Precalculus A 2016 Graphs of Rational Functions

Transcription

1 3-7 Precalculus A 2016 Graphs of Rational Functions Determine the equations of the vertical and horizontal asymptotes, if any, of each function. Graph each function with the asymptotes labeled. 1. ƒ(x) 4 2. ƒ(x) 2 x 1 3. g(x) x 3 x 2 1 x 1 (x 1)(x 2) f(x) f(x) f(x) New Vocabulary rational function asymptote vertical asymptote horizontal asymptote slant asymptote O x O x O x Use the parent graph ƒ(x) 1 x to graph each equation. Describe the transformation(s) that have taken place. Identify the new locations of the asymptotes. 4. y 3 x 2 5. y 4 1 x 3 3

2 Determine the slant asymptotes of each equation. 6. y 5x2 10 x 1 7. y x 2 x x 2 x 1 8. Graph the function y x2 x. 6 x 1 9. Physics The illumination I from a light source is given by the formula I k, where k is a constant and d is distance. As the d 2 distance from the light source doubles, how does the illumination change?

3 3.7 Rational Functions and Models This section covers: Rational functions Vertical asymptotes Horizontal asymptotes Slant asymptotes Graphing

4 Rational Functions Rational functions are defined: A function that can be written P(x) Q(x) P x, Q(x) are polynomials where Q(x) 0 and both To determine if a function is a rational function, look at P(x) and Q(x) separately and determine if they are polynomials. If both are polynomials then P(x) is a rational function. Q(x) Domain of a rational function: The domain is all real numbers except where the denominator (Q(x)) equals zero.

5 Examples: Are these functions rational functions? f x = 6 x 2 f x = x2 + 1 x 8 YES NO f x = 4 x + 1 f x = x + 1 x + 1 f x = x3 3x + 1 x 2 5 YES NO YES f x = 5x 3 4x YES

6 Vertical Asymptotes: These occur when the graph is undefined (when Q x = 0). To find, we have to set the denominator = 0 and solve for x. The values you find for x are your vertical asymptotes. The graph will never touch or cross a vertical asymptote! Notice that all vertical asymptotes are written in terms of x.

7 Horizontal Asymptotes: The function is f x = P(x) Q(x) There are three different scenarios for horizontal asymptotes: Degree of P(x) > Q(x), there is no horizontal asymptote Degree of P x < Q x, the horizontal asymptote is y = 0 Degree of P x = Q(x), we take the leading coefficients of both the numerator and the denominator and divide the two. Notice the all horizontal asymptotes are written in terms of y. The graph can cross or intersect the horizontal asymptotes (this is different from vertical asymptotes).

8 Vertical Asymptotes Vertical and Horizontal Asymptotes Vertical and Slant Asymptotes

9 Examples: Find any horizontal or vertical asymptotes: f x = 3x 2 x 2 9 HA: y = 3 VA: x = ±3 f x = f x = x x 3 x x x 2 + 3x 10 HA: y = 0 VA: x = 0, ±1 HA: none VA: x = 5,2 Write a formula for f(x) for a rational function so that its graph has the specified asymptotes: Vertical: x = ±3 Horizontal: y = 5 f x = 5x 2 x 3 (x + 3)

10 Slant Asymptotes A special case of asymptotes that occur when the degree of the numerator (P(x)) is one degree larger than the degree of the denominator (Q(x)). To find these, we divide our rational function using either long division or synthetic division. When dividing, this is the one and only time that you can ignore the remainder.

11 Example: Find the vertical and slant asymptotes: f x = 4x2 2x 1 VA: set the denominator equal to zero and solve: 2x 1 = 0 x = 1 2 Slant A: use long division and the quotient (ignoring the remainder) is the slant asymptote: y = 2x + 1

12 Graphing By Hand To graph by hand it is best to go step by step: 1. Identify any Vertical Asymptotes (draw them in as a dashed line) 2. Identify any Horizontal or Slant Asymptotes (draw them in as a dashed line) 3. Find any y-intercepts 4. Find any x-intercepts 5. Make a t-chart containing values to the left and to the right of your vertical asymptotes 6. Plot the points and graph

13 When we graph, we can also use the transformations rules from previous chapters to visualize rational functions. For instance, just like before, constants that are added or subtracted INSIDE the parenthesis are horizontal shifts (shifts to the left or to the right). Constants that are just hanging out in front or on the end are vertical shifts (shifts up or down). Reflections occur just as before, we reflect the function over the x-axis when we have a negative sign out in front. f x = 1 vertical shift (shift one unit up) reflection (reflect over the x-axis) 2 (x + 3) horizontal shift (shift three units to the left)

14 Special Cases: We can have a hole in the graph. This can occur if and only if P x = 0 and Q x = 0 at exactly the same point. This point is then referred to as a whole, not a vertical asymptote. An example of this is as follows: f x = x+5 x 2 25 We can rewrite this function: f x = (x+5) and then (x+5)(x 5) can cross out the factors that are the same: (x + 5) f x = (x + 5)(x 5) Therefore our function reduced to: f x = 1 (x 5) and we conclude that there is a hole, not a vertical asymptote when x 5 = 0 which occurs at x = 5.

15 Another special case occurs in terms of horizontal asymptotes. There are some instances where the graph might intersect or cross over their horizontal asymptote and then re-approach it from below. An example of this is as follows: f x = 3x2 3x 6 x 2 +8x+16 We still graph this by the same process as before.