Tüm alıştırma soruları

Soru 1641Soru →

The profit $P(x)$ , in dollars, a company makes from selling $x$ units of a product is given by the function $P(x) = -3x^2 + bx - 1,500$ , where $b$ is a positive constant. If the maximum profit the company can make is $1,200$ dollars, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The maximum profit of the quadratic profit function occurs at its vertex. The x-coordinate of the vertex of

P(x) = -3x^2 + bx - 1,500

is given by

x = -\frac{b}{2(-3)} = \frac{b}{6}

. Substituting this into the profit equation and setting it equal to

1,200

yields

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

. Simplifying this expression results in

\frac{b^2}{12} = 2,700

, which gives

b^2 = 32,400

. Since

b

is positive,

b = 180

.

Adım Adım Çözüm

1

Identify the x-coordinate of the vertex of the quadratic function in terms of

b

.

The x-coordinate of the vertex is

x = \frac{b}{6}

.

A quadratic function in the form

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

has its maximum value at the vertex when the coefficient of the squared term,

a

, is negative. The x-coordinate of the vertex is given by

x = -\frac{b}{2a}

. For this function,

a = -3

, so the x-coordinate of the vertex is

x = -\frac{b}{2(-3)} = \frac{b}{6}

.

2

Substitute

x = \frac{b}{6}

back into the profit equation

P(x)

and set it equal to the maximum profit of

1,200

dollars.

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

The maximum profit of

1,200

dollars is the value of the function at the vertex.

3

Simplify the equation to solve for the positive constant

b

.

b = 180

Simplifying the term

-3\left(\frac{b}{6}\right)^2

gives

-3\left(\frac{b^2}{36}\right) = -\frac{b^2}{12}

. Simplifying the term

b\left(\frac{b}{6}\right)

gives

\frac{b^2}{6}

. Substituting these back in gives

-\frac{b^2}{12} + \frac{b^2}{6} - 1,500 = 1,200

. Combining the

b^2

terms gives

\frac{b^2}{12} - 1,500 = 1,200

. Adding

1,500

to both sides gives

\frac{b^2}{12} = 2,700

. Multiplying both sides by

12

yields

b^2 = 32,400

. Taking the square root of both sides gives

b = \pm 180

. Since

b

must be a positive constant,

b = 180

.

Anahtar Kavram

Finding the maximum value of a quadratic function by using the vertex formula

x = -\frac{b}{2a}

and evaluating the function at that point.

Soru 1642Soru →

For all $x > 3$ , the expression $\frac{2x^2 - 11x + 15}{x - 3} + \frac{3x^2 - 12}{x - 2}$ is equivalent to $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

a + b

is

6

.

The correct answer is

6

. By factoring the numerators of both rational expressions, we can simplify them. The first term is

\frac{(2x-5)(x-3)}{x-3} = 2x-5

. The second term is

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

. Adding these simplified terms gives

(2x-5) + (3x+6) = 5x+1

. Comparing this to

ax+b

yields

a=5

and

b=1

, so

a+b = 5+1=6

.

Adım Adım Çözüm

1

Factor the numerator of the first rational expression and simplify.

\frac{2x^2 - 11x + 15}{x - 3} = 2x - 5

Since

2x^2 - 11x + 15 = (2x - 5)(x - 3)

, the factor

(x - 3)

divides out for all

x > 3

.

2

Factor the numerator of the second rational expression and simplify.

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

Since

3x^2 - 12 = 3(x - 2)(x + 2)

, the factor

(x - 2)

divides out for all

x > 3

.

3

Combine the simplified terms by adding them.

5x + 1

Adding

(2x - 5)

and

(3x + 6)

yields

(2x + 3x) + (-5 + 6) = 5x + 1

.

4

Identify the values of

a

and

b

and calculate their sum.

a = 5

,

b = 1

, and

a + b = 6

Comparing

5x + 1

with

ax + b

gives

a = 5

and

b = 1

, so

a + b = 5 + 1 = 6

.

Anahtar Kavram

Simplifying rational expressions by factoring the numerator and dividing out common factors.

Soru 1643Soru →

The passage below contains a blank. Based on the grammatical context of the passage, what form of the noun 'star' correctly completes the blank?

Aşağıdaki boşlukları doldurun

In 1912, astronomer Henrietta Swan Leavitt analyzed photographic plates of the Magellanic Clouds to study variable stars. While observing these celestial bodies, she noted a pattern linking the time it took for each one to complete a cycle of brightness to its actual light output. Ultimately, Leavitt's discovery of the relationship between these luminosity and pulsation period revolutionized modern astrophysics by allowing researchers to measure cosmic distances for the first time.

