Tüm alıştırma soruları

Soru 1801Soru →

In a certain company, the number of entry-level employees is $60\%$ of the total number of employees, and the remaining employees are managers. Over a year, the number of entry-level employees increases by $x\%$ , and the number of managers decreases by $y\%$ . As a result, the total number of employees in the company decreases by $4\%$ , and the ratio of the number of managers to the number of entry-level employees becomes $1$ to $3$ . What is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

x

is

20

.

To find the value of

x

, represent the initial and final quantities of entry-level employees and managers algebraically. Using the

4\%

decrease in total employees and the final manager-to-entry-level ratio of

1

to

3

, set up a system of linear equations:

2y - 3x = 20

and

2y = 100 - x

. Solving this system for

x

yields

20

.

Adım Adım Çözüm

1

Represent the initial employee counts algebraically.

Initial entry-level employees =

0.60T

, initial managers =

0.40T

, where

T

is the initial total.

Establishing initial quantities based on the given percentages.

2

Represent the final employee counts after percent changes.

Final entry-level =

0.60T(1 + 0.01x)

, final managers =

0.40T(1 - 0.01y)

.

Applying the percent increase and decrease to the respective initial quantities.

3

Use the total percent change to create a linear equation.

2y - 3x = 20

Setting the sum of the final counts equal to

0.96T

and simplifying.

4

Use the final ratio of managers to entry-level employees to create a second equation.

2y = 100 - x

Setting the ratio of final managers to final entry-level employees equal to

1/3

and simplifying.

5

Solve the system of equations for

x

.

x = 20

Substituting the second equation into the first to solve for the target variable.

Anahtar Kavram

Percents and Percent Change

Soru 1802Soru →

An agricultural drone is used to spray liquid fertilizer on a field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, can be modeled by the equation $F = 120 - 4.8t$ , where $t$ is the number of minutes after the drone begins spraying. What is the best interpretation of the number $4.8$ in this context?

The volume of fertilizer, in liters, in the drone's tank before the spraying operation begins.

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

The number of minutes it takes for the drone to spray one liter of fertilizer.

The number of minutes the drone can spray fertilizer before the tank is completely empty.

Cevabı ve açıklamayı göster

Cevap: The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

Cevap

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

In the linear equation

F = 120 - 4.8t

, the variable

F

represents the remaining fertilizer in liters, and

t

represents the elapsed time in minutes. The value 4.8 is the coefficient of

t

, which represents the rate of change of the fertilizer in the tank. The negative sign in front of 4.8 indicates a decrease over time, meaning that the drone sprays 4.8 liters of fertilizer from the tank each minute.

Adım Adım Çözüm

1

Identify the role of the constant 4.8 in the given linear equation

F = 120 - 4.8t

.

The coefficient of the independent variable

t

is

-4.8

, which represents the slope of the linear relationship.

In the slope-intercept form of a linear equation, the coefficient of the variable represents the rate of change (slope).

2

Interpret the meaning of the slope in the context of the problem.

Since

F

is measured in liters and

t

is measured in minutes, the slope represents a change in liters per minute. The negative sign indicates that the volume of fertilizer is decreasing.

The rate of change is the change in the dependent variable (liters of fertilizer) divided by the change in the independent variable (minutes).

3

Select the option that correctly describes this rate of change.

The rate of decrease is 4.8 liters per minute, which means 4.8 liters of fertilizer are sprayed from the tank each minute.

This matches the definition of the rate at which fertilizer is leaving the tank.

Anahtar Kavram

Interpreting the slope of a linear equation in a real-world context.

Soru 1803Soru →

A city's annual budget is divided into education, public safety, and other expenses. In 2025, the budget allocated for education was $12$ million dollars. For 2026, the education budget is increased by $15\%$ , and the public safety budget is decreased by $8\%$ from its 2025 value. If the public safety budget in 2026 is equal to the education budget in 2026, what was the public safety budget, in millions of dollars, in 2025?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The public safety budget in 2025 was

15

million dollars.

To find the 2025 public safety budget, we first determine the 2026 education budget. A

15\%

increase on the 2025 education budget of

12

million yields

12 \times 1.15 = 13.8

million. Since the 2026 public safety budget is equal to this amount, it is also

13.8

million. Given that the 2026 public safety budget is an

8\%

decrease from its 2025 value (

P

), we set up the equation

P \times (1 - 0.08) = 13.8

. Solving for

P

gives

P = \frac{13.8}{0.92} = 15

million.

Adım Adım Çözüm

1

Calculate the education budget for 2026.

The education budget in 2026 is

13.8

million dollars.

The education budget in 2026 is a

15\%

increase from the 2025 education budget of

12

million dollars:

12 \times (1 + 0.15) = 12 \times 1.15 = 13.8

million dollars.

2

Express the relationship between the 2025 and 2026 public safety budgets.

