algebra 2a - unit 5: rational equations lesson 19-24
simplify the expression (2x^2-50)//(x+5) by canceling any common factors.
2x - 10
find the quotient of 6x^3 − 12x^2 + 15x ÷ 3x
2x^2 - 4x + 5
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237360 if f(x) = 1/x, what is the function rule of f'(x)?
1/3x
let x represent the number of games remaining, and let y represent the number of games terry needs to win. which inequality represents this situation?
18+y/35+x > 24/36+x
how many milliliters should sam add?
40
how much water does jin need to add?
40 fluid ounces
if 12 games are left, what is the least amount of additional games that terry needs to win to have the higher winning percentage?
6 games
if 9 games are left in the tournament, what is the lowest number of games martina needs to win to beat kerry?
6 games
how many fluid ounces of water are in the first mixture?
7 fluid ounces
two mixtures have the same percentage of an acid solution. the first mixture has 7 fluid ounces of solution with x fluid ounces of water. the second mixture has 3x fluid ounces of solution with 21 fluid ounces of water. which equation will help find the correct value of x?
7/x = 3x/21
use the inequality to find how many months it will take before the average monthly cost of company a is less than company b.
8
how many days does it take for the cost of each company to be the same?
8 days
let r represent the number of games remaining, and let w represent the number of games martina needs to win.
8+w/25+r > 15/29+r
which equation will help determine how many days it will take for the average charge per day for company a to be the same as the average charge per day for company b?
80+22d/d = 32
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237359 if f'(x) = a * f(1/bx), which statement is true?
a = -3
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237358 if f'(x) = a * f(1/bx), which statement is true?
b = 4
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237363 if f(x) = 5/x(x-5), what is the function rule of f(x)?
f'(x) = 15/x(x-5)
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237361 if f(x) = 1/x, what is the function rule of f'(x)?
f'(x) = 4/x
examine the transformation that takes the preimage f(x) to the image f′(x) in the graph. https://cdstools.flipswitch.com/asset/media/1237362 if f(x) = 5/x(x-5), what is the function rule of f'(x)?
f'(x) = 5/-1/2x(-1/2x-5)
lesson 22
graphing rational functions
what can santos do next to solve the equation?
he can factor the quadratic expression x^2 + x − 12 to find the solutions.
which graph represents the function f(x) = (x-10)(x+1)/(x+3)(x-3)?
https://cdstools.flipswitch.com/asset/media/1237281
examine the graph of f(x) = 1/x. if f'(x) = f(x + 4), which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237318
examine the graph of f(x) = 1/x-2 - 3 if f'(x) = 1/x-2 + 4, which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237323
examine the graph of f(x) = 1/x(x+2) + 4. if f'(x) = f(x) - 3, which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237328
examine the graph of f(x) = 1/x(x+4) - 2 if f'(x) = f(x - 5), which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237330
if f'(x) = -f(3x), which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237338
if f'(x) = 1/4f(x), which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237349
if f'(x) = f(-x), which is the correct graph of f(x) and f'(x)?
https://cdstools.flipswitch.com/asset/media/1237354
lesson 23
rational equations and inequalities in context
what does the vertical asymptote represent in the context of this situation?
the vertical asymptote at x = -29 has no meaning in the context of this situation because martina can't play a negative number of games
lesson 21
solving rational equations
the function f(x) = x^3 − 5x^2 + 7x − 35 can be rewritten as f(x) = (x − 5)(x^2 + 7) which statements must be true? select all that apply.
the binomial x - 5 is a factor of f(x) the remainder is 0 when f(x) is divided by x - 5
terry has won 18 of 35 chess games. what is the domain and range of the situation?
the domain can be represented by the whole numbers over the interval [6, inf) because terry needs to play a minimum of 6 additional games to win enough games to have a higher winning percentage than sandra the range can be represented by the whole numbers over the interval [6, inf) because terry needs to win a minimum of 6 additional games to have a higher winning percentage than sandra
what is the domain and range according to the situation?
