Resources

Resources

Transformations of functions

What kind of transformation converts the graph of

f(x) = 8|x+1| - 6

into the graph of

g(x) = 8|x+3| - 7

What kind of transformation converts the graph of

f(x) = 8|x - 2| - 7

into the graph of

g(x) = 8|x + 2| + 2

A B C D

is graphed in the coordinate plane. The following are the vertices of the rectangle:

A(-7,-5), B(-7,6), C(-4,6)

D(-4,-5)

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Given these coordinates, what is the length of side

C D

of this rectangle?

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\square

What kind of transformation converts the graph of

f(x)=7|x-4|+5

into the graph of

g(x)=7|x-2|+3

A manufacturing facility records the time its employees work each week in fractions of an hour.

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29 \frac{1}{2} \mathrm{hr} \quad

3 \frac{3}{4} \mathrm{hr} \quad

44 \frac{1}{4} \mathrm{hr}

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How many more hours (h) did Claire work than the combined hours of Taliyah and Avery?

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The equation is:

h=

\square

-\square+

\square

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\square

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\square

more hours than Taliyah and Avery combined.

Which of the following equations represent functions where

x

is the input and

y

is the output? Select all that apply.

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Multi-select Choices:

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y = \frac{x}{2}

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y = -x

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y - 2x = 2

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y = 2|x|

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x = \frac{1}{2}

Suppose that the polynomial function

f

is defined as follows.

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f(x)=2x^{2}(x-4)^{2}(x+7)^{3}

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List each zero of

f

according to its multiplicity in the categories below.

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If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, click on ＂N＂

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Zero(s) of multiplicity one:

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Zero(s) of multiplicity two:

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Zero(s) of multiplicity three:

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Select all of the true statements.

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Multi-select Choices:

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6 \times 10^5

20

times as much as

3 \times 10^4

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7 \times 10^3

100,000

times as much as

7 \times 10^{-3}

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9 \times 10^{-5}

0.01

times as much as

9 \times 10^{-3}

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3 \times 10^4

20

0

times as much as

20

1

x=2y

x\in (2, 3, 4, 5)

y\in (1, 2, 3, 4)

, what is the range

10

Eva sees a dolphin

3.2

meters below sea level and a bird

\frac{47}{10}

meters above sea level.

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2

of the following expressions represents the vertical distance between the dolphin and the bird?

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2

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(A) |\(\frac{47}{10}+3.2|\)

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|-3.2

\(-\frac{47}{10}|\)

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|3.2

\(-\frac{47}{10}|\)

g(w)=(w+13)^{3}(w+19)^{2}

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The polynomial function

g

is defined. When

g(w)

(w+16)

, the remainder is

r

. What is the value of

|r|

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At the school carnival, Nicole is in charge of a spinner game where students can win prizes. The spinner is divided into

4

unequal sections labeled book, sticker, eraser, and keychain. Nicole keeps track of what prizes the spinner lands on for the first

8

students who play. Here are her results:

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sticker, book, eraser, keychain, sticker, eraser, sticker, keychain

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Based on the data, what is the probability of the spinner landing on keychain?

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Write your answer as a fraction or whole number.

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A piece of paper is to display

128

square inches of text. If there are to be one-inch margins on both sides and two-inch margins at the bottom and top, what are the dimensions of the smallest piece of paper (by area) that can be used?

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1

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8 \prime \prime \times 16 \prime \prime

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10 \prime \prime \times 15 \prime \prime

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10 \prime \prime \times 18 \prime \prime

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10 \prime \prime \times 20 \prime \prime

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(E) None of these

Given a fixed integer

n \geq 2

S

be the set of all factors of

n

. Ellie and Neil play a game where they take turns removing elements from

S

with Ellie going first. After the first turn, a number removed on each turn must be relatively prime to the number removed in the previous turn by the opponent. The first player who is unable to remove a number on their turn (including starting a turn with

S

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What is the smallest composite

n

that Neil has a winning strategy on?

\mathrm{J}

-3

\mathrm{K}

\mathrm{L}

8

units away from Point

\mathrm{J}

\mathrm{K}

\mathrm{L}

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\mathrm{K}=\square \quad \mathrm{L}=\square

Seth was studying a population of fruit flies. The population,

P

t

days of the study is modeled by the function

P(t)=10 e^{0.5 t}

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What function could sech use to find

D

, the day the number of fruit flies reached a given value,

p

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D=\frac{1}{10} \ln (2 p)

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D=\frac{1}{5} \ln (b)

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D=\frac{1}{2} \ln (10 p)

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D=2 \ln \left(\frac{p}{10}\right)

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E. D=10 \ln \left(\frac{p}{2}\right)

A baby mango tree stands

2

6

meters above the ground. Its roots extend

\frac{7}{5}

meters below the ground.

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2

of the following expressions represents the vertical distance between the top of the mango tree and the tips of its roots?

