Resources

Resources

Compare linear and exponential growth

7 \square

Mark for Review

\newline

7 m=5(n+p)

\newline

The given equation relates the positive numbers

m, n

p

. Which equation correctly gives

n

m

p

\newline

n=\frac{5 p}{7 m}

\newline

n=\frac{7 m}{5}-p

\newline

n=5(7 m)+p

\newline

n=7 m-5-p

\left\{\begin{array}{l} y=3+4 x \\ y=9+3 x \end{array}\right.

\newline

(8,35)

\newline

When substituting the point into the first equation you get

y=

\square

\newline

Since the point

(8,35)

\newline

Select an answer

\newline

Select an answer

\newline

When substituting the point into the second

\newline

Select an answer

\newline

a solution to the given system.

\newline

equation you get

y=

\square

\newline

Select an answer

\newline

(11,42)

\newline

When substituting the point into the first equation you get

y=

\square

\newline

Select an answer

\newline

When substituting the point into the second equation you get

y=

\square

\newline

Select an answer

\newline

(6,27)

\newline

When substituting the point into the first equation you get

y=

\newline

\square

\newline

Select an answer

\newline

Since the point

(6,27)

\newline

Select an answer

\newline

When substituting the point into the second

\newline

Select an answer

\newline

a solution to the given system.

\newline

equation you get

y=

\square

\newline

Select an answer

Estimate the solution to the system of equations.

\newline

You can use the interactive graph below to find the solution.

\newline

\left\{\begin{array}{l} y=4 x-2 \\ y=x+3 \end{array}\right.

\newline

1

\newline

x=\frac{2}{3}, y=4 \frac{2}{3}

\newline

x=1 \frac{2}{3}, y=3 \frac{2}{3}

\newline

x=\frac{2}{3}, y=3 \frac{2}{3}

\newline

x=1 \frac{2}{3}, y=4 \frac{2}{3}

\newline

y

Estimate the solution to the system of equations.

\newline

You can use the interactive graph below to find the solution.

\newline

\left\{\begin{array}{l} y=4 x-2 \\ y=x+3 \end{array}\right.

\newline

1

\newline

x=\frac{2}{3}, y=4 \frac{2}{3}

\newline

x=1 \frac{2}{3}, y=3 \frac{2}{3}

\newline

x=\frac{2}{3}, y=3 \frac{2}{3}

\newline

x=1 \frac{2}{3}, y=4 \frac{2}{3}

A cereal company wants to enlarge the volume of the cylindrical container used for one of its products by enlarging the radius of the cylinder. The height must be

20

centimeters. The new volume of the cylinder is given by the equation

V(x)=20 \pi(5+x)^{2}

x

is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?

\newline

1

\newline

\pi\left(20 x^{2}+200 x+500\right)

\newline

20 \pi\left(x^{2}+10 x+10\right)+300 \pi

\newline

20 \pi\left(x^{2}+10 x+25\right)

\newline

20 \pi x^{2}+200 \pi x+500 \pi

Determine whether each expression is equivalent to

81^{0.5} t+0.25

\newline

Equivalent Not Equivalent

\newline

\begin{array}{l} 27^{0.5 t+0.25} \cdot 3^{0.5 t+0.25} \\ \frac{27^{t+0.5}}{3^{t+0.5}} \end{array}

\newline

9^{t} \cdot 3

\frac{3.5}{3 p^{3}}+\frac{7}{4 p}

\newline

Which expression is equivalent to the sum for all

p>7

\newline

1

\newline

\frac{24.5}{12 p^{4}}

\newline

\frac{21 p^{2}+14}{12 p^{3}}

\newline

\frac{21 p^{2}-14}{12 p^{3}}

\newline

\frac{10.5 p^{2}-28}{12 p^{3}}

\frac{12 x-x^{2}}{x^{2}-10 x-24}

\newline

Which expression is equivalent to the given expression for all

x<10

\newline

1

\newline

-\frac{x}{x-2}

\newline

\frac{x}{x-2}

\newline

\frac{x}{x+2}

\newline

-\frac{x}{x+2}

p(x)=3x^3-20x^2+37x-20

has a known factor of

(x-4)

p(x)

as a product of linear factors.

p(x)=

Piecewise functions

f

g

can model the heights (in meters) of two airplanes. Here are the graphs of

f

g

t

is the number of minutes that have passed since noon at a local airport.

