Tüm alıştırma soruları

Soru 341Soru →

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 342Soru →

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 343Soru →

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 344Soru →

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 345Soru →

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 346Soru →

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 347Soru →

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 348Soru →

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 349Soru →

A municipal water reservoir contains 250 million gallons of water. During a dry spell, water is released from the reservoir at a constant rate of 3.5 million gallons per day. Additionally, water evaporates from the reservoir at a constant rate of 0.3 million gallons per day. If no water enters the reservoir, the total amount of water $W$ , in million gallons, remaining in the reservoir after $d$ days of the dry spell is modeled by the equation $W = 250 - r d$ , where $r$ is a constant. What is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 3.8

Cevap

The correct value of

r

is 3.8.

The constant

r

in the linear equation

W = 250 - r d

represents the total rate, in million gallons per day, at which the water volume in the reservoir decreases. Since water is lost through both release (

3.5

million gallons per day) and evaporation (

0.3

million gallons per day), the total rate of decrease is the sum of these two rates, which is

3.5 + 0.3 = 3.8

million gallons per day. Therefore, the value of

r

is

3.8

.

Adım Adım Çözüm

1

Identify the factors causing a decrease in the reservoir's water volume.

Water is lost through release at

3.5

million gallons per day and evaporation at

0.3

million gallons per day.

Both release and evaporation contribute to the total rate of water depletion.

2

Calculate the total daily rate of water loss.

3.5 + 0.3 = 3.8

million gallons per day.

Adding the individual rates of loss yields the overall rate of decrease.

3

Compare the total daily rate of water loss to the model equation

W = 250 - r d

.

r = 3.8

.

In the linear model,

250

represents the initial amount of water, and

r

represents the constant rate at which water decreases per day. Thus,

r

is the total daily rate of water loss.

Anahtar Kavram

Interpreting the slope (rate of change) in a linear equation in context.

Soru 350Soru →

A garden hose discharges water at a constant rate of $8$ quarts per minute. What is this rate, in gallons per hour? (Given that $1\text{ gallon} = 4\text{ quarts}$ )

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

To convert the discharge rate from quarts per minute to gallons per hour, we first convert quarts to gallons. Since

1\text{ gallon} = 4\text{ quarts}

, we divide the rate of

8

quarts per minute by

4

to get

2

gallons per minute. Next, to convert minutes to hours, we multiply this rate by

60

(since there are

60

minutes in

1\text{ hour}

). This gives

2 \times 60 = 120

gallons per hour.

Adım Adım Çözüm

1

Convert quarts per minute to gallons per minute

2

gallons per minute

Since

1\text{ gallon} = 4\text{ quarts}

, divide the flow rate of

8

quarts per minute by the conversion factor of

4

to find the rate in gallons per minute:

\frac{8\text{ quarts/min}}{4\text{ quarts/gallon}} = 2\text{ gallons/min}

.

2

Convert gallons per minute to gallons per hour

120

gallons per hour

Since there are

60

minutes in an hour, multiply the rate of

2

gallons per minute by

60

to find the total gallons discharged in one hour:

2\text{ gallons/min} \times 60\text{ min/hour} = 120\text{ gallons/hour}

.

Anahtar Kavram

Unit Conversions

Soru 351Soru →

In the $xy$ -plane, the graph of a cubic polynomial function $p$ with real coefficients has exactly two $x$ -intercepts, at $(1, 0)$ and $(4, 0)$ . If the graph of $p$ passes through the points $(0, -8)$ and $(2, 2)$ , what is the value of $p(6)$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

A cubic polynomial with real coefficients and exactly two

x

-intercepts at

(1, 0)

and

(4, 0)

must have one root of multiplicity 1 and one root of multiplicity 2. This yields two possible forms:

p(x) = a(x - 1)^2(x - 4)

or

p(x) = a(x - 1)(x - 4)^2

. Substituting the

y

-intercept

(0, -8)

into the first form gives

a = 2

, but the resulting polynomial

p(x) = 2(x - 1)^2(x - 4)

does not pass through

(2, 2)

since

p(2) = -4

. Substituting

(0, -8)

into the second form gives

a = \frac{1}{2}

, and the resulting polynomial

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

correctly passes through

(2, 2)

since

p(2) = 2

. Finally, evaluating this function at

x = 6

yields

p(6) = \frac{1}{2}(6 - 1)(6 - 4)^2 = 10

.