the domain can be represented by the whole numbers over the interval [6, inf), because when there are 6 more games, martina can begin to win enough games to get the higher winning percentage. the number of additional games played can also be higher than 6. the range can be represented by the whole numbers over the interval [6, inf), because the lowest number of games martina needs to win to get the higher winning percentage is 6. the number of additional wins can also be higher than 6
notice the horizontal asymptote at y = 6. what does this represent in the context of the situation?
the horizontal asymptote is at y = 6 because terry will never need to win more than 6 additional games to have a higher winning percentage than sandra, given that sandra doesn't win any more games.
the graph of f(x) = 5/x^2(x-1) + 6 underwent a transformation become f'(x) = 5/(x-2)^2(x-3) + 6. which sentence correctly describes the transformation from f(x) to f'(x)?
the image is 2 units to the right of the preimage.
the graph of f(x) = 3x/x^2(x+4) - 3 underwent a transformation become f'(x) = 3x/x^2(x+4) + 5. which sentence correctly describes the transformation from f(x) to f'(x)?
the image is 8 units above the preimage.
the graph of f(x) = x/x(x+5) - 3 underwent a transformation to become f'(x) = -x/(-x)((-x)+5) - 3. which sentence correctly describes the transformation from f(x) to f'(x)?
the image is reflected over the y-axis.
the graph of f(x) = x/x(x+1) + 7 underwent a transformation to become f'(x) = 3x/x(x+1) + 21. which sentence correctly describes the transformation from f(x) to f'(x)?
the image is stretched away from the x-axis by a factor of 3
the graph of f(x) = 4x/6x^2(2x+4) - 3 underwent a transformation to become f'(x) = 2x/3x^2(x+4) -3. which sentence correctly describes the transformation from f(x) to f'(x)?
the image is stretched away from the y-axis by a factor of 2.
what does the y-intercept, or initial value, of (0, 4.93) represent in the context of the situation?
the ordered pair (0, 4.93) means that if no more games are played, martina would have to win at least 5 more games to have a higher winning percentage than kerry. this is not possible in the situation
which graph has appropriate labels and scales for this situation?
the y-axis: 30% solution the x-axis: 15% solution (the numbers are 10, 20, 30, 40....)
what are the x- and y-intercepts of the function f(x) = x/x^2-3x?
there is neither an x-intercept nor a y-intercept for f(x)
lesson 24
transformations of rational functions
john packages turkeys from a farm before they are sold to stores. which graph has appropriate labels and scales for this situation?
x-axis: 1.5% solution y-axis: 5% solution (the numbers are 0, 5, 10, 15, 20,25)
is the binomial x + 3 a factor of the function f(x) = x^3 + 4x^2 − 3x − 18? why or why not?
yes, because f(-3) = 0
consider the function f(x) = x^2-1/x+1, which is shown in the graph. what are the coordinates of the hole in the graph of this function?
(-1, -2)
what is the correct quotient of 42x^4 ÷ (7x^2 + 14x) when written as a rational expression?
(6x^3)//(x+2)
what is the remainder when x^3 − 2x^2 − 51x − 108 is divided by x − 2?
-210
what is the remainder when 4x^3 + 8x − 32 is divided by x + 10?
-4112
consider the function represented by the graph. https://cdstools.flipswitch.com/asset/media/1237153 when x = -3, what is f(x)?
0
consider the function f(x) = x+4/(x+1)(x^2-9x+20). which statements are true? select all that apply.
0.33 of 1 (this is annoying sorry) a hole is at (-1, -1/18) a vertical asymptote is at x = 5 a vertical asymptote is at x = -1 a vertical asymptote is at x = 4
simplify the expression 5x − 35//9x − 63 by canceling any common factors.
5/9
what is the value of x in the rational equation 18/23 = 45/x
57 1/2
which expression is equivalent to (5x^3-8x^2-3)//(x+2)?