\small{20}

\small{40}

\small{60}

\small{80}

\small{100}

g

0

g

1

g

2

g

3

g

4

g

5

g

6

g

7

g

8

g

9

f

0

f

1

f

2

t

f

4

f

5

The airplanes have the same height about

f

6

minutes after noon. What is the other time the airplanes have the same height? Round your answer to the nearest ten minutes. About minutes after noon

Khan Academy Donate ^{(

3

)} embongw s blazing!

2

week streak Level

15

111

111

skills Production is profitable only when revenue is greater than cost. The monthly profit of a company selling

x

units is given by the quadratic function:

P(x) = -\frac{1}{200}x^{2} + 30x

Which of the following equivalent expressions displays the break-even point as a constant or coefficient? Choose

1

(A) -\frac{1}{200}((x-3,000)^{2}-9,000,000)

(B) -\frac{1}{2.00}(x-3,000)^{2}+45,000

13

13

Skip Check ENG US (j))

2024

-0

Rafael has a solar-powered video game console. He saw and modeled the following relationships. Function Input Output D The time of year,

x

Hours of daylight,

D(x)

E Hours of daylight,

k

Amount of solar energy gained over the day,

E(k)

G Amount of solar energy,

r

Hours of video games Rafael can play,

G(r)

2

of the following composed functions make sense in this context? Choose

2

G(E(k))

E(D(x))

D(E(k))

5

E(G(r))

Rafael has a solar-powered video game console. He saw and modeled the following relationships.

\newline

\begin{tabular}{lll}

\newline

Function & Input & Output \\

\newline

D

& The time of year,

x

& Hours of daylight,

D(x)

\newline

E

& Hours of daylight,

k

& Amount of solar energy gained over the day,

E(k)

\newline

G

& Amount of solar energy,

r

& Hours of video games Rafael can play,

G(r)

\newline

\newline

2

of the following composed functions make sense in this context?

\newline

2

\newline

G(E(k))

\newline

8

x

0

\newline

x

1

\newline

x

2

100

1

1

3

t} Which of the following is an equivalent form of the function

f

in which the exponent is

t

1

1

answer: (Choice A)

f(t)=100(1.331)^t

f(t)=100(1.331)^t

f(t)=100(3.3)^t

f(t)=100(3.3)^t

f(t)=100(4.1)^{t}

f(t)=100(4.1)^{t}

f(t)=300(1.1)^{t}

f(t)=300(1.1)^{t}

\begin{array}{l}g(x)=\left\{\begin{array}{ll}x^{2}-5 & , \quad x \in(-\infty,-7) \\ 9 x-17 & , \quad x \in[-7,2] \\ (x+1)(x-5) & , \quad x \in(2, \infty)\end{array}\right. \\ g(7)= \\end{array}

\newline

35

\newline

40

\newline

45

\newline

50

\newline

10,13,16,19, \ldots

\newline

He comes up with

s(n)=7+3 n

\newline

What domain should David use for

s

so it generates the sequence?