0.92 \times P = 13.8

, where

P

is the 2025 public safety budget.

The public safety budget in 2026 is equal to the education budget in 2026 (

13.8

million dollars) and represents an

8\%

decrease from the 2025 public safety budget

P

:

P \times (1 - 0.08) = 13.8

.

3

Solve for the 2025 public safety budget

P

.

P = 15

million dollars.

Dividing both sides of the equation by

0.92

yields

P = \frac{13.8}{0.92} = 15

million dollars.

Anahtar Kavram

Calculating initial values after a percentage change

Alternatif Yöntem

Instead of converting to decimals immediately, you can use fractions:

12 \times \frac{115}{100} = 12 \times \frac{23}{20} = \frac{69}{5} = 13.8

. Let

P

be the 2025 public safety budget. Then

\frac{92}{100} P = 13.8

, so

P = 13.8 \times \frac{100}{92} = \frac{1380}{92} = 15

.

Tahmini Süre:1m 30s

Soru 1804Soru →

For the function $f$ , it is given that $f(4) = 18$ . The function $g$ is defined by $g(x) = \frac{1}{3}f(x + 2)$ . What is the value of $g(2)$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

g(2)

is 6.

To evaluate

g(2)

, substitute

x = 2

into the definition

g(x) = \frac{1}{3}f(x+2)

, which yields

g(2) = \frac{1}{3}f(2+2) = \frac{1}{3}f(4)

. Since

f(4) = 18

, this simplifies to

\frac{1}{3}(18) = 6

.

Adım Adım Çözüm

1

Substitute

x = 2

into the expression for

g(x)

.

g(2) = \frac{1}{3}f(2 + 2)

To evaluate the function

g

at

x = 2

, we substitute 2 for every occurrence of

x

in the function definition.

2

Simplify the input argument for the function

f

.

g(2) = \frac{1}{3}f(4)

Adding 2 and 2 inside the parentheses simplifies the input of

f

to 4.

3

Substitute the given value of

f(4)

into the simplified expression.

g(2) = \frac{1}{3}(18)

The problem states that

f(4) = 18

.

4

Perform the final multiplication.

6

Multiplying 18 by

\frac{1}{3}

is equivalent to dividing 18 by 3, which yields 6.

Anahtar Kavram

Evaluating a transformed function by substituting a value into function notation.

Tahmini Süre:45s

Soru 1805Soru →

In temperate forest ecosystems, subterranean mycorrhizal networks composed of vast fungal mycelia connect the root systems of geographically distant trees. Through these subterranean highways, mature canopy trees can share vital nutrients like carbon and phosphorus with struggling understory seedlings ______ this cooperative resource distribution significantly increases the health and overall survival rate of the entire forest community.

Which choice completes the text so that it conforms to the conventions of Standard English?

seedlings; this

seedlings, this

seedlings this

seedlings, but this

Cevabı ve açıklamayı göster

Cevap: seedlings; this

Cevap

seedlings; this

The option using a semicolon correctly links two independent clauses. The clause before the blank is independent, and the clause after the blank is also independent. A semicolon is a standard punctuation mark used to connect two closely related independent clauses without a coordinating conjunction.

Adım Adım Çözüm

1

Analyze the structure of the two clauses surrounding the blank.

The clause before the blank ('Through these subterranean highways... seedlings') is independent because it contains a subject and a verb and expresses a complete thought. The clause after the blank ('this cooperative resource distribution... community') is also independent.

Identifying the clause boundaries helps determine what punctuation or conjunction is needed to link them.

2

Evaluate how independent clauses can be joined.

Two independent clauses can be joined by a semicolon, a period, or a comma coupled with a coordinating conjunction (like 'and'). They cannot be joined with just a comma or no punctuation at all.

This narrows down the choices to grammatically correct linking methods.

3

Select the option that conforms to standard English conventions and maintains logical flow.

The option containing a semicolon correctly separates the two clauses. The option with 'but' is grammatically possible but logically incorrect because the relationship between sharing nutrients and increasing survival is supportive, not contrastive.

The correct answer must be both grammatically valid and contextually logical.

Anahtar Kavram

Clause Boundaries and Linking

Soru 1806Soru →

Consider the system of equations below:

\begin{aligned} y &= 3x^2 - 5x - 4 \\ y &= x^2 - 2x + 5 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $x > 0$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

y

is 8.

By setting the two equations equal to each other, we obtain

3x^2 - 5x - 4 = x^2 - 2x + 5

. Simplifying this equation by moving all terms to one side yields

2x^2 - 3x - 9 = 0

. Factoring this quadratic equation gives

(2x + 3)(x - 3) = 0

, which has solutions

x = -1.5

and

x = 3

. Since the problem specifies that

x > 0

, we must use

x = 3

. Substituting

x = 3

into the second equation, we find

y = (3)^2 - 2(3) + 5 = 8

. Substituting into the first equation also yields

y = 3(3)^2 - 5(3) - 4 = 8

. Therefore, the value of

y

is 8.