Cevabı ve açıklamayı göster

Cevap

stars'

The correct answer is the plural possessive noun 'stars''. The plural demonstrative pronoun 'these' indicates that the noun must be plural, and the noun 'luminosity' immediately follows the blank, indicating that the stars possess the luminosity. Therefore, the plural possessive form 'stars'' is required.

Adım Adım Çözüm

1

Analyze the grammatical context preceding the blank to determine number.

The demonstrative pronoun 'these' precedes the blank, which indicates that the following noun must be plural ('stars').

Demonstrative pronouns must agree in number with the nouns they modify.

2

Analyze the grammatical relationship between the blank and the subsequent noun.

The noun following the blank is 'luminosity'. Since the luminosity belongs to the stars, the plural noun 'stars' must be in the possessive case ('stars'').

A noun modifying another noun to show ownership or association requires possessive punctuation.

Anahtar Kavram

Plural and Possessive Nouns

Tahmini Süre:1m 0s

Soru 1644Soru →

For what value of $x$ is the equation $8^{x + 1} = 16^x$ true?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

The correct answer is 3. To find the value of

x

, express both 8 and 16 as powers of 2:

8 = 2^3

and

16 = 2^4

. Substituting these values into the equation gives

(2^3)^{x + 1} = (2^4)^x

. Applying the power of a power rule,

(a^m)^n = a^{mn}

, yields

2^{3(x + 1)} = 2^{4x}

, which simplifies to

2^{3x + 3} = 2^{4x}

. Since the bases are now the same, their exponents must be equal:

3x + 3 = 4x

. Subtracting

3x

from both sides gives

x = 3

.

Adım Adım Çözüm

1

Express the bases 8 and 16 as powers of 2.

(2^3)^{x + 1} = (2^4)^x

Writing both sides of the equation with a common base allows for the equating of exponents.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

2^{3x + 3} = 2^{4x}

This simplifies each side of the equation to a single base with a single exponent.

3

Equate the exponents and solve the resulting linear equation.

3x + 3 = 4x

, which simplifies to

x = 3

Since the bases are equal, their exponents must be equal for the equation to hold true.

Anahtar Kavram

Solving exponential equations by expressing both sides with a common base and equating the exponents.

Alternatif Yöntem

An alternative approach is to write the equation in terms of base 4. While 8 is not an integer power of 4, we can write

8 = 4^{1.5}

and

16 = 4^2

. The equation becomes

(4^{1.5})^{x+1} = (4^2)^x

, which simplifies to

1.5(x+1) = 2x

. Solving this gives

1.5x + 1.5 = 2x \implies 0.5x = 1.5 \implies x = 3

.

Tahmini Süre:45s

Soru 1645Soru →

What is the set of all solutions to the equation $\sqrt{x + 12} = x$ ?

4

only

-3

only

4

and

-3

3

and

-4

Cevabı ve açıklamayı göster

Cevap: $4$ only

Cevap

The solution set containing only

4

The correct answer is the set containing only

4

. Squaring both sides of the equation

\sqrt{x + 12} = x

produces

x + 12 = x^2

. Rearranging this into standard quadratic form gives

x^2 - x - 12 = 0

. Factoring the quadratic gives

(x - 4)(x + 3) = 0

, which yields potential solutions of

x = 4

and

x = -3

. Testing

x = 4

in the original equation yields

\sqrt{4 + 12} = \sqrt{16} = 4

, which is a true statement. Testing

x = -3

yields

\sqrt{-3 + 12} = \sqrt{9} = 3

, which does not equal

-3

. Thus,

x = -3

is extraneous, and

4

is the only valid solution.

Adım Adım Çözüm

1

Square both sides of the equation to eliminate the radical.

x + 12 = x^2

Squaring both sides of

\sqrt{x + 12} = x

removes the square root.

2

Rewrite the equation in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - x - 12 = 0

Subtracting

x

and

12

from both sides moves all terms to one side of the equation.

3

Factor the quadratic equation.

(x - 4)(x + 3) = 0

We search for two numbers that multiply to

-12

and add to

-1

. These numbers are

-4

and

3

.

4

Solve for the potential solutions.

x = 4

or

x = -3

Setting each factor equal to zero yields the prospective values of

x

.

5

Check both potential solutions in the original equation to identify extraneous solutions.

Substituting

x = 4

:

\sqrt{4 + 12} = \sqrt{16} = 4

(valid). Substituting

x = -3

:

\sqrt{-3 + 12} = \sqrt{9} = 3 \neq -3

(extraneous).