5x^2 - 18x + 36 - 76/x+2
what is the remainder when x^2 − 4x − 12 is divided by x + 7?
65
what is the value of x in the rational equation 7/9x = 49/441?
7
which expression is equivalent to 7x^3 − 14x^2 − 7x + 2//x − 3?
7x^2 + 7x + 14 + 44/x-3
find the quotient of 8b^3 + 3b^2 − b ÷ b
8b^2 + 3b - 1
what is the value of x in the rational equation 27/9 = 3x/9?
9
which statement is true about a function that has the zeros x = −1 x = 2, and x = 3?
if the function rule is divided by x − 3, there will not be a remainder.
examine the graph of a function. https://cdstools.flipswitch.com/asset/media/1237153 which statement is true about the function represented in the graph?
if the function rule is divided by x − 4, there will not be a remainder.
roy used long division to divide 6x^2 + 7x + 3 by 3x + 2. his work is shown. which statement correctly describes roy's mistake, if any?
in step 1, roy should have subtracted 4x from 7x instead of adding them.
is the binomial x − 4 a factor of the function f(x) = x^3 + 4x^2 − 3x − 18? why or why not?
no, because f(4) = 98
is the binomial x − 8 a factor of the function f(x) = x^3 − 9x^2 + 12x − 17? why or why not?
no, because f(8) = 15
the function g(x) = x^3 + x^2 − 26x + 24 can be rewritten as g(x) = (x + 6)(x − 4)(x − 1) which equations must be true? select all that apply.
p(1) = 0 p(4) = 0 p(-6) = 0
examine the following graph of a function. https://cdstools.flipswitch.com/asset/media/1237294 what is the domain and range of the function represented by the graph?
range: (−∞, −196) ∪ (−196, 0) ∪ (0, ∞) domain: (−∞, −3) ∪ (−3, 0) ∪ (0, 1) ∪ (1, 5) ∪ (5, ∞)
what is the domain and range of the function f(x) = 3x/x^3-5x^2-14x?
range: (−∞, −3/14) ∪ (−314, 0) ∪ (0, ∞) domain: (−∞, −2) ∪ (−2, 0) ∪ (0, 7) ∪ (7, ∞)
what are the x- and y-intercepts of the function f(x) = x+5/x-2?
x-intercept (-5, 0) y-intercept (0, -5/2)
find the quotient of x^3 + 2x^2 − 4x ÷ x
x^2 + 2x - 4
find the quotient of 4x^3 + 12x^2 − 24x ÷ 4x
x^2 + 3x - 6
which graph represents the function f(x) = x+1/x^2+5x+4?
https://cdstools.flipswitch.com/asset/media/1237285
which graph represents the function f(x) = x+7/x^2-7x?
https://cdstools.flipswitch.com/asset/media/1237289
which graph represents the function f(x) = x/x^2+3x?
https://cdstools.flipswitch.com/asset/media/1237295
examine the following graph of a function. https://cdstools.flipswitch.com/asset/media/1237278 which statements are true about the function represented by the graph? select all that apply.
0.63 of 1 there is no x-intercept there is no y-intercept there is a hole at x = -4 there is a vertical asymptote at x = 3
examine the following graph of a function. https://cdstools.flipswitch.com/asset/media/1237293 which statements are true about the function represented by the graph? select all that apply.
0.8 of 1 a hole is at x = -3 a vertical asymptote is at x = 5 a vertical asymptote is at x = 1 a horizontal asymptote is at y = 0
solve for r. 5r-0.6//2.5r+6.7 = 4/9
0.92
what is the value of x in the rational equation x+2/x+3 = 18/24?
1
what is the value of x in the rational equation x/5 = 6/18?
1 2/3
what is the remainder when 3x^2 − 7x + 5 is divided by x + 5?
115
what is the remainder when 2x^3 − 9x^2 + 11 is divided by x − 6?
