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Algebra 2 > Rational Expressions, Equations, and Functions > Flashcards

Rational Expressions, Equations, and Functions Flashcards

1
Q

What is a ratio?

A

a fraction

numerator ÷ denominator

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2
Q

What is a rational expression?

A

a ratio of polynomials

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3
Q

What is an expression?

A

a combination of numbers, variables, and operations

examples:

5 + 23

4x

6x + y²

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4
Q

How do you multiply rational expressions?

A
  1. factor numerators and denominators (find excluded values)
  2. cancel
  3. multiply
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5
Q

How do you simplify rational expressions?

A
  1. combine like terms
  2. factor numerators and denominators
  3. cancel
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6
Q

What are excluded values?

A

values of x that make the denominator = 0

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7
Q

What is a complex fraction?

A

a fraction that contains a fraction in the numerator and/or denominator (a.k.a. nested fraction)

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8
Q

How do you simplify complex fractions?

A
  1. add or subtract fractions if necessary
  2. rewrite as division: numerator ÷ denominator
  3. rewrite as multiplication: first fraction times reciprocal of second fraction
  4. factor numerators and denominators
  5. cancel
  6. multiply
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9
Q

How do you divide rational expressions?

A
  1. rewrite as multiplication: first fraction times reciprocal of second fraction
  2. factor numerators and denominators (note excluded values)
  3. cancel
  4. multiply
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10
Q

How do you add or subtract rational expressions?

A
  1. factor denominators
  2. get common denominator by multiplying the numerator and denominator by any missing factors
  3. add or subtract the numerators; leave the denominator factored
  4. simplify
    1. combine like terms
    2. factor
    3. cancel
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11
Q

What are factors?

A

the polynomials being multiplied

factor × factor = product

example:

(x – 2)(x + 2) = x2 – 4

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12
Q

What is a product?

A

the result of multiplying two or more polynomials

factor × factor = product

example:

(x – 2)(x + 2) = x2 – 4

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13
Q

When subtracting, make sure to…

A

…distribute the negative.

example:

5 - (x + 6) = 5 - x - 6

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14
Q

Dividing is the same as…

A

…multiplying by the reciprocal.

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15
Q

You can cancel only if…

A

…numerators and denominators are factored.

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16
Q

When adding or subtracting fractions, you need a…

A

…common denominator.

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17
Q

What is a quotient?

A

the result of dividing two or more quantities

dividend ÷ divisor = quotient

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18
Q

How do you factor a trinomial?

A

GCF(trinomial)

–and/or–

(binomial)(binomial)
“multiplies to c and adds to b”

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19
Q

How do you factor a binomial?

A

GCF(binomial)

–and/or–

Difference of Squares

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20
Q

How do you multiply a binomial times a binomial?

A

distributive property

(a + b)(c + d)

a(c + d) + b(c + d)

ac + ad + bc + bd

example:

(x + 5)(x - 3)

x(x -3) + 5(x - 3)

x2 - 3x +5x - 15

x2 + 2x - 15

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21
Q

How do you multiply a monomial times a binomial?

A

distributive property

a(b + c)

ab + ac

example:

2bd(3bc + 5ad)

6b2cd + 10abd2

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22
Q

What is a constant?

A

a number

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23
Q

What is a binomial?

A

a polynomial with two terms

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24
Q

What is a trinomial?

A

a polynomial with three terms

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25
Q

What is a polynomial?

A

an expression of one or more terms

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26
Q

What is a term (in an expression)?

A

a number, a variable, or numbers and variables joined by multiplication or division

terms are separated by addition or subtraction in an expression

27
Q

What is a proportion?

A

an equation with a fraction on both sides

28
Q

How do you solve a proportion?

A
  1. factor denominators
  2. find excluded values
  3. multiply just the numerator on both sides by factors to cancel factors in denominators; remember to distribute
  4. get everything on one side (0 on the other side) and solve by factoring or quadratic formula
  5. check for extraneous solutions (zeros and excluded values)
29
Q

What is an extraneous solution of a function?

A

a zero (numerator = 0) that is also an excluded value (denominator = 0)

30
Q

In an equation, if the variable is on both sides…

A

…get the variable on one side.

31
Q

What is an equation?

A

two expressions that are equal to each other

left expression = right expression

32
Q

What is a hole (a.k.a. removable discontinuity)?

A

a break in the graph found at canceled excluded values

33
Q

What is an asymptote?