Adım Adım Çözüm

1

Set the quadratic expressions equal to each other.

3x^2 - 5x - 4 = x^2 - 2x + 5

Since both equations are solved for

y

, their right-hand sides must be equal at any point of intersection.

2

Rearrange the terms to set the quadratic equation to zero.

2x^2 - 3x - 9 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

allows us to solve it by factoring.

3

Factor the quadratic expression to find the roots.

(2x + 3)(x - 3) = 0

, which yields

x = -1.5

or

x = 3

.

Factoring shows the values of

x

that satisfy the system.

4

Select the positive root and substitute it back to find

y

.

y = 8

The problem specifies

x > 0

, so we use

x = 3

. Substituting

x = 3

into

y = x^2 - 2x + 5

gives the corresponding

y

-value.

Anahtar Kavram

Solving a system of nonlinear equations by setting the equations equal to each other and solving the resulting quadratic equation.

Soru 1807Soru →

An environmental study monitors the populations of two different plant species in a conservation area. The population of Species A is modeled by a linear function, $A(t) = 120 + 15t$ , where $t$ represents the number of years since the start of the study. The population of Species B is modeled by an exponential function, $B(t) = c \cdot d^t$ , where $c$ and $d$ are constants. At the start of the study ( $t = 0$ ), the population of Species A is $4$ times the population of Species B. After $2$ years, the population of Species A is equal to the population of Species B. What is the population of Species B after $4$ years?

Cevabı ve açıklamayı göster

Cevap: 750

Cevap

The population of Species B after

4

years is

750

.

To find the population of Species B after

4

years, we evaluate the models at the given points. At

t = 0

,

A(0) = 120

. Since the population of Species A is

4

times that of Species B at

t = 0

, the initial population of Species B is

30

, which gives

c = 30

. At

t = 2

,

A(2) = 120 + 15(2) = 150

. Since the populations are equal at

t = 2

, we have

B(2) = 30 \cdot d^2 = 150

, which simplifies to

d^2 = 5

. The population of Species B at

t = 4

is given by

B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2 = 750

.

Adım Adım Çözüm

1

Find the population of Species A at the start of the study (

t = 0

)

A(0) = 120

This establishes the baseline population of Species A to find the corresponding initial population of Species B.

2

Determine the constant

c

, which represents the initial population of Species B

c = 30

Since the population of Species A is

4

times that of Species B at

t = 0

, we solve

120 = 4c

.

3

Calculate the population of Species A after

2

years (

t = 2

)

A(2) = 150

This value is needed because the population of Species B equals the population of Species A at

t = 2

.

4

Solve for the growth factor term

d^2

d^2 = 5

Using the equality

B(2) = 150

, we solve

30 \cdot d^2 = 150

.

5

Calculate the population of Species B after

4

years (

t = 4

)

B(4) = 750

Using the exponential model

B(t) = 30 \cdot d^t

, we find

B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2

.

Anahtar Kavram

Solving systems involving linear and exponential models using initial conditions and key points.

Soru 1808Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 9)$ . What is the positive difference between the two $x$ -intercepts of the graph?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The correct answer is 6.

The quadratic function can be represented in vertex form as

f(x) = -(x - 4)^2 + 9

since the leading coefficient is

-1

and the vertex is at

(4, 9)

. Setting the function equal to zero to find the

x

-intercepts gives

(x - 4)^2 = 9

, which yields

x = 7

and

x = 1

. The positive difference between these intercepts is

7 - 1 = 6

.

Adım Adım Çözüm

1

Write the quadratic function in vertex form using the given vertex

(4, 9)

and the leading coefficient.

f(x) = -(x - 4)^2 + 9

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex. Since the coefficient of

x^2

is

-1

, we have

a = -1

,

h = 4

, and

k = 9

.

2

Set

f(x) = 0

to find the

x

-intercepts.

(x - 4)^2 = 9

The

x

-intercepts of a graph are the points where

f(x) = 0

.

3

Solve for

x

.

x = 7

and

x = 1

Taking the square root of both sides gives

x - 4 = \pm 3

, which results in

x = 7

and

x = 1

.

4

Find the positive difference between the two

x

-intercepts.

6

Subtract the smaller

x

-intercept from the larger

x

-intercept:

7 - 1 = 6

.

Anahtar Kavram

Finding the intercepts of a quadratic function using its vertex form

Alternatif Yöntem

Alternatively, expand the vertex form

f(x) = -(x - 4)^2 + 9

to get

f(x) = -(x^2 - 8x + 16) + 9 = -x^2 + 8x - 7

. Factoring this expression gives

f(x) = -(x - 7)(x - 1)

. The roots are

x = 7

and

x = 1

, and their difference is

7 - 1 = 6

.

