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REVIEW

REVIEW. What is a polynomial with a single root at x = -4 and a double root at x=7? (you need not multiply it out). Single root at x=-4 looks like (x--4) Double root at x=7 looks like (x-7)(x-7) The polynomial form is: a(x+4)(x-7)(x-7) I can put any number I want in for a.

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REVIEW

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  1. REVIEW What is a polynomial with a single root at x = -4 and a double root at x=7? (you need not multiply it out) • Single root at x=-4 looks like (x--4) • Double root at x=7 looks like (x-7)(x-7) • The polynomial form is: a(x+4)(x-7)(x-7) • I can put any number I want in for a. • B) 2(x+4)(x-7)(x-7)

  2. Rational Functions

  3. Poles • Also called “vertical asymptotes” • Values of x where y shoots of to positive or negative infinity • These happen when you have a non-zero number in the numerator and a zero in the denominator • Poles are not part of the domain of the function

  4. Domain • Has a pole at x=1/4 • Poles happen when you divide by zero • It doesn’t make sense to divide by zero • The domain is x≠1/4

  5. Roots • Also known as zeroes • Values of x that cause y to be zero • A rational function is zero whenever the numerator P(x) is zero and the denominator Q(x) is not zero

  6. Summary so far • Poles happen when you have not-zero/zero • Set denominator = 0 and solve for x. • Roots happen when you have zero/not-zero • Set numerator = 0 and solve for x. • If you have 0/0, you need more math.

  7. What are all the poles (vertical asymptotes) of the following rational function: • x = -1/2, -3, -1 • x = - 3, x = -1 • x = -3, x = -1, x = 1 • x = -3, x = -1, x = 0 • None of the above

  8. D

  9. Asymptote • The line (or curve) that your function looks like when x is very big or very small • By very small, I mean very negative.

  10. Example

  11. Zoomed way out Looks like a line!

  12. Asymptote • The line (or curve) that your function looks like when x is very big or very small • By very small, I mean very negative. • The line (or curve) that your function looks like when you zoom out a lot.

  13. The basic principle of asymptotes • The highest power always wins • When looking for a general idea of the asymptote, you can ignore everything except the leading terms • Highest power in the numerator • Highest power in the denominator

  14. Asymptote • The line (or curve) that your function looks like when x is very big or very small • By very small, I mean very negative. • The line (or curve) that your function looks like when you zoom out a lot. • Remember, an asymptote is a function, so start it with y= or f(x)=

  15. What is the horizontal asymptote of the following rational function: • x = -1/2 • y = 1 • y= 1/2 • All of the above • Both (a) and (b)

  16. What is the horizontal asymptote of the following rational function: The leading terms are x2 and 2x2. When x is very big, y ≈ (x2)/(2x2)=1/2. c) y =1/2

  17. Summary • Poles happen when you have not-zero/zero • Roots happen when you have zero/not-zero • Horizontal Asymptotes are what you get when you imagine x is very big or very small. • Highest power wins principle: divide the two leading terms.

  18. What is a rational function having poles at x = 4 and x = -2, having zeros at x = 6 and x = 1, and having a horizontal asymptote at y=9? a) b) x + 2 c) x d) (a) and (b) e) None of the above

  19. What is a rational function having poles at x = 4 and x = -2, having zeros at x = 6 and x = 1, and having a horizontal asymptote at y=9? Poles at x=4, x=-2:  bottom looks like a(x-4)(x+2)=a(x2-2x-8) Roots at x =6 and x =1  top looks like b(x-6)(x-1)=b(x2-7x+6) So far we have: Horizontal asymptote at y=9, means (bx2)/(ax2)=b/a=9 So the function should look like: E

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