Find the solution(s) to each equation. 1)(x-3)(x+3) = 0 2) (x-4)(x+1) = 0 3) x(x 2 – 1) = 04) x 2 – 4x – 5 = 0 x – 3 = 0 and x +3 = 0 x = 3 and x = -3 x(x – 1)(x + 1) = 0 x – 4 = 0 and x +1 = 0 x = 4 and x = -1 (x – 5)(x + 1) = 0 x – 5 = 0 and x +1 = 0x = 0, x – 1 = 0, and x +1 = 0 x = 0, x = 1, and x = -1 x = 5 and x = -1

f(x) = = Has the following characteristics: vertical asymptote is at each real zero of q(x). 9.3a Rational Functions and Their Graphs GRAPHS OF RATIONAL FUNCTIONS Objective – To be able to graph general rational functions. p(x) a x m + …… q(x) b x n + ……

Example 1 Find any points of discontinuity (vertical asymptotes) y = 3 x 2 – x – 12 (x – 4)(x + 3) x – 4 = 0 and x + 3 = 0 +4 x = 4 and x = -3 -3

On White Board Find any points of discontinuity (vertical asymptotes) y = 1 x 2 – 16 (x – 4)(x + 4) x – 4 = 0 and x + 4 = 0 +4 x = 4 and x = -4 -4

On White Board Find any points of discontinuity (vertical asymptotes) y = x + 1 x 2 + 2x – 8 (x + 4)(x – 2) x + 4 = 0 and x – 2 = 0 -4 x = -4 and x = 2 +2

On White Board Find any points of discontinuity (vertical asymptotes) y = x 2 – 1 x x = 0 x 2 = None

f(x) = = Has the following characteristics: GRAPHS OF RATIONAL FUNCTIONS p(x) a x m + …… q(x) b x n + …… horizontal asymptote: - If m < n, the line y = 0 is a horizontal asymptote. - If m = n, the line y = is a horizontal asymptote. - If m > n, the graph has no horizontal asymptote. abab

Example 2 Find the horizontal asymptote of: y = – 4x + 3 2x + 1 **Remember that if the exponent is the same you use the coefficients. y = -4 / 2 y = – 2

On White Board Find the horizontal asymptote of: y = – 2x + 6 x – 1 **Remember that if the exponent is the same you use the coefficients. y = -2 / 1 y = – 2

On White Board Find the horizontal asymptote of: y = 2x + 5 x y = 0

On White Board Find the horizontal asymptote of: y = 4x x None

f(x) = = Has the following characteristics: 1.x-intercepts are the real zeros of p(x). 2.vertical asymptote is at each real zero of q(x). 3.horizontal asymptote: - If m < n, the line y = 0 is a horizontal asymptote. - If m = n, the line y = is a horizontal asymptote. - If m > n, the graph has no horizontal asymptote. ambnambn 9.3b Rational Functions and Their Graphs GRAPHS OF RATIONAL FUNCTIONS Objective – To be able to graph general rational functions. p(x) a m x m + …… q(x) b n x n + ……

Example 1 Graph y = x y /461/4 4x 2 x 2 – 9 -1 / 2 0 Where are the asymptotes? VA: x = 3, x = -3 HA: y = 4/1 HA: y = 4 –5–4–3–2– –5 –4 –3 –2 – /461/4

Example 2 Graph the function y = x y / 6 x + 1 (x – 3)(x + 2) -1 / 3 0 –5–4–3–2– –5 –4 –3 –2 – VA:x = 3 and x = / 14 HA:y = 0 5/65/6 6 / 14

Example 3 Graph y = x y / 2 x 2 – 2x +1 x – 2 -4 / 3 4 9/29/2 Where are the asymptotes? x = 2 –5–4–3–2– –5 –4 –3 –2 – / 3

Sec. 9.4 Multiplying and Dividing Rational Expressions Objective: Multiply/Divide and Simplify Rational Expressions

Example 1 Simplify x 2 – 7x – 18 x 2 – 8x -9

First Factor top and bottom (x – 9)(x + 2) (x – 9)(x + 1) Then cancel like terms and get (x + 2) (x + 1) Numerator Bottom

Example 2 Simplify

See if you can cross cancel anything =

Example 3 Simplify

First do like example 1 and factor everything (x + 6)(x – 2) (x – 7)(x + 5) (x + 6)(x + 5) (x + 4) Cancel all appropriate parts We Get (x – 2)(x – 7) or x 2 – 9x + 14 x + 4x st Numerator 1 st Denominator nd Numerator

Example 4 Simplify

What do we ever do when we divide by a fraction. Yes we multiply by the reciprocal

Do the same steps from before and we get (x + 5)(x – 5) (x - 6)(x + 3) (x + 3)(x – 1)x + 5 After Cross Cancelling we end up with (x – 5)(x – 6)orx 2 – 11x + 30 x – 1x - 1

Last Example:Complex Fraction Simplify

Just like the previous problem we write the bottom fraction by the reciprocal and mult. When you factor you get.