\newline

1

\newline

n \geq 0

n

\newline

n \geq 0

n

\newline

n \geq 1

n

\newline

n \geq 1

n

Google Classroom

\newline

Microsoft Teams

\newline

\begin{array}{l} p(n)=\left(n^{3}-12 n^{2}+47 n-60\right)(n-4) \\ q(n)=(n+13)(n-4) \end{array}

\newline

p

q

are defined. One of the functions has a zero at

n=5

. What is the value of

p(5)+q(5)

\newline

\square

\newline

a=2200

\newline

\mathrm{sec} \quad 2250

\newline

=+230

\newline

200,250,300,350 \ldots

\newline

\begin{aligned} a & =2 a \quad d=50 \\ n & =30 \quad 5 n \\ S_{n} & =\frac{n}{2}(2 a+(n-1, d) \end{aligned}

\newline

200

\newline

250

\newline

300

\newline

350

used for one of its products by enlarging the radius of the cylinder. The height must be

20

centimeters. The new volume of the cylinder is given by the equation

V(x)=20 \pi(5+x)^{2}

x

is the additional length of the radius in centimeters. Which of the following equivalent expressions displays the current volume of the cylinder as a constant or coefficient?

\newline

\pi\left(20 x^{2}+200 x+500\right)

\newline

20 \pi\left(x^{2}+10 x+10\right)+300 \pi

\newline

20 \pi\left(x^{2}+10 x+25\right)

\newline

20 \pi x^{2}+200 \pi x+500 \pi

9

\newline

Mark for Review

\newline

\begin{array}{c} y=x+9 \\ y=x^{2}+16 x+63 \end{array}

\newline

A solution to the given system of equations is

(x, y)

. What is the greatest possible value of

x

\newline

-6

\newline

7

\newline

9

\newline

63

Fred is \\[\(4\)\\] times as old as Nathan and is also \\[\(27\)\\] years older than Nathan. Let \\[f\\] be Fred's age and let \\[n\\] be Nathan's age. Which system of equations represents this situation? Choose \(1\) answer: Choose \(1\) answer:\(\newline\)(Choice A) \\[\begin{cases} n=\(4\)f \\\\ n=f+\(27\) \end{cases}\\] A \\[\begin{cases} n=\(4\)f \\\\ n=f+\(27\) \end{cases}\\]\(\newline\)(Choice B) \\[\begin{cases} \(4\)f=n \\\\ f=n+\(27\) \end{cases}\\] B \\[\begin{cases} \(4\)f=n \\\\ f=n+\(27\) \end{cases}\\]\(\newline\)(Choice C) \\[\begin{cases} f=\(4\)n \\\\ f=n+\(27\) \end{cases}\\] C \\[\begin{cases} f=\(4\)n \\\\ f=n+\(27\) \end{cases}\\]\(\newline\)(Choice D) \\[\begin{cases} f=\(4\)n \\\\ f=n\(-27\) \end{cases}\\] D \\[\begin{cases} f=\(4\)n \\\\ f=n\(-27\) \end{cases}\\]

\newline

\begin{array}{l} y=\frac{1}{4} x+2 \\ y=-x^{2}+4 x+3 \\ x^{2}+y^{2}=25 \end{array}

\newline

A system of three equations and their graphs in the

x y

-plane are shown. How many solutions does the system have?

\newline

1

\newline

\newline

\newline

\newline

Which of the following values is equal to the value?

\newline

1

\newline

2^{407}

\newline

3\left(2^{203}\right)

\newline

3\left(2^{204}\right)

\newline

2^{202}+2^{205}

You might need: 囲 Calculator

\newline

Jacob distributed a survey to his fellow students asking them how many hours they spent on the Internet in the past day. He also asked them to rate their mood on a scale from

0

10

10

being the happiest. A line was fit to the data to model the relationship.

\newline

Which of these linear equations best describes the given model?

\newline

1

\newline

\hat{y}=x+7.5

\newline

\hat{y}=-x+7.5

\newline

\hat{y}=-\frac{1}{2} x+7.5

\newline

Based on this equation, estimate the mood rating for a student that spent

5

5

\newline

\square

AD

PM

are the medians of triangles

\triangle ABC

\triangle PQR

, respectively where

\triangle ABC \sim \triangle PQR

\frac{AB}{PQ} = \frac{AD}{PM}

10

. Find the unknown entries

a, b, c

d

in the following distribution of heights of students in the class. Height (in cm)

f

cf

150

-155

PM

0

12

155

-160

13

PM

1

160

-165

10

35

165

-170

PM

2

43

170

-175

7

d

PM

4

Solve the following system of equations by elimination method:

PM

5

PM

6

2

2

\begin{aligned} t-5 & =\frac{1}{3}(u+15)-5 \\ t & =\frac{2}{3}(u+9) \end{aligned}

\newline

Which of the following accurately describes all solutions to the system of equations shown?