119
find the value(s) of x in the rational equation x/36 = 4/x
12 and -12
dominic is looking to buy a cell phone, and he's comparing cellular plans. company a charges a $175 start-up fee and $75 per month for service. company b does not have a start-up fee, but it charges $97 per month. he wants to know how many months it will take before the average monthly cost of company a is less than company b. which inequality will help dominic figure this out?
175+75m/m < 97
what is the remainder when x^3 + 4x^2 − 7x − 10 is divided by x + 4?
18
sam needs a 20% acid solution, but she has only an 18% solution and a 25% solution. she decides to use 100 ml of the 18% solution, and she needs to know how much of the 25% solution she should add. which equation represents this situation? let x represent the number of milliliters she should add.
18+0.25x/100+x = 0.2
which equation shows the first step when solving for the value of x in the rational equation x+2/x+3 = 18/24?
18x + 54 = 24x + 48
what is the value of x in the rational equation x/4 = 9/15?
2.4
jin is a veterinarian tech. she needs a 20% solution of a cleaning fluid to disinfect examination tables. if 10 ounces of the cleaning fluid is already in a bucket, which equation can she use to know how much water she should add?
20 = 10/w+10 * 100
find the value(s) of x in the rational equation x/25 = 16/x
20 and -20
find the quotient of 2a^2 − 3a − 9 ÷ a
2a - 3 - 9/a
find the quotient of 10t^3 − 15t^2 − 20t + 35 ÷ 5t
2t^2 - 3t - 4 + 7/t
which expression is equivalent to (6x^2-2x-28)//(3x-7)?
2x + 4
what is the value of x in the rational equation 21/45 = 7x/45?
3
what is the value of x in the rational equation 4/7 = 20/x?
35
what is the quotient when you divide 6x^3 + 13x^2 − 2x − 12 by 2x + 3?
3x^2 + 2x - 4
find the quotient of 18x^3 + 12x^2 − 30 ÷ 6x
3x^2 + 2x - 5/x
find the quotient of 3x^3 + 4x − 9 ÷ x + 2
3x^2 - 6x + 16 - 41/x+2
the expression (3x+2)(4x-4)//(3x+2)(x-1) can be simplified to an integer value. what is the quotient after canceling common factors and simplifying?
4
what is the quotient when you divide 16x^3 − 3x + 5 by 4x + 3?
4x^2 - 3x + 3/2 + 1/2(4x+3)
simplify the expression 15x + 5//3x + 1 by canceling any common factors.
5
what is the value of x in the rational equation 8/2x = 12/15?
5
consider the function f(x) = x+4/x^3+x^2-12x. which statements are true? select all that apply.
a hole is at (-4, 1/28) a vertical asymptote is at x = 3 a vertical asymptote is at x = 0
consider the function f(x) = 5x-10/x-2. which statement about the function is true?
a hole is at (2, 5)
consider the function f(x) = x-5/x^2-2x-15. which statements are true? select all that apply.
a hole is at (5, 1/8) a vertical asymptotes is at x = -3
examine the following graph of a function. https://cdstools.flipswitch.com/asset/media/1237279 what are the asymptotes of the function represented by the graph? select all that apply.
a vertical asymptote is at x = 2 a vertical asymptote is at x = -4 a horizontal asymptote is at y = 0
consider the function f(x) = 5x-7/3x+9. which statement about the function is true?
a vertical asymptote is at x = −3 because −3 makes the denominator 0.
part a: use the remainder theorem to find the remainder when p(x) = x^3 + 4x^2 + 5x + 2 is divided by x + 1 part b: is x + 1 a factor of p(x) = x^3 + 4x^2 + 5x + 2?
a: the remainder is 0 b: yes, x + 1 is a factor of p(x)
what is the end behavior of the function f(x) = x-1/(x-1)(x+4)?
as x approaches −∞, f(x) approaches 0, and as x approaches ∞, f(x) approaches 0
what is the end behavior of the function f(x) = 5x^2/7x(x+4)?