Tahmini Süre:1m 30s

Soru 1809Soru →

A parabola in the $xy$ -plane has vertex $(2, 11)$ and passes through the point $(5, -7)$ . The equation of the parabola is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The value of

a + b + c

is

9

.

The correct value of

a + b + c

is

9

. This is found by writing the parabola's equation in vertex form as

y = -2(x - 2)^2 + 11

and expanding it to standard form

y = -2x^2 + 8x + 3

, which gives the coefficients

a = -2

,

b = 8

, and

c = 3

. Alternatively, substituting

x = 1

directly into the vertex form gives

f(1) = a(1)^2 + b(1) + c = -2(1 - 2)^2 + 11 = 9

.

Adım Adım Çözüm

1

Write the equation of the parabola in vertex form using the given vertex

(2, 11)

.

y = a(x - 2)^2 + 11

The vertex form of a quadratic function is

y = a(x - h)^2 + k

, where

(h, k)

is the vertex of the parabola.

2

Substitute the coordinates of the point

(5, -7)

into the vertex form equation to solve for the constant

a

.

a = -2

Since the point

(5, -7)

lies on the parabola, substituting

x = 5

and

y = -7

allows us to solve the linear equation

-7 = a(5 - 2)^2 + 11

for

a

.

3

Expand the vertex form equation

y = -2(x - 2)^2 + 11

into standard form

y = ax^2 + bx + c

to identify the coefficients

a

,

b

, and

c

.

y = -2x^2 + 8x + 3

, which gives

a = -2

,

b = 8

, and

c = 3

.

Expanding the squared term and distributing the coefficient

-2

converts the equation to standard form, making it easy to read off the coefficients.

4

Calculate the sum of the coefficients

a + b + c

.

a + b + c = 9

Adding the identified coefficients:

-2 + 8 + 3 = 9

.

Anahtar Kavram

Writing and converting quadratic functions between vertex form

y = a(x - h)^2 + k

and standard form

y = ax^2 + bx + c

.

Alternatif Yöntem

Instead of expanding the vertex form equation to find the individual coefficients

a

,

b

, and

c

, recognize that the expression

a + b + c

is equal to

f(1)

for the quadratic function

f(x) = ax^2 + bx + c

. After finding

a = -2

using the vertex form

f(x) = a(x - 2)^2 + 11

, directly evaluate

f(1) = -2(1 - 2)^2 + 11 = 9

.

Tahmini Süre:2m 30s

Soru 1810Soru →

During a weekend sale, a store reduced the price of a bicycle by $20\%$ . If the sale price of the bicycle is $\$160$ , what was the original price, in dollars, of the bicycle?

\$128

\$180

\$192

\$200

Cevabı ve açıklamayı göster

Cevap: $\$200$

Cevap

The original price of the bicycle was

\$200

.

The correct answer is

\$200

. A discount of

20\%

means that the sale price of

\$160

is

80\%

of the original price (

100\% - 20\% = 80\%

). Setting up the equation

0.80 \times P = 160

, where

P

is the original price, and solving for

P

gives

P = \frac{160}{0.80} = 200

.

Adım Adım Çözüm

1

Express the relationship between the original price and the sale price using the discount percentage.

The sale price represents

80\%

of the original price because a

20\%

discount was applied (

100\% - 20\% = 80\%

).

Establishing the percentage value of the sale price relative to the original price is necessary to set up the equation.

2

Set up an equation with the original price, represented by

P

.

0.80 \times P = 160

Since the sale price is

80\%

of the original price, multiplying the original price by

0.80

must equal the sale price of

\$160

.

3

Solve for the original price,

P

, by dividing both sides of the equation by

0.80

.

P = \frac{160}{0.80} = 200

Dividing the sale price by the decimal representation of its percentage of the original price isolates the variable and calculates the original price.

Anahtar Kavram

Percents and Percent Change

Soru 1811Soru →

\begin{aligned} k x - 5y &= 8 \\ 3x - 2y &= 12 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has no solution, what is the value of $k$ ?

\frac{6}{5}

\frac{10}{3}

\frac{15}{2}

-\frac{15}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{15}{2}$

Cevap

\frac{15}{2}

The correct answer is the value that makes the two lines parallel. Rewriting both equations in slope-intercept form gives the slopes

\frac{k}{5}

and

\frac{3}{2}

. Equating these slopes yields

\frac{k}{5} = \frac{3}{2}

, which simplifies to

\frac{15}{2}

. Since their

y

-intercepts (

-\frac{8}{5}

and

-6

) are distinct, the lines are parallel and never intersect, meaning the system has no solution.

Adım Adım Çözüm

1

Convert the first equation to slope-intercept form (

y = mx + b

).

kx - 5y = 8 \implies -5y = -kx + 8 \implies y = \frac{k}{5}x - \frac{8}{5}

This identifies the slope (

m_1 = \frac{k}{5}

) and the

y

-intercept (

b_1 = -\frac{8}{5}

) of the first line.