Squaring both sides of an equation can introduce extraneous roots that must be discarded.

Anahtar Kavram

Solving radical equations and verifying solutions to eliminate extraneous roots

Soru 1646Soru →

Read the passage and determine which punctuation mark should fill the blank to properly connect the clauses according to the rules of Standard English.

Aşağıdaki boşlukları doldurun

In the field of architectural acoustics, carefully managing reverberation time—the time it takes for sound to decay in a closed space—is crucial for designing functional environments. A concert hall requires a relatively long reverberation time to blend the sounds of various musical instruments conversely, a lecture hall must have a short reverberation time to ensure that spoken words remain distinct and easily intelligible to the audience.

Cevabı ve açıklamayı göster

Cevap

A semicolon (;)

A semicolon is the correct punctuation mark to place before the conjunctive adverb 'conversely' when joining two independent clauses. This punctuation successfully links the two closely related but complete thoughts without creating a comma splice or run-on sentence.

Adım Adım Çözüm

1

Analyze the clause structures on both sides of the blank.

The clause preceding the blank ('A concert hall requires a relatively long reverberation time to blend the sounds of various musical instruments') and the clause following the blank ('conversely, a lecture hall must have a short reverberation time to ensure that spoken words remain distinct and easily intelligible to the audience') are both independent clauses.

Determining whether the clauses are independent or dependent guides the choice of punctuation.

2

Examine the connection between the two independent clauses.

The second independent clause begins with the conjunctive adverb 'conversely'.

Two independent clauses joined by a conjunctive adverb must be separated by a semicolon to maintain correct sentence structure and avoid run-on sentences or comma splices.

Anahtar Kavram

Using semicolons to link independent clauses separated by conjunctive adverbs.

Tahmini Süre:1m 0s

Soru 1647Soru →

The graph of the quadratic function $f$ in the $xy$ -plane has a vertex at $(4, 7)$ . If the function $g$ is defined by $g(x) = f(x) + 5$ , what is the $y$ -coordinate of the vertex of the graph of $y = g(x)$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The correct answer is 12.

The vertex of the graph of the quadratic function

f

is

(4, 7)

. The function

g(x) = f(x) + 5

adds

5

to every output value of

f(x)

, which shifts the entire graph vertically upward by

5

units. Consequently, the vertex of the graph of

g

is translated from

(4, 7)

to

(4, 7 + 5)

, which simplifies to

(4, 12)

. Therefore, the

y

-coordinate of the vertex of the graph of

y = g(x)

is

12

.

Adım Adım Çözüm

1

Identify the y-coordinate of the vertex of the graph of the function f.

The y-coordinate of the vertex of the graph of f is 7.

The vertex of the graph of f is given as (4, 7).

2

Determine the vertical translation from f(x) to g(x).

The graph of g(x) is shifted upward by 5 units relative to the graph of f(x).

The term + 5 in the definition g(x) = f(x) + 5 increases all y-values of the function by 5.

3

Calculate the y-coordinate of the vertex of the graph of g.

The new y-coordinate is 7 + 5 = 12.

Applying the vertical shift of 5 units to the original y-coordinate of 7 gives the new vertex's y-coordinate.

Anahtar Kavram

Vertical translations of quadratic functions and their graphs

Soru 1648Soru →

A community art studio offers clay sculpting classes. The studio charges a one-time registration fee plus a fee for each pound of clay used. A student who uses $12$ pounds of clay is charged a total of $\$106$ . A student who uses $25$ pounds of clay is charged a total of $\$197$ . If the relationship between the total charge, $C$ , in dollars, and the amount of clay used, $p$ , in pounds, is linear, which of the following equations models this relationship?

C = 22p + 7

C = 7p + 22

C = 7p + 12

C = \frac{1}{7}p + 22

Cevabı ve açıklamayı göster

Cevap: $C = 7p + 22$

Cevap

C = 7p + 22

The correct equation is

C = 7p + 22

. To find this, we first determine the rate of change, which is the cost per pound of clay. This is calculated as the change in total cost divided by the change in the amount of clay used:

\frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

dollars per pound. Next, we use the linear equation form

C = mp + b

, where

m

is the slope (

7

) and

b

is the y-intercept (the registration fee). Substituting

C = 106

and

p = 12

gives

106 = 7(12) + b

, which simplifies to

106 = 84 + b

. Solving for

b

yields

b = 22

. Therefore, the linear relationship is represented by

C = 7p + 22

.