\newline

1

\newline

u=-18

t=-6

\newline

u=-3

t=4

\newline

(C) There are infinite solutions to the system.

\newline

(D) There are no solutions to the system.

\newline

Show calculator

\newline

Solve in removal method.

\newline

\begin{array}{l} 3 x+8 y=15 \\ 2 x-8 y=10 \\ x=\square \\ y=\square \end{array}

Solve the pair of equations with

\newline

\begin{array}{l} \left\{\begin{array}{l} \frac{1}{y}+\frac{1}{2 x}=1 \\ \frac{2}{5 y}+\frac{4}{5 x}=1 \end{array}\right. \\ x=\square \\ y=\square \end{array}

/ \times 6418 \mathrm{~b} 49 \mathrm{df} b \mathrm{c} 9 \mathrm{~d} 0 \mathrm{c}

:quadratic-quations-part

2

6418

49

9

0

9

:apply-quadratio-formula-part

2

\newline

\newline

\newline

\newline

3754

\newline

Tori and Gavin were trying to solve the equation:

\newline

(x+1)^{2}-3=13

\newline

\frac{3}{x^{2}-x-20}+\frac{1}{x^{2}+9 x+20}

\newline

Which expression is equivalent to the sum for all

x>5

\newline

1

\newline

\frac{4}{2 x^{2}+8 x}

\newline

\frac{4 x+10}{\left(x^{2}-25\right)(x+4)}

\newline

\frac{4 x+10}{(x-5)^{2}(x+4)}

\newline

\frac{4 x+8}{\left(x^{2}-16\right)(x+5)}

\newline

\newline

x

. Enter the solution

\newline

\begin{array}{l} 2 x^{2}-24 x+54=0 \\ \text { lesser } x=\square \\ \text { greater } x=\square \end{array}

Operations with rational expres

3

7

:advanced-math-easier/x

0

98

58

3

7

:operations-with-rational-expressions-ea... Khan Academy Donate [(u+

7

7

-7

7

) Which of the following is equivalent to the given expression? Choose

1

answer: (A) (u-u)/(

7

7

7

7

7

7

-7

7

4

\newline

Google Classroom

\newline

Microsoft Teams

\newline

Nate predicted that voter turnout for the next citywide election will be

23 \%

, with a margin of error of

3 \%

. Which of the following represents the plausible range of voter turnout based on Nate's prediction and margin of error?

\newline

1

\newline

20 \%

26 \%

\newline

21.5 \%

24.5 \%

\newline

23 \%

26 \%

\newline

23 \%

29 \%

The equations below relate the distinct positive reabrumbers

a

b

c

d

. Which equation correctly expresses

b

d

\newline

\begin{cases} c = 2a^{2} \ a^{3b} = \left(\frac{c}{2}\right)^{d+1} \end{cases}

The equations below relate the distinct positive reabnumbers

a, b, c

d

. Which equation correctly expresses

b

d

\newline

\begin{aligned} c & =2 a^{2} \\ a^{3 b} & =\left(\frac{c}{2}\right)^{d+1} \end{aligned}

Give the following system of linear equations:

\newline

\left\{\begin{aligned} -21 x_{1}-73 x_{2} & =-9 \\ 69 x_{2}-28 x_{3} & =45 \\ 39 x_{1}-58 x_{3} & =-81 \end{aligned}\right.