as x approaches −∞, f(x) approaches 5/7, and as x approaches ∞, f(x) approaches 5/7.
examine the following graph of a function. https://cdstools.flipswitch.com/asset/media/1237278 what is the end behavior of the function represented by the graph?
as x approaches −∞, the function approaches 0, and as x approaches ∞, the function approaches 0.
which statement correctly describes the end behavior of the function represented by the graph? https://cdstools.flipswitch.com/asset/media/1237293
as x approaches −∞, the function approaches 0, and as x approaches ∞, the function approaches 0.
lesson 19
dividing polynomials
what is the domain and range of the function represented by the graph? https://cdstools.flipswitch.com/asset/media/1237278
domain: (−∞, −4) ∪ (−4, 3) ∪ (3, ∞) range: (−∞, −1/3) ∪ (−1/3, 0) ∪ (0, ∞)
what is the domain and range of the function f(x) = x+5/x^2+3x-10?
domain: (−∞, −5) ∪ (−5, 2) ∪ (2, ∞) range: (−∞, −17) ∪ (−17, 0) ∪ (0, ∞)
if x + 4 is a factor of a function, which statements must be true? select all that apply.
f(-4) = 0 x = -4 is a zero of the function the graph of the function intersects the x-axis at −4
examine the graph of a function. https://cdstools.flipswitch.com/asset/media/1237154 which statement is true about the function represented in the graph?
f(2) = 0
which problem correctly shows 3x^3 + 8x^2 − x − 10 divided by x + 2 using long division?
the quotient is: 3x^2 + 2x - 5
lesson 20
the remainder theorem
which statements are true about the function represented by the graph? https://cdstools.flipswitch.com/asset/media/1237279
there is a hole at x = 0 there are no x- or y-intercepts the domain is (−∞, −4) ∪ (−4, 0) ∪ (0, 2) ∪ (2, ∞)
simplify the expression x^2 − 2x − 15// x^2 − 13x + 40 by canceling any common factors.
x + 3 // x - 8
simplify the expression (x^2+8x+16)//(x+4) by canceling any common factors.
x + 4
which expressions are factors of a function that has the zeros x = −5, x = −2, and x = 7? select all that apply.
x + 5, x + 2, x - 7
examine the graph of a function. https://cdstools.flipswitch.com/asset/media/1237154 which expressions are factors of the function represented in the graph? select all that apply.
x + 5, x - 2
what is the common factor of the numerator and denominator in the expression (x+1)(x+7)/((x+7)(x-1)?
x + 7
which expressions are factors of the function x^3 − 2x^2 − 19x + 20? select all that apply.
x - 1, x - 5, x + 4
examine the graph of a function. https://cdstools.flipswitch.com/asset/media/1237153 which expressions are factors of the function represented in the graph? select all that apply.
x - 2, x + 3, x - 4
which expressions are factors of a function that has the zeros x = −4, x = 3, and x = 5? select all that apply.
x - 3, x - 5, x + 4
what is the common factor of the numerator and denominator in the expression (2x+3)(x-4)//(x-4)(x+4)?
x - 4
which expression is a factor of h(x) = x^2 − 3x − 28?
x - 7
what is the extraneous solution to the equation x^2/x+3 = 9/x+3?
x = -3
find the solution(s) to 16/x−4 = x^2/x−4
x = -4 is the only valid solution
find the solution(s) to the rational equation x-2/6 = 1/x+3
x = -4, x = 3
solve the rational equation 18/x-6 = x+2/36
x = 2 ± 2√166
what is the solution to the rational equation 4x-1/2.1x+1.425 = 2/3?
x = 3/4
which equations represent the vertical asymptotes of the function f(x) = x+3/x^2-3x-10? select all that apply.
x = 5, x = -2
find the solution(s) to x^2/x+6 = 36/x+6
x = 6 is the only valid solution