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Unit 2: RVEs

Husam Abdulnabi , Yahya Adam, Yash Dave, Neena Raj. Unit 2: RVEs. Rational Variable Expressions. Quotients where the numerator and denominator both contain polynomials. . Exponent Laws. When multiplying powers with the same base, you add the exponents.

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Unit 2: RVEs

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  1. HusamAbdulnabi, Yahya Adam, Yash Dave, Neena Raj Unit 2: RVEs

  2. Rational Variable Expressions • Quotients where the numerator and denominator both contain polynomials.

  3. Exponent Laws • When multiplying powers with the same base, you add the exponents. • When dividing powers with the same base, you subtract the exponents. • When a power is raised to another exponent, you multiply the exponents.

  4. Exponent Laws Negative and Zero Exponents • When a base is raised to a zero exponent, the result is always 1. • When a base is raised to a negative exponent, rewrite the reciprocal of the base; the power is written as 1 over the power to a positive exponent

  5. Rational Exponents Rational Numbers • Numbers represented as a quotient of two integers Rational Exponents • Exponents expressed as ratios

  6. Rational Exponents to Radicals • n – Index • a – Natural number • Must be positive

  7. Examples a) b)c) d)

  8. Restrictions • Since rational variable expressions are quotients, many restrictions are applied • Given that no value is divisible by 0(resulting in undefined), possible variables in the denominator in the expression act as restrictions to avoid such error

  9. Simplifying RVE’s 1. Factor the numerator and denominator 2. State all restrictions 3. Reduce the RVE • When you’re simplifying an RVE the simplified expression still has the same value as the original expression, meaning the restrictions are the same!

  10. Examples a) b)

  11. Multiplying RVE’s 1. Factor the numerator and denominator 2. State all restrictions 3. Reduce the RVE 4. Multiply the numerators and denominators

  12. Examples a) b)

  13. Dividing RVE’s 1. Factor the numerator and denominator 2. State all restrictions 3. Find the reciprocal of the second RVE 4. Find the reciprocal of the second RVE 5. State any NEW restrictions • ADD to existing 6. Reduce the RVE

  14. Examples a) b)

  15. M.I.A. • Jane is kidnapped by a herd of gorillas that were hired by devious John Clayton. Terk witnessed the crime, and wants to help Tarzan out by telling him the general area of where they might be hiding; modeled after a triangle. Given the rough dimensions, what is the total area where Jane could be? Simplify expressions and make sure to add restrictions.

  16. Adding/Subtracting RVE’s 1. Factor both the numerator and denominator 2. State all possible restrictions 3. Determine the Lowest Common Denominator (LCD) 4. Multiply each expression by the missing factor(s) 5. Add or subtract the numerators, but keep the denominator the same

  17. Examples a) b)

  18. Elephant Derby! • Tantor, the 4-time reigning champ is primed to win yet again this year. His average speed is modelled after the following (t is time (s), S is speed (m/s): 1) Simplify the expression with all restrictions 2) If t represents 15s, what is Tantor’s speed in m/s?

  19. The Rational Function • The base rational function is modeled after: • If x is positive, y is positive as well, appearing in the 1st quadrant and vice-versa • If x is negative, y is negative as well, appearing in the 3rd quadrant and vice-versa

  20. Asymptotes • A line that continually approaches a given curve but does not meet it at any finite distance.  Vertical Asymptote • Since nothing is divisible by 0: Undefined • V.A – x = o x = 0

  21. Asymptotes Horizontal Asymptote • Since any value of x will not satisfy y = o: • H.A – y = o y = 0

  22. General Info Asymptotes • H.A – y = 0 • V.A – x = o Key Points • (1,1) • (-1,-1) Domain Range y = 0 x = 0

  23. Transformations • The expanded rational function is modelled after: • a - Vertical stretch/compression, reflection in x-axis if negative • b - Horizontal stretch/compression, reflection on y-axis if negative • h- Horizontal shift left or right • Represents vertical asymptote • k- Vertical shift up or down • Represents horizontal asymptote

  24. Restrictions • When simplifying expressions that would be graphed, restrictions present would appear to be holes in the graph, since they do not exist.

  25. Sum Up Asymptotes • H.A – y = k • V.A – x = h Key Points • (b+h,a+k) • (-b+h,-a+k) Domain Range

  26. Base Graphs in Disguise! • Professor needs your help trying to figure out the graph of a tree trunk with the following information: vertical stretch by factor of 2, horizontal stretch by factor of 3, vertical shift up 3 units, horizontal shift the left 6 units, and restrictions at x=-4, and x=6. • Label asymptotes, restrictions, domain and range

  27. A HOLE lot of Trouble

  28. A HOLE lot of Trouble • Tarzans quest to find Jane is coming to an end! However, the only way to get there is through a tree slide has a hidden gapping hole! The slope is modeled after the rational function: 1) If Jane is 10 meters below, will Tarzan ever reach her through the slide? 2) At what time and height should Tarzan jump off to avoid the danger and reach his loved one?! • Provide BOTH algebraic and graphic evidence!

  29. YAY!

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