2-3 Graphing Rational Functions

Transcription

1 2-3 Graphing Rational Functions

2 Factor What are the end behaviors of the Graph? Sketch a graph

3 How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph of a rational function. Interpret rational function graphs from real world situations.

4 Domain: Range: Intercepts: Decreasing: Positive: Negative: Maximums/Minimums Symmetries: End Behavior:

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6 The x-intercepts are the real zeros of the numerator. The y-intercept is where x = 0. The graph has a vertical asymptote at each real zero of the denominator unless the numerator and the denominator share a factor in which it would have a hole. The graph has at most one horizontal asymptote: If the degree of p(x) (numerator) is less than the degree of q(x) (denominator), then the line y=0 is the horizontal asymptote. If the degree of p(x) (numerator) is equal to the degree of q(x) (denominator), then the horizontal asymptote is the ratio of the leading coefficients. If the degree of p(x) (numerator) is greater than the degree of q(x) (denominator) then there may be an oblique asymptote which is found. using long division. It will be in the form y = mx+b

7 Factor the numerator and the denominator Find the x-intercept and y-intercepts Find the horizontal or oblique and vertical asymptotes and any holes Create a sign array to determine where the function is positive or negative or make a table. Sketch the graph Identify the domain and range

8 A Vertical asymptote is a line which corresponds to the zeroes of the denominator. The domain is the set of all x values EXCEPT those values that make the denominator zero. To find the vertical asymptote, you set the denominator equal to zero. What happens when a fraction has a zero in the denominator?

9 Because 0 would make the denominator zero, we draw a vertical asymptote at 0.

10 We are ready to find the x intercepts. This happens when we find the value of x that makes the numerator zero. x= -3. The zero is located at x=-3, so we place a point at (-3,0). To find the y intercept you plug 0 in for x. What happens in this example?

11 Horizontal asymptotes are a bit more tricky because not all rational functions have horizontal asymptotes. Vertical asymptotes may not be touched or crossed. Horizontal asymptotes may be crossed and are used to guide the end behavior of a graph. Horizontal asymptotes show us where the graph will be with very large and very small values.

12 Since the degree of the numerator is the same as the degree of the denominator you look at the leading coefficients. You would get

13 We discovered that the vertical asymptotes was at x=0, the zero was at x=-3, and the horizontal asymptote was at y= Now to sketch in the graph we draw a sign array. Include vertical asymptotes and x intercepts.

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15 Vertical asymptote Can NEVER be crossed. It is found by setting the denominator equal to zero. Horizontal asymptote shows end behaviors. (N)umerator = (D)enominator then horizontal at the IF D > N, then the Horizontal asymptote is at the x-axis. If N>D, then we have a slant line found by long division. The domain is all the x values except where there is a vertical asymptote.

16 First, see if we can simplify the problem by factoring. We find the vertical asymptotes by setting (x+1) =0 and (x-2) =0 so x = -1, 2 After factoring the rational equation, we find that the numerator and denominator have (x+1) in common, so we can factor it out. Remember that we cannot have a zero in the denominator, so (x+1)/(x+1) will leave a hole in our graph. That is, x= -1

17 The denominator is greater than the numerator, so as the numbers get very large, our graph will approach the x-axis. Notice the hole at x=-1

18 1. Factor the numerator and denominator Find the x and y intercepts (1, 0), (0, )

19 3. Find the vertical asymptotes and holes x = 3, x = Find the horizontal asymptote Deg of numerator is less than deg of denominator. y = 0

20 5. Make a sign array to see where positive or negative. Use the intervals of the intercepts and the vertical asymptotes.

21 This is an example where the graph crosses the horizontal asymptote. 6. Find the domain and range

22 1. Factor the numerator and denominator Find the x and y intercepts (1, 0), (-1, 0), (0, )

23 3. Find the vertical asymptotes and holes x = 2, x = Find the horizontal asymptote Deg of numerator is the same as the deg of denominator. y = 2

24 5. Make a sign array to see where positive or negative. Use the intervals of the intercepts and the vertical asymptotes. (Or make a table) 6. Find the domain and range

25 1. Factor the numerator and denominator Numerator does not factor 2. Find the x and y intercepts Using the quadratic formula the solutions are not real for the x intercepts. (0, )

26 3. Find the vertical asymptotes and holes x = Find the oblique asymptote Deg of numerator is greater than the deg of denominator. Use long division. y = x - 4

27 5. Make a sign array to see where positive or negative. Use the intervals of the intercepts and the vertical asymptotes. (Or make a table) 6. Find the domain

28 To check your graphs