2

Convert the second equation to slope-intercept form.

3x - 2y = 12 \implies -2y = -3x + 12 \implies y = \frac{3}{2}x - 6

This identifies the slope (

m_2 = \frac{3}{2}

) and the

y

-intercept (

b_2 = -6

) of the second line.

3

Equate the two slopes to find the value of

k

that makes the lines parallel.

\frac{k}{5} = \frac{3}{2} \implies k = 5 \times \frac{3}{2} = \frac{15}{2}

A system of linear equations has no solution if the lines are parallel (equal slopes) and distinct (different

y

-intercepts). Since the

y

-intercepts

-\frac{8}{5}

and

-6

are different, setting the slopes equal guarantees no solution.

Anahtar Kavram

A system of two linear equations has no solution if the lines represented by the equations have the same slope but different

y

-intercepts (parallel lines).

Tahmini Süre:1m 30s

Soru 1812Soru →

The polynomial function $p$ is defined by $p(x) = x^3 + 4x^2 - 7x - 10$ . If $p(2) = 0$ , which of the following expressions must be a factor of $p(x)$ ?

x + 2

x - 2

x - 1

x - 5

Cevabı ve açıklamayı göster

Cevap: $x - 2$

Cevap

The expression

x - 2

must be a factor of

p(x)

.

According to the Factor Theorem, if a polynomial

p(x)

evaluates to

0

at a certain value

x = c

(meaning

p(c) = 0

), then

(x - c)

is a factor of the polynomial. Since we are given that

p(2) = 0

, the value

2

is a root of

p(x)

, which means the expression

x - 2

must be a factor of the polynomial.

Adım Adım Çözüm

1

Identify the given root of the polynomial.

The problem states that

p(2) = 0

, which means that

x = 2

is a root of the polynomial

p(x)

.

A root is any value of

x

for which the polynomial evaluates to

0

.

2

Apply the Factor Theorem to write the corresponding factor.

According to the Factor Theorem, if

c

is a root of a polynomial

p(x)

, then

(x - c)

is a factor of

p(x)

. Substituting

c = 2

gives

(x - 2)

as a factor.

The Factor Theorem directly relates the roots of a polynomial to its linear factors.

Anahtar Kavram

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Soru 1813Soru →

In the dry environment of the Taklamakan Desert, ancient textiles have remained remarkably well-preserved for thousands of years. The arid conditions prevent the growth of bacteria and fungi that normally cause organic decay _______ archaeologists have been able to recover silk garments with their vibrant colors still intact.

Which choice completes the text so that it conforms to the conventions of Standard English?

, as a result,

; as a result,

as a result

although

Cevabı ve açıklamayı göster

Cevap: ; as a result,

Cevap

The correct choice is '; as a result,', which uses a semicolon and a comma-delineated transition phrase to properly connect two independent clauses and establish a logical cause-and-effect relationship.

The correct option correctly joins two independent clauses using a semicolon followed by the conjunctive adverb 'as a result' and a comma. This conforms to Standard English conventions for linking independent clauses and logically reflects the cause-and-effect relationship between the prevention of decay and the recovery of preserved textiles.

Adım Adım Çözüm

1

Analyze the clauses surrounding the blank.

The clause before the blank ('The arid conditions prevent the growth of bacteria and fungi that normally cause organic decay') and the clause after the blank ('archaeologists have been able to recover silk garments with their vibrant colors still intact') are both independent clauses because they can each stand alone as complete sentences.

Identifying clause types is necessary to determine the appropriate punctuation and linking words needed to join them.

2

Identify the logical relationship between the two clauses.

The first clause describes a cause (arid conditions preventing decay) and the second clause describes the direct effect or consequence of that cause (the recovery of well-preserved garments). Therefore, a transition expressing cause-and-effect or consequence is required.

Determining the logical connection helps eliminate transition words that indicate an incorrect relationship, such as contrast.

3

Evaluate the grammar and punctuation of the choices.

To link two independent clauses, standard English conventions require a period, a semicolon, or a comma paired with a coordinating conjunction. The transition phrase 'as a result' is a conjunctive adverb, which must be preceded by a semicolon and followed by a comma when linking independent clauses.

Applying grammatical rules ensures the sentence avoids errors like comma splices, run-ons, and fragments.

Anahtar Kavram

Clause Boundaries and Linking

Soru 1814Soru →

The graph of the function $f$ in the $xy$ -plane passes through the point $(5, -2)$ . The function $g$ is defined by $g(x) = f(x) + 6$ . Which of the following points must lie on the graph of $g$ ?