Adım Adım Çözüm

1

Calculate the slope (

m

), which represents the price per pound of clay, using the coordinate points

(12, 106)

and

(25, 197)

.

m = \frac{197 - 106}{25 - 12} = \frac{91}{13} = 7

The slope of a linear equation represents the constant rate of change between the independent variable (pounds of clay) and the dependent variable (total cost).

2

Substitute the slope

m = 7

and one of the points, such as

(12, 106)

, into the slope-intercept form

C = mp + b

to solve for the y-intercept

b

.

106 = 7(12) + b \implies 106 = 84 + b \implies b = 22

The y-intercept represents the flat registration fee, which is the cost when

0

pounds of clay are used.

3

Write the final equation by substituting the calculated slope and y-intercept back into the slope-intercept form.

C = 7p + 22

Combining the constant rate of change and the initial value yields the linear model.

Anahtar Kavram

Linear Equations in Two Variables

Soru 1649Soru →

The quadratic function $h$ is defined by $h(x) = (x + 2)^2 - 5$ . What are the coordinates of the vertex of the graph of $y = h(x)$ in the $xy$ -plane?

(-2, -5)

(2, -5)

(-2, 5)

(2, 5)

Cevabı ve açıklamayı göster

Cevap: $(-2, -5)$

Cevap

The coordinates of the vertex of the graph are

(-2, -5)

.

The correct answer is the coordinate pair

(-2, -5)

. The vertex form of a quadratic function is given by

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

, where

(h, k)

is the vertex of the parabola. For the given equation

h(x) = (x + 2)^2 - 5

, we rewrite the squared expression as

(x - (-2))^2

to match the standard form. This shows that

h = -2

. The value of

k

is the constant term

-5

. Therefore, the vertex of the graph of

h

is

(-2, -5)

.

Adım Adım Çözüm

1

Identify the general vertex form of a quadratic function.

The vertex form is

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

, where the vertex of the parabola is at the point

(h, k)

.

This general equation allows us to directly read the coordinates of the vertex by comparing the coefficients.

2

Compare the given function

h(x) = (x + 2)^2 - 5

to the vertex form.

By matching the terms, we get

x - h = x + 2

, which implies

h = -2

. The constant term is

k = -5

.

This identifies the values of

h

and

k

that make up the coordinates of the vertex

(h, k)

.

3

Write the vertex coordinates.

The vertex coordinates are

(h, k) = (-2, -5)

.

Combining the identified values gives the final coordinates of the vertex.

Anahtar Kavram

Identifying the vertex of a quadratic function directly from its vertex form equation,

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

.

Soru 1650Soru →

If a polynomial $p(x)$ is divided by $x - 5$ , the remainder is $12$ . Which of the following equations must be true?

p(5) = 12

p(-5) = 12

p(5) = -12

p(12) = 5

Cevabı ve açıklamayı göster

Cevap: $p(5) = 12$

Cevap

p(5) = 12

According to the Remainder Theorem, dividing a polynomial

p(x)

by a linear expression

x - c

results in a remainder equal to

p(c)

. Here, the divisor is

x - 5

, so

c = 5

. Since the remainder is

12

, the equation that must be true is

p(5) = 12

.

Adım Adım Çözüm

1

Identify the divisor and its corresponding value of

c

using the Remainder Theorem.

The divisor is

x - 5

, so we set

x - 5 = 0

, which gives

c = 5

.

The Remainder Theorem states that the remainder of a polynomial

p(x)

divided by

x - c

is

p(c)

.

2

Set

p(c)

equal to the given remainder.

Since the remainder is

12

and

c = 5

, we get

p(5) = 12

.

This directly applies the relation

p(c) = \text{remainder}

.

Anahtar Kavram

The Remainder Theorem

Soru 1651Soru →

For the exponential function $g(x) = k \cdot a^x$ , where $k$ and $a$ are positive constants, the value of $g(x + 1)$ is $20\%$ greater than the value of $g(x - 1)$ for all real numbers $x$ . If $g(2) = 15$ , what is the value of $g(6)$ ?

21.0

21.6

31.104

36.0

Cevabı ve açıklamayı göster

Cevap: 21.6

Cevap

21.6

The correct value of 21.6 is obtained by setting up the ratio

g(x+1)/g(x-1) = a^2

. Since

g(x+1)

is

20\%

greater than

g(x-1)

, this ratio is equal to 1.20, so

a^2 = 1.20

. Since

g(6) = g(2) \cdot a^4 = 15 \cdot (a^2)^2

, substituting

a^2 = 1.20

gives

15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

.

Adım Adım Çözüm

1

Express the values of the function at x + 1 and x - 1 using the definition of g(x).

g(x + 1) = k \cdot a^{x + 1}

and

g(x - 1) = k \cdot a^{x - 1}

To set up the mathematical relationship between the two values based on the function definition.