Adım Adım Çözüm

1

Determine the possible forms of the cubic polynomial based on its

x

-intercepts.

Two possible cases: Case 1:

p(x) = a(x - 1)^2(x - 4)

or Case 2:

p(x) = a(x - 1)(x - 4)^2

, where

a

is a non-zero constant.

Since the polynomial has degree 3 and exactly two distinct

x

-intercepts, one intercept must be a root of multiplicity 2 (tangent to the

x

-axis) and the other must be a root of multiplicity 1 (crossing the

x

-axis).

2

Solve for the constant

a

in Case 1 using the

y

-intercept

(0, -8)

.

a = 2

, yielding the candidate function

p(x) = 2(x - 1)^2(x - 4)

.

Substituting

x = 0

and

p(0) = -8

into the equation for Case 1 allows us to solve for

a

.

3

Test the point

(2, 2)

in the Case 1 candidate function.

p(2) = -4 \neq 2

, which means Case 1 is incorrect.

The correct function must satisfy all given points, including

(2, 2)

.

4

Solve for the constant

a

in Case 2 using the

y

-intercept

(0, -8)

.

a = \frac{1}{2}

, yielding the candidate function

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

.

Substituting

x = 0

and

p(0) = -8

into the equation for Case 2 allows us to solve for

a

.

5

Test the point

(2, 2)

in the Case 2 candidate function.

p(2) = 2

, which matches the given point.

Confirming that the Case 2 function is consistent with the point

(2, 2)

.

6

Evaluate the correct function at

x = 6

.

p(6) = 10

.

Substituting

x = 6

into the verified function

p(x) = \frac{1}{2}(x - 1)(x - 4)^2

to find the final answer.

Anahtar Kavram

Identifying the algebraic form of a polynomial from its

x

-intercepts and multiplicities, and determining unknown coefficients using coordinate points.

Soru 352Soru →

A chemist mixes a $10\%$ acid solution and a $30\%$ acid solution to create $200$ milliliters of a mixture that is $18\%$ acid. How many milliliters of the $30\%$ acid solution are in the mixture?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

The correct answer is

80

milliliters. By translating the problem into a system of linear equations, we let

x

be the volume of the

10\%

acid solution and

y

be the volume of the

30\%

acid solution. Since the total volume is

200

milliliters,

x + y = 200

. The amount of acid in the solutions must sum to the amount of acid in the final mixture, so

0.10x + 0.30y = 0.18(200)

, which simplifies to

0.10x + 0.30y = 36

. Multiplying this equation by

10

gives

x + 3y = 360

. Subtracting the first equation from this yields

(x + 3y) - (x + y) = 360 - 200

, or

2y = 160

. Solving for

y

gives

80

milliliters of the

30\%

solution.

Adım Adım Çözüm

1

Define variables and set up the system of equations.

Let

x

be the number of milliliters of the

10\%

acid solution and

y

be the number of milliliters of the

30\%

acid solution. The total volume equation is

x + y = 200

. The total acid content equation is

0.10x + 0.30y = 0.18(200)

.

This sets up the system of linear equations representing the physical constraints of the mixture.

2

Simplify the acid content equation and prepare for elimination.

0.10x + 0.30y = 36

. Multiplying the entire equation by

10

yields

x + 3y = 360

.

Eliminating decimals simplifies the coefficients and makes it easier to solve the system using integer arithmetic.

3

Eliminate

x

by subtracting the total volume equation from the simplified acid content equation.

(x + 3y) - (x + y) = 360 - 200 \implies 2y = 160 \implies y = 80

.

