8 - 3 Graphing Rational Functions
- 1. UNIT 4: rational functions 8-3 Graphing Rational functions
- 2. Rational Functions • Define – Rational Function: is a function with two polynomials (one in the numerator and one in the denominator) • Define- point of discontinuity: Value that makes the denominator zero. (holes / asymptotes) • Hole: point of discontinuity that can be removed (cancelled out with the numerator) • Vertical asymptote: point of discontinuity that can not be removed (doesn’t cancel with numerator) • Horizontal asymptote: determine by the degree of numerator and denominator. (more on that later)
- 3. Graphing rational functions • When graphing rational functions you must find points of discontinuity (holes / asymptotes) 1st -Factor numerator and denominator 2nd – determine points of discontinuity 3rd – graph by making a table ( x − 4)( x + 1) x+3 x+3 x 2 − 3x − 4 f ( x) = y= 2 g ( x) = x −5 x − 4x + 3 x−4 ( x − 3)( x − 1) V.A. when: x=5 V.A. when x=1 & x=3 Hole when x=4 x y x y x y -3 0 -3 0 0 1 1 -1 0 1 2 3 7 5 2 -5 5 6 9 3 5 1 6 7
- 4. Practice x+6 x 2 − 3x + 2 x−3 y= f ( x) = g ( x) = x+4 x−2 3 x 2 − 11x + 6 V.A. x=-4 Hole: x=2 V.A. x=2/3 Hole: x=3 x −5 x+2 2x y= f ( x) = 2 g ( x) = x x − x − 12 3x − 1 V.A. x=0 V.A. x=-3 V.A. x=1/3 V.A. x=4
- 5. Horizontal asymptotes • To find horizontal asymptotes compare the degree of the numerator “M” to the degree of the denominator “N” • If M < N, then y=0 is horizontal asymptote • If M > N, then No horizontal asymptote • If M=N, then divide leading coefficients
- 6. Horizontal asymptotes • Determine the horizontal asymptotes 2x x+3 x 2 − 3x − 4 f ( x) = y= g ( x) = x −5 ( x − 3)( x − 1) x−4 M=1 M=1 M=2 N=1 N=2 N=1 H.A. y=2 H.A. y=0 NO H.A.
- 7. Practice V.A. Holes H.A • Pg 521 # 17-22 23-28
- 8. Word problem • You earn a 75% on the first test of the quarter how many consecutive 100% test scores do you need to bring your test average up to a 95%? 75 + 100 x Write a rational function. Answer: f ( x) = x +1 Find when the rational function will be 95% You will need to make 100% on the next 4 test to bring your test average up to a 95%
- 9. Word problem • A Basketball player have made 5 out of the last 7 free throws. How many more consecutive free throws do they need to make to have an average of 80%? 5+ x Answer: f ( x) = Write a rational function. 7+ x Find when the rational You will need to make the function will be 80% next 3 free throws for average to be an 80%
- 10. Practice word problems • Pg 521 #39,40
- 11. Word problem • The function below gives the concentration of the saline solution after adding x milliliters of 0.5% solution to 100 milliliters of 2% solution. 100(0.02) + x(0.005) y= 100 + x • How many ML of the 0.5% solution must be added to have a combined concentration of 0.9%? Answer: (search table) X=275