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2

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\left|-\frac{7}{5}-2.6\right|

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\left|2.6+\frac{7}{5}\right|

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\left|2.6-\frac{7}{5}\right|

Which expressions are equivalent to

7^{-2} \cdot 7^{6}

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2

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\frac{7^{2}}{7^{-2}}

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\frac{7^{6}}{7^{-2}}

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7^{-12}

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\left(7^{2}\right)^{2}

y=|x|

is reflected across the

x

-axis and then scaled vertically by a factor of

\left(\frac{3}{8}\right)

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What is the equation of the new graph?

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1

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y=\left(\frac{8}{3}\right)|x|

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y=-\left(\frac{8}{3}\right)|x|

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y=\left(\frac{3}{8}\right)|x|

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y=-\left(\frac{3}{8}\right)|x|

What is a point-slope equation of the line with slope

-10

that goes through the point

(1,4)

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y+4=-10(x+1)

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y-4=-10(x-1)

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y+4=10(x+1)

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y-4=10(x-1)

y=|x|

is reflected across the

x

-axis and then scaled vertically by a factor of

7

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What is the equation of the new graph?

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1

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y=7|x|

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y=-7|x|

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y=\frac{1}{7}|x|

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y=-\frac{1}{7}|x|

m^{2}+p^{2}=x

4mp=y

, which of the following is equivalent to

4x+2y

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1

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(2m+p)^{2}

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(2m+2p)^{2}

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(4m+2p)^{2}

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(4m+8p)^{2}

What is the discriminant of the quadratic equation

7 x^{2}-7 x-3=0

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-133

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35

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-35

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133

h(n) = -15\cdot 6^{n}

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Complete the recursive formula of

h(n)

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h(1)=\square

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h(n)=h(n-1) \cdot \square

m^{2}+p^{2}=x

4mp=y

, which of the following is equivalent to

4x+2y

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1

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(2m+p)^{2}

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(2m+2p)^{2}

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(4m+2p)^{2}

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(4m+8p)^{2}

|2x-7| \leq 8

, what is the greatest possible value of

|x-7|

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-\frac{1}{2}

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0

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\frac{1}{2}

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\frac{15}{2}

Jonael dropped a sandbag from a hot air balloon at a target that is

250

meters below the balloon. At this moment, the sandbag is

75

meters below the balloon.

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2

of the following expressions represents the distance between the sandbag and the target?

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2

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|-250+75|

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|250+75|

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|-75+250|

Rakesh tried to solve the differential equation

\frac{d y}{d x}=2 y \cdot \cos (x)

. This is his work:

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\frac{d y}{d x}=2 y \cdot \cos (x)

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1

\quad \int \frac{1}{y} d y=\int 2 \cos (x) d x

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2

\quad \ln |y|=2 \sin (x)

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3

\quad e^{\ln |y|}=e^{2 \sin (x)}

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4

\quad|y|=e^{2 \sin (x)}

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5

\quad y= \pm e^{2 \sin (x)}+C

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Is Rakesh's work correct? If not, what is his mistake?

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1

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(A) Rakesh's work is correct.

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1

is incorrect. The separation of variables wasn't done correctly.

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2

is incorrect. The right-hand side of the equation should be

2 \sin (x)+C

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4

e^{\ln |y|}

|y|

Aditya tried to solve the differential equation

\frac{d y}{d x}=6 x-30

. This is his work:

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\frac{d y}{d x}=6 x-30

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1

\quad \int 30 d y=\int 6 x d x

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2

\quad 30 y=3 x^{2}+C

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3

\quad y=\frac{x^{2}}{10}+C

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Is Aditya's work correct? If not, what is his mistake?

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1

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(A) Aditya's work is correct.

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1

is incorrect. The separation of variables wasn't done correctly.

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2

is incorrect. Aditya didn't integrate

30

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3

is incorrect. The right-hand side of the equation should be

\frac{30}{3 x^{2}+C}

Tess tried to solve the differential equation

\frac{d y}{d x}=2 x y-6 x

. This is her work:

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\frac{d y}{d x}=2 x y-6 x

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1

\quad \frac{d y}{d x}=(y-3) 2 x

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2

\quad \int \frac{1}{y-3} d y=\int 2 x d x

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3

\quad \ln |y-3|=x^{2}+C_{1}

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4

\quad e^{\ln |y-3|}=e^{x^{2}}+C_{1}

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5

\quad|y-3|=e^{x^{2}}+C_{1}

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6

\quad y-3= \pm e^{x^{2}}+C_{2}

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7

\quad y= \pm e^{x^{2}}+C

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Is Tess's work correct? If not, what is her mistake?

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1

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(A) Tess's work is correct.

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1

is incorrect. We're not allowed to factor an expression when solving differential equations.

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4

is incorrect. The right-hand side of the equation should be

e^{x^{2}+C_{1}}

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7

is incorrect. Tess didn't account for adding

3

to the right side.

g(x)

g(x)

is the reflection across the

y

f(x) = -4x + 1

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\text{[g(x) = 4x - 1]}

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\text{[g(x) = -4x + 1]}

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\text{[g(x) = 4x + 1]}

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\text{[g(x) = -4x - 1]}