\newline

in matrix form:

\newline

\boldsymbol{A} \boldsymbol{x}=\boldsymbol{b} \Longleftrightarrow\left(\begin{array}{c|c|c} \square & \\ \square & & \\ \square \end{array}\right) \cdot\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right)=\left(\begin{array}{l} \square \\ \square \\ \hline \end{array}\right.

Solving systems of linear equations: advanced

\newline

\begin{aligned} x-y & =y-4+2(4.5-2 x) \\ 4(x+2 y) & =-p+7 y \end{aligned}

\newline

In the system of equations,

p

is a constant. For which value of

p

is there exactly one solution

(x, y)

x=-1

\newline

1

\newline

-5

\newline

9

\newline

(c) Any real number

\newline

D) None of the above

\newline

Find the solution to the system of equations.

\newline

You can use the interactive graph below to find the solution.

\newline

\left\{\begin{array}{l} y=x-4 \\ y=4 x+2 \end{array}\right.

\newline

x=

\square

\newline

y=

\square

y

sixth-grade-math/cc

-6

th-ratios-prop-topic/cc

-6

th-ratio-word-problems/test/cc

-6

th-ratios-prop-topic-..

\newline

Salem ortho Email

\newline

Arabic Keyboard

{ }^{2}

\newline

\newline

3

\newline

\newline

\newline

6 \because

\newline

\newline

Tiriaq is making ornaments. They need

7

6

ribbons to make one ornament.

\newline

Complete the double number line to show the other values of beads and ribbons.

\newline

1

\newline

\newline

\begin{array}{lllll} n & 7 & 14 & 91 & 98 \end{array}

\newline

2

11

\newline

\newline

\newline

12.42 \mathrm{PM}

\newline

6 / 17 / 2024

(p \circ q)(x)

(q \circ p)(x)

\newline

\begin{array}{l} p(x)=x^{2} \\ q(x)=x+3 \end{array}

\newline

Write your answer as a polynomial in simplest form.

\newline

\begin{array}{l} (p \circ q)(x)=\square \\ (q \circ p)(x)=\square \end{array}

1

1

shows part of the curve with equation

y = e^{\frac{1}{3^2}}

x \geq 0

The finite region

R

, shown shaded in Figure

1

, is bounded by the curve, the

y

x

-axis, and the line with equation

x = 2

The table below shows corresponding values of

x

y

y = e^{\frac{1}{x^2}}

\newline

x

\newline

0

\newline

0

5

\newline

1

\newline

1

5

\newline

2

\newline

y

\newline

1

\newline

x \geq 0

1

\newline

x \geq 0

2

\newline

x \geq 0

3

\newline

x \geq 0

4

\newline

(a) Use the trapezium rule, with all the values of

y

in the table, to find an estimate for the area of

R

, giving your answer to

2

decimal places. (

3

) (b) Use your answer to part (a) to deduce an estimate for (i)

x \geq 0

7

x \geq 0

8

giving your answers to

2

decimal places. a)

\begin{cases}0.5(8w+2v)=3\8w=2-v+4w\end{cases}\Which of the following accurately describes all solutions to the system of equations shown?

1) \$v=1

w=\frac{1}{4}

2) \$v=4

w=\frac{1}{4}

\(3) There are infinite solutions to the system.\(4) There are no solutions to the system.

33

In which example is the current

1.0\,\text{A}

\newline

10\,\text{V}

lamp with input power

0.10\,\text{W}

\newline

I^{2}

I

\newline

B a resistor of resistance

0.50\,\Omega

with a potential difference (P.d.) of

2.0\,\text{V}

\newline

C two resistors with a current of

0.50\,\text{A}

connected in series

Direction (Q. Nos.

25

-30

) This section contains

6

questions. When fom

0

9

(both inclusive).