(5, -8)

(11, -2)

(5, 4)

(-1, -2)

Cevabı ve açıklamayı göster

Cevap: $(5, 4)$

Cevap

The point

(5, 4)

Since the point

(5, -2)

is on the graph of

f

, we know that

f(5) = -2

. The function

g

is defined by

g(x) = f(x) + 6

, which represents a vertical shift upward by

6

units. Substituting

x = 5

into the equation for

g

yields

g(5) = f(5) + 6 = -2 + 6 = 4

. Therefore, the point

(5, 4)

must be on the graph of

g

.

Adım Adım Çözüm

1

Determine the value of

f(5)

using the given point on the graph of

f

.

f(5) = -2

Since the point

(5, -2)

lies on the graph of

f

, the input

x = 5

corresponds to the output

y = -2

.

2

Use the definition of

g(x)

to evaluate

g(5)

.

g(5) = f(5) + 6 = -2 + 6 = 4

Substitute

x = 5

into the definition

g(x) = f(x) + 6

to find the corresponding

y

-value for the graph of

g

.

Anahtar Kavram

Vertical translations of function graphs

Tahmini Süre:45s

Soru 1815Soru →

The table below shows the remaining balance, $B$ , in dollars, on a transit card after a user has taken $r$ rides on a city bus.

Number of rides ( $r$ )	Remaining balance ( $B$ , in dollars)
$3$	$22.50$
$8$	$10.00$
$12$	$0.00$

Which of the following equations represents the relationship between $B$ and $r$ ?

B = 22.50 - 2.50r

B = 30.00 - 0.40r

B = 30.00 - 2.50r

B = 30.00 + 2.50r

Cevabı ve açıklamayı göster

Cevap: $B = 30.00 - 2.50r$

Cevap

B = 30.00 - 2.50r

The correct equation has a rate of change of

-2.50

dollars per ride and a starting balance of

30.00

dollars. The rate of change (slope) can be calculated by finding the change in balance divided by the change in the number of rides between two points, such as

(3, 22.50)

and

(8, 10.00)

, which gives

\frac{10.00 - 22.50}{8 - 3} = -2.50

. Using the slope-intercept form

B = mr + B_0

and the point

(12, 0.00)

, we get

0.00 = -2.50(12) + B_0

, which simplifies to

B_0 = 30.00

. Therefore, the equation is

B = 30.00 - 2.50r

.

Adım Adım Çözüm

1

Calculate the slope (rate of change) using two points from the table, such as

(3, 22.50)

and

(8, 10.00)

.

The slope is

m = \frac{10.00 - 22.50}{8 - 3} = \frac{-12.50}{5} = -2.50

.

The slope represents the constant change in the remaining balance per ride.

2

Use the slope-intercept form

B = mr + B_0

and one point to find the initial balance

B_0

.

Substituting

m = -2.50

and the point

(12, 0.00)

gives

0.00 = -2.50(12) + B_0

, which simplifies to

0.00 = -30.00 + B_0

, so

B_0 = 30.00

.

The initial balance

B_0

is the y-intercept, which is the value of

B

when

r = 0

.

3

Write the final linear equation representing the relationship.

The final equation is

B = 30.00 - 2.50r

.

This equation models the remaining balance on the card as a function of the number of rides taken.

Anahtar Kavram

Finding a linear equation in two variables from a table of values.

Soru 1816Soru →

What is the larger solution to the equation below?

\frac{x}{x - 2} + \frac{3}{x} = 2

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The larger solution to the equation is 6.

To solve the rational equation, we clear the denominators by multiplying both sides by

x(x - 2)

, which yields the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

(x - 6)(x - 1) = 0

, leading to the solutions

x = 6

and

x = 1

. Both values are valid because they do not make any denominator of the original expression equal to zero. The larger of the two values is 6.

Adım Adım Çözüm

1

Multiply both sides of the equation by the common denominator

x(x - 2)

to clear the fractions.

x^2 + 3(x - 2) = 2x(x - 2)

Clearing denominators simplifies the rational equation into a polynomial equation.

2

Expand both sides of the equation.

x^2 + 3x - 6 = 2x^2 - 4x

Distributing the multiplication allows us to combine like terms.

3

Move all terms to one side of the equation to write it in the standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 7x + 6 = 0

Setting the quadratic expression to zero prepares it for factoring.

4

Factor the quadratic equation.

(x - 6)(x - 1) = 0

Finding two numbers that multiply to

6

and sum to

-7

gives

-6

and

-1

.

5

Identify the values of

x

that satisfy the factored equation and check for extraneous solutions.

x = 6

or

x = 1

Neither solution makes the denominators in the original equation,

x-2

or

x

, equal to zero, so both are valid. The larger of these two solutions is 6.

Anahtar Kavram

Solving rational equations by clearing denominators to form a quadratic equation, and verifying solutions against the domain constraints.