An object is 24 cm from a camera lens. The object is in focus on the film when the lens is 12 cm from the film. Find the focal length of the lens. =+ Use the lens equation. 1f1f 1di1di 1do1do =+ Substitute. 1f1f =+Write equivalent fractions with the LCD == Add and simplify Since =, the focal length of the lens is 8 cm. 1f1f 1818 Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

Find the least common multiple of 2x 2 – 8x + 8 and 15x 2 – 60. Step 1: Find the prime factors of each expression. 2x 2 – 8x + 8 = (2)(x 2 – 4x + 4) = (2)(x – 2)(x – 2) 15x 2 – 60 = (15)(x 2 – 4) = (3)(5)(x – 2)(x + 2) Step 2:Write each prime factor the greatest number of times it appears in either expression. Simplify where possible. (2)(3)(5)(x – 2)(x – 2)(x + 2) = 30(x – 2) 2 (x + 2) The least common multiple is 30(x + 2)(x – 2) 2. Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

Simplify x x +30 4x 3x + 15 = +Multiply. 1 3(x + 2)(x + 5) 4x(x + 2) 3(x + 2)(x + 5) = Add x(x + 2) 3(x + 2)(x + 5) = Simplify the numerator. 4x 2 + 8x +1 3(x + 2)(x + 5) = Simplify the denominator. 4x 2 + 8x +1 3x x +30 =+Identity for Multiplication. 1 3(x + 2)(x + 5) 4x 3(x + 5) x + 2 +=+Factor the denominators. 1 3x x +30 4x 3x (x + 2)(x + 5) 4x 3(x + 5) Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

Simplify – 2x x 2 – 2x – 3 = Simplify. 4(2x) – (3)(x – 3) 4(x + 1)(x – 3) = Simplify. 5x + 9 4x 2 – 8x – 12 Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples 3 4x + 4 –=–Factor the denominators. 2x x 2 – 2x – 3 3 4x + 4 2x (x – 3)(x + 1) 3 4(x + 1) = – Identity for Multiplication. x – 3 2x (x – 3)(x + 1) 3 4(x + 1) 4444.

Simplify. 1x1x 1y1y 2y2y 1x1x + – 1x1x 1y1y 2y2y 1x1x + – 1x1x 1y1y 2y2y 1x1x + – xy = The LCD is xy. Multiply the numerator and denominator by xy. 2 xy y 1 xy x 1 xy x 1 xy y + – = Use the Distributive Property. y + x 2x – y = Simplify. Method 1:First find the LCD of all the rational expressions. Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

(continued) = Write equivalent expressions with common denominators. 1x1x 1y1y 2y2y 1x1x + – + – y xy x xy 2x xy x xy 2x – y xy x + y xy = Add. Method 2: First simplify the numerator and denominator. =÷ Divide the numerator fraction by the denominator fraction. x + y xy 2x – y xy = Multiply by the reciprocal. x + y xy 2x – y = x + y 2x – y Adding and Subtracting Rational Expressions LESSON 9-5 Additional Examples

Solve =. Check each solution. 1 x – 3 6x x 2 – 9 x 2 – 9 = 6x(x – 3)Write the cross products. x 2 – 9 = 6x 2 – 18xDistributive Property –5x x – 9 = 0Write in standard form. 5x 2 – 18x + 9 = 0Multiply each side by –1. (5x – 3)(x – 3) = 0Factor. 5x – 3 = 0 or x – 3 = 0 x = or x = 3Zero-Product Property x – 3 6x x 2 – 9 = Solving Rational Equations LESSON 9-6 Additional Examples

(continued) Check:When x = 3, both denominators in the original equation are zero. The original equation is undefined at x = 3. So x = 3 is not a solution. When is substituted for x in the original equation, both sides equal – Solving Rational Equations LESSON 9-6 Additional Examples

Solve – =. 3 5x 4 3x x 4 3x 1313 – =. 9 – 20 = 5xSimplify. 15x– = 15xMultiply each side by the LCD, 15x. 3 5x 4 3x 1313 –= Distributive Property 45x 5x 60x 3x 15x 3 – = x 11 5 Since – makes the original equation true, the solution is x = – Solving Rational Equations LESSON 9-6 Additional Examples

Josefina can row 4 miles upstream in a river in the same time it takes her to row 6 miles downstream. Her rate of rowing in still water is 2 miles per hour. Find the speed of the river current. Relate:speed with the current = speed in still water + speed of the current, speed against the current = speed in still water – speed of the current, time to row 4 miles upstream = time to row 6 miles downstream Write: = 6 (2 + r ) 4 (2 – r ) Solving Rational Equations LESSON 9-6 Additional Examples Define: Distance (mi) Rate (mi/h) Time (h) With current r Against current 4 2 – r 4 (2 – r ) 6 (2 + r )

(continued) (2 – r )(6) = (2 + r )(4)Simplify. 4 = 10rSolve for r. 0.4 = rSimplify. The speed of the river current is 0.4 mi/h. = 6 (2 + r ) 4 (2 – r ) (2 + r )(2 – r ) = (2 + r )(2 – r )Multiply by the LCD (2 + r )(2 – r ). 6 (2 + r ) 4 (2 – r ) Solving Rational Equations LESSON 9-6 Additional Examples 12 – 6r = 8 + 4rDistributive Property

Find the LCD LCD is (k+1)(k-2) Multiply everything by LCD

(continued) Combine Like Terms Combine 2k 2 and k’s to left side