This isolates the variable

y

, which directly represents the volume of the

30\%

acid solution requested in the problem.

Anahtar Kavram

Solving systems of linear equations in context

Soru 353Soru →

If the graph of $y = f(x)$ contains the point $(3, 7)$ , and the function $g$ is defined by $g(x) = f(x + 4) - 2$ , what is the value of $g(-1)$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Since the graph of

y = f(x)

contains the point

(3, 7)

, we have

f(3) = 7

. The function

g

is defined as

g(x) = f(x + 4) - 2

. To find the value of

g(-1)

, we substitute

x = -1

into the definition of

g

:

g(-1) = f(-1 + 4) - 2 = f(3) - 2

. Substituting

f(3) = 7

gives

g(-1) = 7 - 2 = 5

.

Adım Adım Çözüm

1

Translate the point

(3, 7)

on the graph of

f(x)

into function notation.

f(3) = 7

By definition, if a point

(a, b)

is on the graph of

y = f(x)

, then

f(a) = b

.

2

Substitute

x = -1

into the expression for

g(x)

to evaluate

g(-1)

.

g(-1) = f(-1 + 4) - 2

To find

g(-1)

, replace every occurrence of

x

with

-1

in the function definition of

g(x)

.

3

Simplify the input of the function

f

and compute the final value.

g(-1) = f(3) - 2 = 7 - 2 = 5

Simplify

-1 + 4

to

3

, then substitute the known value

f(3) = 7

and subtract

2

.

Anahtar Kavram

Evaluating a transformed function using function notation and given coordinate points.

Soru 354Soru →

A retailer purchases an item at a wholesale price. The retailer marks up the wholesale price by $p\%$ to establish the retail price. During a clearance sale, the retailer discounts the retail price by $(p - 10)\%$ . If the clearance sale price of the item is $8\%$ greater than the original wholesale price, and $p > 10$ , what is the value of $p$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

p

is

20

.

The correct answer is

20

. By representing the markup and discount as decimal multipliers, we can write the equation for the final price as a function of the wholesale price:

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08W

. Dividing by

W

and letting

y = \frac{p}{100}

, we get

(1 + y)(1.1 - y) = 1.08

. Expanding this gives

1.1 + 0.1y - y^2 = 1.08

, which rearranges to the quadratic equation

y^2 - 0.1y - 0.02 = 0

. Factoring this equation yields

(y - 0.2)(y + 0.1) = 0

. Since

p > 10

,

y

must be positive, which means

y = 0.2

. Therefore,

p = 20

.

Adım Adım Çözüm

1

Express the retail price in terms of the wholesale price

W

and the markup percentage

p\%

.

Retail Price =

W \left(1 + \frac{p}{100}\right)

A markup of

p\%

increases the base price

W

by a factor of

\left(1 + \frac{p}{100}\right)

.

2

Express the clearance sale price after applying a discount of

(p - 10)\%

to the retail price.

Clearance Price =

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right)

A discount of

(p - 10)\%

decreases the retail price by a factor of

\left(1 - \frac{p - 10}{100}\right)

.

3

Set the clearance price equal to

1.08W

, which represents an

8\%

increase over the wholesale price, and simplify the equation by dividing both sides by

W

.

\left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08

The final clearance price is

8\%

greater than the wholesale price

W

, so we equate it to

1.08W

and divide both sides by

W

to eliminate the variable.

4

Substitute

y = \frac{p}{100}

into the simplified equation and expand the terms.

(1 + y)(1.1 - y) = 1.08 \Rightarrow 1.1 + 0.1y - y^2 = 1.08

Writing the equation in terms of

y

simplifies the algebraic expansion. The term

1 - \frac{p - 10}{100}

becomes

1 - (y - 0.1) = 1.1 - y

.