Soru 1817Soru →

A laboratory technician prepared a chemistry solution containing only liquid A and liquid B. Initially, liquid A made up $35\%$ of the total volume of the solution. After $150$ milliliters of liquid B was removed from the solution, the volume of liquid A became $42\%$ of the total volume of the remaining solution. What was the initial volume of liquid A, in milliliters?

315

378

750

900

Cevabı ve açıklamayı göster

Cevap: 315

Cevap

315

The correct answer is 315. Let

V

represent the initial total volume of the solution. Initially, liquid A's volume is

0.35V

. After removing

150

milliliters of liquid B, the total volume becomes

V - 150

. Since only liquid B was removed, the volume of liquid A remains unchanged. This volume is now

42\%

of the new total volume, which gives the equation

0.35V = 0.42(V - 150)

. Expanding the right side yields

0.35V = 0.42V - 63

. Subtracting

0.35V

and adding

63

to both sides results in

0.07V = 63

, which simplifies to

V = 900

milliliters. To find the initial volume of liquid A, we calculate

35\%

of

900

, which is

315

milliliters.

Adım Adım Çözüm

1

Let

V

represent the initial total volume of the solution in milliliters. Express the initial volume of liquid A in terms of

V

.

The initial volume of liquid A is

0.35V

milliliters.

Liquid A constitutes

35\%

of the initial total volume

V

.

2

Express the final total volume of the solution after removing

150

milliliters of liquid B.

The final total volume is

V - 150

milliliters.

Subtracting the volume of liquid B that was removed from the initial total volume gives the remaining total volume.

3

Set up an equation representing the volume of liquid A in the remaining solution and solve for

V

.

0.35V = 0.42(V - 150) \implies 0.35V = 0.42V - 63 \implies 0.07V = 63 \implies V = 900

milliliters.

The volume of liquid A remains constant, but it now represents

42\%

of the remaining total volume

V - 150

.

4

Calculate the initial volume of liquid A.

0.35 \times 900 = 315

milliliters.

Substitute the value of

V

back into the expression for the initial volume of liquid A.

Anahtar Kavram

Solving multi-step percent problems involving a change in the baseline total volume.

Tahmini Süre:2m 0s

Soru 1818Soru →

For all real values of

x

that satisfy the equation below, what is the value of

x + 1

?

\sqrt{5x + 39} - 3 = x

-5

-4

6

7

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The correct answer is

6

.

By adding

3

to both sides, we get

\sqrt{5x + 39} = x + 3

. Squaring both sides yields

5x + 39 = x^2 + 6x + 9

. Setting this quadratic equation to zero gives

x^2 + x - 30 = 0

, which factors as

(x + 6)(x - 5) = 0

. The potential solutions are

x = 5

and

x = -6

. Checking these values in the original equation shows that only

x = 5

is valid:

\sqrt{5(5) + 39} - 3 = 8 - 3 = 5

, which satisfies the equation. The value

x = -6

is extraneous since

\sqrt{5(-6) + 39} - 3 = 3 - 3 = 0 \neq -6

. Therefore, the value of

x + 1

is

5 + 1 = 6

.

Adım Adım Çözüm

1

Isolate the radical expression on one side of the equation.

\sqrt{5x + 39} = x + 3

Isolating the radical allows us to square both sides to eliminate the square root.

2

Square both sides of the equation.

5x + 39 = (x + 3)^2 \implies 5x + 39 = x^2 + 6x + 9

Squaring a square root removes the radical, allowing us to solve the resulting quadratic equation.

3

Rearrange the terms to form a standard quadratic equation of the form

ax^2 + bx + c = 0

.

x^2 + x - 30 = 0

Subtracting

5x

and

39

from both sides sets the quadratic expression equal to zero.

4

Factor the quadratic equation to find potential solutions.

(x + 6)(x - 5) = 0 \implies x = -6 \text{ or } x = 5

Finding two numbers that multiply to

-30

and add up to

1

gives

6

and

-5

.

5

Substitute both candidate solutions back into the original equation to check for extraneous solutions.

For

x = 5

:

\sqrt{5(5) + 39} - 3 = 8 - 3 = 5

(Valid). For

x = -6

:

\sqrt{5(-6) + 39} - 3 = 3 - 3 = 0 \neq -6

(Extraneous).

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original relation.

6

Use the valid solution to calculate the requested expression.

x + 1 = 5 + 1 = 6

The question asks for the value of

x + 1

.

Anahtar Kavram

Solving radical equations and identifying extraneous solutions

Soru 1819Soru →

If $\frac{16^{x+1}}{2^{x-3}} = 64$ , what is the value of $x$ ?