\newline

25

f:[2, \infty) \rightarrow Y

f(x)==x^{2}-4 x+5

is both one-one and onto, if

Y \in[a, \infty)

, then the value of

a

\newline

26

f(x)=\left(\frac{4 a-7}{3}\right) x^{3}+(a-3) x^{2}+x+5

is one-one function, where

a \in[\lambda, \mu]

, then the value of

|\lambda-\mu|

\newline

27

f

be a one-one function with domain

\{x, y, z\}

\{1,2,3\}

. It is given the exactly one of the following statements is true and remaining two are false,

f(x)==x^{2}-4 x+5

0

f(x)==x^{2}-4 x+5

1

\newline

28

f(x)==x^{2}-4 x+5

2

, number of functions from

f(x)==x^{2}-4 x+5

3

f(x)==x^{2}-4 x+5

4

such that range contains exactly

3

f(x)==x^{2}-4 x+5

5

, then the value of

f(x)==x^{2}-4 x+5

6

\newline

29

f(x)==x^{2}-4 x+5

7

f(x)==x^{2}-4 x+5

8

\newline

30

f(x)==x^{2}-4 x+5

9

be a polynomial of degree

4

with leading coefficient

1

Y \in[a, \infty)

0

Y \in[a, \infty)

1

Y \in[a, \infty)

2

, then the value of

Y \in[a, \infty)

3

Write the equation

2(x+3)=2 x+6

as a system of equations.

\newline

\begin{array}{l} y=\square x+\square \newline y=\square x+\square \end{array}

Graphs and Functlons

\newline

Combining functions: Advanced

\newline

Suppose that the functions

f

g

are defined as follows.

\newline

\begin{array}{l} f(x)=\sqrt{x+4} \\ g(x)=5 x^{2}+2 \end{array}

\newline

f \cdot g

f+g

. Then, give their domains using interval notation.

\newline

(f \cdot g)(x)=

\newline

f \cdot g:

\newline

(f+g)(x)=

\newline

f+g

Graphs and Functions

\newline

Combining functions: Advanced

\newline

Suppose that the functions

f

g

are defined as follows.

\newline

\begin{array}{l} f(x)=\sqrt{4 x-5} \\ g(x)=x-4 \end{array}

\newline

f \cdot g

f+g

. Then, give their domains using interval notation.

\newline

(f \cdot g)(x)=

\newline

\square

\newline

f \cdot g:

\square

\newline

(f+g)(x)=

\newline

\square

\newline

f+g:

\square

Compotition of two finctions: Adrenced

\newline

For the real-yalued functions

f(x)=\sqrt{3 x+30}

g(x)=x-5

, find the composition

f \circ g

and specfy its domain using interyel notation

\newline

(f \circ z)(x)=\square

\newline

f \circ g:

\newline

\begin{tabular}{|ccc}

\newline

\frac{}{}

^{P}

\sqrt{\square}

\newline

(0,, \square)

[0, \square]

\newline

(0, \square]

g(x)=x-5

0

\newline

g(x)=x-5

1

g(x)=x-5

2

g(x)=x-5

3

\newline

g(x)=x-5

4

3

\newline

\newline

Suppose that the functions

f

g

are defined as follows.

\newline

\begin{array}{l} f(x)=\frac{9}{2 x}, x \neq 0 \\ g(x)=5 x-4 \end{array}

\newline

Find the compositions

f \circ f

g \circ g

\newline

Simplify your answers as much as possible.

\newline

(Assume that your expressions are defined for all

x

\newline

\begin{array}{l} (f \circ f)(x)=\square \\ (g \circ g)(x)=\square \end{array}

\newline

\begin{array}{l} f(x)=\frac{1}{6 x}, x \neq 0 . \\ g(x)=\frac{1}{6 x}, x \neq 0 . \\ f(g(x))=\square \\ g(f(x))=\square \end{array}

\newline

f

g

are inverses of each other

\newline

f

g

are not inverses of each other

\newline

f(x)=x+2

\newline

\begin{array}{l} g(x)=x+2 \\ f(g(x))=\square \\ g(f(x))=\square \end{array}

\newline

f_{\text {and }} \boldsymbol{g}

are inverses of each other

\newline

f

g

are not inverses of each other

...