5

Rearrange the quadratic equation into standard form, factor it, and solve for

y

.

y^2 - 0.1y - 0.02 = 0 \Rightarrow (y - 0.2)(y + 0.1) = 0 \Rightarrow y = 0.2

(since

p > 10

,

y > 0.1

)

Factoring the quadratic yields

y = 0.2

and

y = -0.1

. Since

p > 10

,

y

must be positive, which leaves

y = 0.2

as the only valid solution.

6

Convert the value of

y

back to

p

.

p = 20

Since

y = \frac{p}{100} = 0.2

, multiplying both sides by

100

gives

p = 20

.

Anahtar Kavram

Compounding percent changes algebraically using variable markups and discounts.

Soru 355Soru →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(2, 15)$ and $(6, 7)$ . What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

The value of

b

is

19

.

The slope of the line is found using the two given points:

m = \frac{7 - 15}{6 - 2} = -2

. Substituting the slope

m = -2

and the point

(2, 15)

into the equation

y = mx + b

gives

15 = -2(2) + b

, which simplifies to

15 = -4 + b

. Adding

4

to both sides yields

b = 19

.

Adım Adım Çözüm

1

Calculate the slope of the line passing through

(2, 15)

and

(6, 7)

.

m = \frac{7 - 15}{6 - 2} = \frac{-8}{4} = -2

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Substitute the slope

m = -2

and the coordinates of one point, such as

(2, 15)

, into the equation

y = mx + b

to find

b

.

15 = -2(2) + b \implies 15 = -4 + b

Since the point lies on the line, its coordinates must satisfy the equation of the line.

3

Solve the equation for

b

.

b = 19

Add

4

to both sides of the equation to isolate the variable

b

.

Anahtar Kavram

Determining the equation of a line given two points.

Soru 356Soru →

The function $f$ is defined for all real numbers, and the graph of $y = f(x)$ in the $xy$ -plane has a single minimum at the point $(5, -2)$ . The function $g$ is defined by $g(x) = -3f(2x - 4) + 7$ . What is the $y$ -coordinate of the maximum point on the graph of $y = g(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The correct answer is 13.

The graph of

y = f(x)

has a minimum at

(5, -2)

, which means

f(5) = -2

and

f(x) \ge -2

for all

x

. The function

g(x) = -3f(2x-4) + 7

includes a vertical stretch by a factor of

3

, a vertical reflection across the

x

-axis, and a vertical shift upward by

7

units. Because of the vertical reflection, the minimum value of the original function becomes the maximum value of the transformed function. Applying the vertical transformations to the

y

-coordinate of the minimum point yields

-3(-2) + 7 = 6 + 7 = 13

.

Adım Adım Çözüm

1

Identify the minimum point and minimum value of the original function

f(x)

.

f(5) = -2

, and

f(x) \ge -2

for all real numbers

x

.

The problem states that the graph of

y = f(x)

has a single minimum at the point

(5, -2)

.

2

Determine the transformed

x

-coordinate corresponding to the original input of

5

.

2x - 4 = 5 \implies 2x = 9 \implies x = 4.5

.

Setting the argument of the function

f(2x-4)

equal to the original minimum input of

5

allows us to find the corresponding input

x

for the function

g

.

3

Apply the vertical transformations to find the output value of

g(x)

at

x = 4.5

.

g(4.5) = -3f(5) + 7 = -3(-2) + 7 = 6 + 7 = 13

.

Substituting

f(5) = -2

into the definition of

g(x)

gives the vertical transformation of the point.

4

Confirm that the point is indeed the maximum of the transformed function

g(x)

.

Since

f(2x-4) \ge -2

, multiplying by

-3

yields

-3f(2x-4) \le 6

. Adding

7

yields

g(x) \le 13

, confirming that

13

is the maximum value.

Multiplying a function by a negative number reflects its graph vertically, changing a minimum point into a maximum point.

Anahtar Kavram

Applying horizontal and vertical transformations to function coordinates, and understanding how vertical reflections affect the extrema (minima and maxima) of a graph.

