known as a removable discontinuity. Where did we get the term removable from? • When you graph the function on your calculator you won’t be able to see the hole but the function is still discontinuous (has a break or jump). How do the pixels on the Calculator work?

  29. Finding a HoleExample 8 Remember the function We were able to cancel the (x + 3) in the numerator and denominator before finding the vertical asymptote. Why don’t common factors affect the shape of the graph? What happens when we choose numbers close to 3? Because (x + 3) is a common factor there will be a hole at the point where

  30. Graph of Example 8 Notice there is a hole in the graph at the point where x = -3. You would not be able to see this hole if you graphed the curve on your calculator (but it’s there just the same.)

  31. Finding a HoleExample 9 • If • Factor both numerator and denominator to see if there are any common factors. • Because there is a common factor of x - 2 there will be a hole at x = 2. This means the function is undefined at x = 2. For every other x value the function looks like

  32. Graph of Example 9 There is a hole in the curve at the point where x = 2. This curve also has a vertical asymptote at x = -2 and a slant asymptote y = x.

  33. Problems Find the vertical asymptotes, horizontal asymptotes, slant asymptotes and holes for each of the following functions. (Click mouse to see answers.) Vertical: x = -2 Horizontal : y = 1 Slant: none Hole: at x = - 5 Vertical: x = 3 Horizontal : none Slant: y = 2x +11 Hole: none

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