A

a line that a graph approaches but never intersects

34
Q

What is a horizontal asymptote?

A

an invisible horizontal line (y = constant) that a graph approaches but never intersects as x approaches -∞ or ∞

35
Q

What is the equation of a horizontal line?

A

y = constant

36
Q

What is the equation of a vertical line?

A

x = constant

37
Q

What is a vertical asymptote?

A

an invisible vertical line (x = constant) that a graph approaches but never intersects as x approaches a remaining excluded value

38
Q

How do you graph a rational function?

A
  1. factor numerator and denominator
  2. find excluded values (denominator = 0)
  3. cancel
  4. find hole (canceled excluded value)
  5. find vertical asymptote (remaining excluded value) and graph
  6. identify domain (all real #’s except excluded values)
  7. find horizontal asymptote (none, y = 0, or y = ratio) and graph
  8. identify range (all real #’s except horizontal asymptote)
  9. sketch graph from calculator (don’t forget holes!)
39
Q

How do you find a vertical asymptote?

A
  1. factor numerators and denominators
  2. find excluded values (denominator = 0)
  3. cancel
  4. canceled excluded values are holes
  5. remaining excluded values are vertical asymptotes
40
Q

How do you find a horizontal asymptote?

A

Compare the degree of the numerator with the degree of the denominator:

top heavy: no horizontal asymptote

bottom heavy: y = 0 (plus or minus any vertical shifts)

equal degree: y = ratio of coefficients

41
Q

What is the domain of a rational function?

A

all real numbers except excluded values (denominator = 0: holes AND vertical asymptote)

42
Q

What is the range of a rational function?

A

all real numbers except the horizontal asymptote

43
Q

How do you find a hole (a.k.a. removable discontinuity)?

A
  1. factor numerators and denominators
  2. find excluded values (denominator = 0)
  3. cancel
  4. canceled excluded values are holes
44
Q

What operation does a fraction bar represent?

45
Q

What is a difference?

A

the result of subtracting two polynomials

46
Q

What is a quotient?

A

the result of dividing two polynomials

47
Q

How do you factor a difference of squares?

A

(sum of square roots)(difference of square roots)

48
Q

What is a monomial?

A

a polynomial with one term

49
Q

What is the domain of a function?

A

the set of all possible values of x

50
Q

What is the range of a function?

A

the set of all possible values of y

51
Q

What is a variable?

A

a letter representing a quantity that takes on different values

52
Q

Which is the independent variable?

53
Q

Which is the dependent variable?

54
Q

What can you do after factoring all numerators and denominators?

A

cancel common factors

55
Q

What’s the best way to solve this equation?

A
  1. find excluded values: x ≠ -4 or 9
  2. add the right side to get one fraction
  3. multiply both sides by factors in denominators to cancel them; remember to distribute
  4. get everything on one side (0 on the other side) and solve by factoring or quadratic formula
  5. check for extraneous solutions (zeros and excluded values)
56
Q

What does it mean to solve an equation?

A

get the variable by itself to find its value

57
Q

What’s the first step to solve this equation?

A
  1. factor numerators and denominators
  2. cancel

===

then, multiply both sides by fewest factors necessary to cancel denominators

58
Q

If both fractions of a proportion have the same denominator, what can you do?

A

multiply both sides by that denominator to cancel them

59
Q

What’s the best way to solve this equation?

A
  1. factor denominators
  2. find excluded values: k ≠ 2 or 3
  3. multiply just the numerator on both sides by factors to cancel factors in denominators: (k - 3) and (k - 2)
  4. get everything on one side (0 on the other side) and solve by factoring or quadratic formula
  5. check for extraneous solutions (zeros and excluded values)
60
Q

When multiplying fractions, you multiply…

A

…numerators with numerators and denominators with denominators (straight across).

61
Q

When factoring, first look for a…

A

…GCF.

example:

3x2 + 6x - 45 >> 3 is the GCF

3(x2 + 2x - 15) >> factor by distributive property

3(x - 5)(x + 3) >> keep factoring if possible

62
Q

What is a zero (a.k.a. solution) of a function?

A

an x-intercept of the graph

where f(x) = 0, which is when the numerator = 0

63
Q

In a function, what do excluded values tell you?

A

canceled: hole (a.k.a. removable discontinuity)

remaining: vertical asymptote

64
Q

What does f(x) mean?

A

it’s another way of writing the y variable (the dependent variable)

Algebra 2 (3 decks)

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