Analyzing Graphs of Rational Functions

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Presentation transcript:

Analyzing Graphs of Rational Functions Sec. 2.7b

Directly in with some Practice Problems Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the given rational function. Numerator is zero when x = 1  x-intercept is 1 f (0) = 1/6  y-intercept is 1/6 Denominator factors as (x – 3)(x + 2)  V.A.: x = 3, x = –2 Degree of Numerator < Degree of Denominator  H.A.: y = 0 Graph:

Directly in with some Practice Problems Find the intercepts, asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the given rational function. Domain: H.A.: Range: V.A.: Continuity: Continuous on D Inc/Dec: Dec: Symmetry: None End Behavior: Boundedness: Unbounded Local Extrema: None

More Practice Problems Find the intercepts and analyze and draw the graph of the given rational function. Numerator factors as 2(x – 1)(x + 1)  x-intercepts: –1, 1 f (0) = 1/2  y-intercept: 1/2 Denominator factors as (x – 2)(x + 2)  V.A.: x = –2, x = 2 Degree of Numerator = Degree of Denominator  H.A.: y = 2 Graph:

More Practice Problems Find the intercepts and analyze and draw the graph of the given rational function. Domain: Range: H.A.: Continuity: Continuous on D V.A.: Inc/Dec: Inc: Dec: End Behavior: Symmetry: y-axis (even) Boundedness: Unbounded Local Extrema: Local Max of 1/2 at x = 0

More Practice Problems Find the end behavior asymptote of the given rational function. and graph it together with f in two windows: (a) one showing the details around the V.A. of f, (b) one showing a graph of f that resembles its end behavior asymptote. Denominator is zero at x = 1  V.A.: x = 1 Rewrite f using polynomial division: End Behavior Asymptote:

More Practice Problems Find the end behavior asymptote of the given rational function. and graph it together with f in two windows: (a) one showing the details around the V.A. of f, (b) one showing a graph of f that resembles its end behavior asymptote. First, graph these two functions in [–4.7, 4.7] by [–8, 8] Now, change your viewing window to [–40, 40] by [–500, 500]

More Practice Problems Find the intercepts and analyze and draw the graph of the given rational function. Use a calculator to find the x-intercept = – 0.260 f (0) = y-intercept = –1 V.A.: x = 1 H.A.: none E.B.A.: Graph:

More Practice Problems Find the intercepts and analyze and draw the graph of the given rational function. Domain: H.A.: None Range: V.A.: Continuity: Continuous on D E.B.A.: Inc/Dec: Dec: Inc: End Behavior: Symmetry: None Boundedness: Unbounded Local Extrema: Local Min of 3 at x = 2