Goal: Graph rational expressions by hand and identify all important features

Transcription

1 Goal: Graph rational expressions by hand and identify all important features Why are we doing this? Rational expressions can be used to model many things in our physical world. Understanding the features of the graphs help us to see things in the physical world differently and can help us make predictions about the future. Warm up Find the domain of f(x) algebraically and graphically. 6 f(x)= x 2 2x 24 note 2 a it a w th r o 1

2 Graphing Rational Expressions by hand Graph with no calculator. Vertical Asymptotes the line at x = a is a vertical asymptote of the graph of f if f (x) or f (x) - as x a from the right and from the left Horizontal asymptotes OR oblique asymptotes affects the far ends of the graph. It is what happens to the graph as x approaches positive infinity or as x approaches negative infinity. Sometimes you can graph a rational expression by thinking how the parent function has moved. parent function has shifted left 2 the vertical asymptotote will be at x = 2 the horizontal asymptote stays at y = 0 there will be a y intercept. To find it set x = 0 2

3 Sometimes you can graph a rational expression by thinking how the parent function has moved. parent function has shifted up 3 the vertical asymptotote will be at x = 0 the horizontal asymptote will be at y = 3 there will be a x intercept. To find it set y = 0 Sometimes you can graph a rational expression by thinking how the parent function has moved. parent function has shifted up 3 and left 2 the vertical asymptote will be at x = 2 the horizontal asymptote will be at y = 3 there will be a x intercept. To find it set y = 0 there will be a y intercept. To find it set x = 0 3

4 All other rational functions will need more advanced strategies for graphing by hand. We need to find: Domain you already know set the denominator = 0 and remove those values. Holes I will teach you this next week Vertical asymptotes what ever made the denominator zero is a vertical asymptote (or hole) Horizontal asymptote Is decided by comparing the degree on top of the fraction to the degree on the bottom.. Range all real numbers excluding the horizontal asym. and holes X intercepts find by letting y = 0 and solve. Y intercepts find by letting x = 0 and solve. Steps to graphing rational functions: 1. Factor the numerator and denominator. 2. State the domain. Remember the denominator = 0 3. Holes: Look for a common factor in the numerator and denominator. a. If there is a common factor then we will have a hole in the graph Find it's location by setting the common factor equal to zero, solve, plug that 'x' number into the reduced fraction to find the function value. Make a note of the location of the hole in (, ) form. b. If there is not a common factor, then we will not have a hole in the graph. 4. Vertical asymptotes: set the denominator of the reduced fraction = 0 and solve. 5. Horizontal (or Slant) asymptote: compare the degree of the numerator to the degree of the denominator. a) bottom heavy Just like the graph of b) balanced c) top heavy horizontal asymptote at y = 0 horizontal asymptote at y = 3/4 use the leading coef. from the top and bottom 6. 'x' intercepts: let y = 0 (means switch f(x) to 0). Start solving the equation by cross multiplying. When you cross multiply it will then be 0 = numerator. Finish solving to find the x intercepts. 7. 'y' intercepts: let x = 0 (Put zero everywhere there is an x and solve the equation. 8. Put all the information collected so far onto a graph. No horizontal asymptote. IF THE TOP IS ONE DEGREE HIGHER THAN THE BOTTOM, THEN USE LONG DIVISION TO FIND THE EQUATION OF THE SLANT ASYMPTOTE. 9. Plug in 1 5 more points to decide what is happening between and beyond the vertical asymptotes. 4

5 Horizontal Asymptotes 3 Scenarios Find if possible: domain, vertical asymptotes, horizontal asymptotes, holes, x intercepts, y intercepts and graph. f(x)= x 4 x 2 4x 12 domain: holes: vertical asym: horiz. OR slant asym: x int: y int: extra points needed: 5

6 Find if possible: domain, vertical asymptotes, horizontal asymptotes, holes, x intercepts, y intercepts and graph. domain: holes: vertical asym: horiz. OR slant asym: x int: y int: extra points needed: 6