Soru 357Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = x^2 - 4x - 5$ intersects the $x$ -axis at the points $(p, 0)$ and $(q, 0)$ and has vertex $(h, k)$ . What is the area of the triangle with vertices at $(p, 0)$ , $(q, 0)$ , and $(h, k)$ ?

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The area of the triangle is 27.

To find the area of the triangle, we first determine the coordinates of its vertices. The base of the triangle lies on the

x

-axis, with endpoints at the

x

-intercepts of the function

f(x) = x^2 - 4x - 5

. Solving

x^2 - 4x - 5 = 0

by factoring gives

(x - 5)(x + 1) = 0

, so the intercepts are at

x = -1

and

x = 5

. The distance between these two points is

5 - (-1) = 6

, which is the base of the triangle. The third vertex is the vertex of the parabola. The

x

-coordinate of the vertex is

h = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2

. Substituting

x = 2

into the function gives the

y

-coordinate:

k = f(2) = 2^2 - 4(2) - 5 = -9

. The height of the triangle is the vertical distance from the

x

-axis to the vertex, which is

|-9| = 9

. The area of the triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

.

Adım Adım Çözüm

1

Find the

x

-intercepts of the parabola.

The intercepts are

(-1, 0)

and

(5, 0)

.

Setting

f(x) = 0

gives

x^2 - 4x - 5 = 0

. Factoring the quadratic equation yields

(x - 5)(x + 1) = 0

, which gives

x = 5

and

x = -1

.

2

Calculate the base of the triangle.

The base length is

6

.

The base of the triangle is the segment along the

x

-axis between the two intercepts. The distance between

(-1, 0)

and

(5, 0)

is

5 - (-1) = 6

.

3

Find the vertex

(h, k)

of the parabola.

The vertex is at

(2, -9)

.

The

x

-coordinate of the vertex is the midpoint of the intercepts:

h = \frac{-1 + 5}{2} = 2

. The

y

-coordinate is

k = f(2) = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9

.

4

Calculate the area of the triangle.

The area is

27

.

The height of the triangle is the distance from the

x

-axis to the vertex, which is

|k| = |-9| = 9

. Using the formula for the area of a triangle,

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

.

Anahtar Kavram

Finding the

x

-intercepts and vertex of a quadratic function to solve geometric problems in the coordinate plane.

Soru 358Soru →

The function $f$ is defined by $f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ is tangent to the $x$ -axis at the point $(3, 0)$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

For the graph of a polynomial function to be tangent to the

x

-axis at

(3, 0)

, the root

x = 3

must have an even multiplicity (at least 2). The function is defined as

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

. Since there is already one factor of

(x - 3)

explicitly defined, the remaining cubic factor

g(x) = x^3 - kx^2 + 5x - 15

must also have a factor of

(x - 3)

to make the total multiplicity of the root

x = 3

at least 2. According to the Factor Theorem, if

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

. Substituting

x = 3

into

g(x)

gives

3^3 - k(3)^2 + 5(3) - 15 = 0

, which simplifies to

27 - 9k = 0

. Solving this equation for

k

yields

k = 3

.

Adım Adım Çözüm

1

Identify the relationship between graph tangency and factor multiplicity.

For the graph of a polynomial function to be tangent to the

x

-axis at a point

(c, 0)

, the factor

(x - c)

must have an even multiplicity of at least 2 in the polynomial's factored form.

An odd multiplicity root causes the graph to cross the

x

-axis, while an even multiplicity root causes the graph to touch the

x

-axis and turn around (tangency).

2

Apply the multiplicity requirement to the given function.

Since

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

already contains one factor of

(x - 3)

, the cubic expression

g(x) = x^3 - kx^2 + 5x - 15

must also contain

(x - 3)

as a factor to ensure the total multiplicity of the root

x = 3

is at least 2.

This guarantees that

(x - 3)^2

is a factor of

f(x)

.

3

Apply the Factor Theorem to the cubic expression.

Since

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

.

The Factor Theorem states that a polynomial

P(x)

has a factor

(x - c)

if and only if

P(c) = 0

.