\frac{2}{3}

\frac{5}{3}

-\frac{1}{3}

-\frac{7}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{1}{3}$

Cevap

-\frac{1}{3}

The correct answer is

-\frac{1}{3}

. By expressing all terms with base 2, the equation

\frac{16^{x+1}}{2^{x-3}} = 64

becomes

\frac{2^{4x+4}}{2^{x-3}} = 2^6

. Applying exponent division rules yields

2^{(4x+4)-(x-3)} = 2^6

, which simplifies to

2^{3x+7} = 2^6

. Setting the exponents equal gives

3x + 7 = 6

, which simplifies to

x = -\frac{1}{3}

.

Adım Adım Çözüm

1

Rewrite all terms in the equation using a common base of 2.

The equation becomes

\frac{(2^4)^{x+1}}{2^{x-3}} = 2^6

.

Expressing bases 16 and 64 as powers of 2 allows us to apply exponent laws to simplify the equation.

2

Apply the power of a power rule to the numerator, then apply the quotient rule for exponents.

The left side simplifies to

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

.

The expression

(2^4)^{x+1}

becomes

2^{4x+4}

. Using the quotient rule

\frac{a^m}{a^n} = a^{m-n}

, we subtract the denominator's exponent

(x-3)

from the numerator's exponent

(4x+4)

, distributing the negative sign to get

3x+7

.

3

Set the exponents equal to each other and solve the resulting linear equation.

3x + 7 = 6 \implies 3x = -1 \implies x = -\frac{1}{3}

.

Since the bases on both sides of the equation are equal, their exponents must be equal.

Anahtar Kavram

Solving exponential equations by rewriting with a common base

Tahmini Süre:1m 30s

Soru 1820Soru →

A researcher is drying a sample of wet soil in a laboratory. Initially, water makes up $40\%$ of the total weight of the sample. During the first stage of drying, $25\%$ of the water in the sample evaporates. During the second stage of drying, an additional quantity of water evaporates such that the water in the sample now makes up $20\%$ of the sample's total weight. What percent of the water that remained after the first stage evaporated during the second stage?

20\%

37.5\%

50\%

33.3\%

Cevabı ve açıklamayı göster

Cevap: $50\%$

Cevap

50\%

The correct answer is the option stating

50\%

. After the first stage of drying, the sample contains

30

grams of water and has a total weight of

90

grams. In the second stage,

15

grams of water evaporates, leaving

15

grams of water and a total weight of

75

grams. Since

15

is exactly

20\%

of

75

, the final condition is met. The

15

grams of water that evaporated in the second stage represents exactly

50\%

of the

30

grams of water that remained after the first stage.

Adım Adım Çözüm

1

Define the initial weights using a convenient total weight.

Let the initial total weight of the wet soil sample be

100

grams. Since water makes up

40\%

of this weight, the initial water weight is

40

grams and the dry soil weight is

60

grams.

Choosing a concrete value of

100

grams simplifies the percent calculations.

2

Calculate the water remaining after the first stage of drying.

In the first stage,

25\%

of the water evaporates. The evaporated water is

0.25 \times 40 = 10

grams. The remaining water weight is

40 - 10 = 30

grams. The total weight of the sample is now

60 \text{ (dry soil)} + 30 \text{ (water)} = 90

grams.

We must establish the new baseline amount of water and total sample weight before the second drying stage.

3

Set up an equation for the second stage of drying.

Let

x

be the weight of water in grams that evaporates during the second stage. The final water weight is

30 - x

grams, and the final total weight of the sample is

90 - x

grams. Since the final water weight is

20\%

of the final total weight, we write:

\frac{30 - x}{90 - x} = 0.20

.

This relates the unknown quantity of water evaporated to the given final water percentage.

4

Solve the equation for the evaporated water weight

x

.

Multiplying both sides by

90 - x

gives

30 - x = 0.20(90 - x) \Rightarrow 30 - x = 18 - 0.20x

. Rearranging terms yields

12 = 0.80x

, which gives

x = 15

grams.

Solving the algebraic equation finds the exact amount of water that evaporated during the second stage.

5

Calculate the percent of the remaining water that evaporated.

Divide the water evaporated in the second stage (

15

grams) by the water remaining after the first stage (

30

grams):

\frac{15}{30} \times 100\% = 50\%

.

This answers the specific question asked using the correct base weight.

Anahtar Kavram

Multi-step percent change with a shifting base

Alternatif Yöntem

Instead of assuming a weight of

100

grams, the problem can be solved using variables. Let the initial total weight of the wet soil sample be

W

. The initial water weight is

0.40W

and the dry soil is

0.60W

. After the first stage, remaining water is

0.75 \times 0.40W = 0.30W

, and total weight is

0.90W

. Let

E

be the water evaporated in the second stage. The final water content is

0.30W - E

, and the final total weight is

0.90W - E

. The equation is

0.30W - E = 0.20(0.90W - E)

, which solves to

E = 0.15W

. The ratio of evaporated water to remaining water is

\frac{0.15W}{0.30W} = 0.50

, or

50\%

.

Tahmini Süre:3m 0s

Cevap

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