4

Solve for the constant

k

by substituting

x = 3

into

g(x)

.

3^3 - k(3)^2 + 5(3) - 15 = 0 \implies 27 - 9k + 15 - 15 = 0 \implies 27 - 9k = 0 \implies 9k = 27 \implies k = 3

.

Arithmetic simplification yields the value of the constant.

Anahtar Kavram

The relationship between polynomial factors, root multiplicities, and the behavior of the graph at

x

-intercepts.

Soru 359Soru →

An equation is shown below.

\frac{x}{x - 1} - \frac{2}{x + 2} = \frac{6}{x^2 + x - 2}

What is the value of the real solution to the equation?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

The correct answer is 2. Multiplying both sides by the least common denominator

(x - 1)(x + 2)

results in the quadratic equation

x(x + 2) - 2(x - 1) = 6

. Simplifying this equation yields

x^2 + 2 = 6

, which has solutions

x = 2

and

x = -2

. However, substituting

x = -2

into the original equation results in division by zero, making it an extraneous solution. Therefore,

2

is the only valid real solution.

Adım Adım Çözüm

1

Multiply the entire equation by the least common denominator,

(x - 1)(x + 2) = x^2 + x - 2

, to clear the denominators.

x(x + 2) - 2(x - 1) = 6

This simplifies the rational equation into a polynomial equation.

2

Expand the terms and simplify the equation.

x^2 + 2x - 2x + 2 = 6

, which simplifies to

x^2 + 2 = 6

.

Distributing the terms allows us to group like terms and solve for the variable.

3

Solve the quadratic equation for

x

.

x^2 = 4

, which gives

x = 2

or

x = -2

.

Subtracting 2 from both sides isolates the squared variable.

4

Check for extraneous solutions by substituting the potential solutions back into the original denominators.

For

x = -2

, the denominator

x + 2

becomes 0, which is undefined. For

x = 2

, all denominators are non-zero.

Solutions that make any denominator in the original equation equal to zero are extraneous and must be excluded.

Anahtar Kavram

Solving rational equations by finding a common denominator and checking for extraneous solutions.

Soru 360Soru →

A digital photography archive consists of RAW files and JPEG files. Initially, the RAW files account for $80\%$ of the total storage space used by the archive. To reduce the storage space, the photographer compresses $40\%$ of the RAW files into JPEGs, which reduces the storage space of those specific files by $62.5\%$ . If the remaining uncompressed RAW files now account for $x\%$ of the updated total storage space of the archive, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The remaining uncompressed RAW files occupy 48% of the initial total storage space, and the updated total storage space is 80% of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

The remaining uncompressed RAW files occupy

48\%

of the initial total storage space, and the updated total storage space is

80\%

of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

Adım Adım Çözüm

1

Define variables for initial storage space.

Initial RAW space is

0.80T

; initial JPEG space is

0.20T

.

Establishes the baseline values relative to the initial total storage space

T

.

2

Calculate the space occupied by the portion of RAW files to be compressed and the remaining uncompressed RAW files.

Compressed RAW space is

0.32T

; remaining uncompressed RAW space is

0.48T

.

Splits the RAW files into the group that changes size and the group that remains unchanged.

3

Determine the new space occupied by the compressed files after the

62.5\%

reduction.

Reduction is

0.20T

; new space of compressed files is

0.12T

.

Calculates the impact of the compression percent change on the target subpopulation.

4

Calculate the updated total storage space of the archive.

Updated total storage space is

0.80T

.

Summing all updated file spaces (

0.48T

uncompressed RAW +

0.20T

original JPEG +

0.12T

newly compressed files) provides the new base for the final percentage calculation.

5

Compute the final percentage of the updated total storage space occupied by the uncompressed RAW files.

x = 60

.

Dividing the remaining uncompressed RAW space (

0.48T

) by the new total space (

0.80T

) gives the updated percentage.

Anahtar Kavram

Multi-step percent change and relative base calculations.

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