2

Apply the condition that g(x + 1) is 20% greater than g(x - 1).

k \cdot a^{x + 1} = 1.20 \cdot k \cdot a^{x - 1}

, which simplifies to

a^2 = 1.20

To find the factor of growth over an interval of 2 units of x.

3

Relate the value of g(6) to the given value of g(2).

g(6) = g(2) \cdot a^4 = g(2) \cdot (a^2)^2

To write the unknown value in terms of the known value and the determined factor

a^2

using exponent rules.

4

Substitute the known values into the equation to calculate the result.

g(6) = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

To calculate the final numerical value of g(6).

Anahtar Kavram

Exponential Functions and Equations

Tahmini Süre:2m 30s

Soru 1652Soru →

A municipal water treatment facility utilizes 3 identical backup filtration units to process stormwater runoff. The average volume of water, $V(t)$ , in thousands of gallons, remaining to be filtered per unit $t$ hours after the units are activated is modeled by the equation:

3V(t) + 14.4t = 162

where $0 \le t \le 10$ . Which of the following is the best interpretation of the number $14.4$ in this context?

The initial total volume of water, in thousands of gallons, remaining to be filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.8.

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.6.

Cevabı ve açıklamayı göster

Cevap: The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

Cevap

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The correct answer is the option stating that 14.4 represents the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined. In the given equation

3V(t) + 14.4t = 162

, we can isolate the total volume remaining,

3V(t)

, as

3V(t) = 162 - 14.4t

. In this form, the rate of change of the total volume is

-14.4

, indicating that the combined remaining volume decreases by 14.4 thousand gallons each hour.

Adım Adım Çözüm

1

Express the total volume of water remaining to be filtered in terms of the variable

V(t)

.

Since

V(t)

is the average volume of water remaining per unit and there are 3 identical units, the total volume of water remaining to be filtered by all 3 units combined is

3V(t)

.

This allows us to analyze the total filtration system rather than the average per unit.

2

Isolate the term representing the total volume remaining,

3V(t)

, in the given equation.

3V(t) = 162 - 14.4t

This puts the equation into a standard linear form

y = mx + b

where

y = 3V(t)

is the dependent variable (total volume remaining) and

x = t

is the independent variable (time in hours).

3

Interpret the slope and the y-intercept of the isolated linear equation.

The constant term 162 is the initial total volume of water (in thousands of gallons) remaining at

t = 0

. The coefficient of

t

, which is

-14.4

, represents the change in the total volume remaining per hour.

In a linear model

y = mx + b

, the slope

m

represents the rate of change of the dependent variable per unit increase of the independent variable.

4

Determine the contextual meaning of the absolute value of the rate of change, 14.4.

A rate of change of

-14.4

thousand gallons per hour means that the total volume of water remaining decreases by 14.4 thousand gallons each hour. Therefore, 14.4 is the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate at which the remaining volume decreases is equal to the rate at which the system filters the water.

Anahtar Kavram

Interpreting coefficients and constants in a linear relationship within a real-world context, particularly when variables represent averages or totals.

Soru 1653Soru →

A ride-share driver uses a mobile application to track their daily net earnings. The driver's net earnings, $E$ , in dollars, after completing $n$ rides in a day can be modeled by the equation $E = 18n - 25$ . Which of the following is the best interpretation of the number $25$ in this context?

The driver's net earnings, in dollars, for each completed ride.

The driver's daily starting cost, in dollars, before any rides are completed.

The driver's net earnings, in dollars, after completing exactly one ride.

The number of completed rides for which the driver's net earnings are equal to 0 dollars.

Cevabı ve açıklamayı göster

Cevap: The driver's daily starting cost, in dollars, before any rides are completed.

Cevap

The driver's daily starting cost, in dollars, before any rides are completed.

In the linear equation

E = 18n - 25

, the variable

E

represents net earnings and

n

represents the number of rides. The constant term

-25

represents the value of

E

when

n = 0

. In this context, starting with

-25

dollars in net earnings means the driver has a daily starting cost or fee of

25

dollars before completing any rides.

Adım Adım Çözüm

1

Identify the component of the linear equation

E = 18n - 25

associated with the number

25

.

The number

25

is part of the constant term

-25

in the equation.

Linear equations in context are typically written in the form

y = mx + b

, where

m

is the slope and

b

is the y-intercept.

2

Determine the contextual meaning of the constant term (y-intercept) in this model.

The constant term

-25

represents the value of

E

when

n = 0

.

Setting the independent variable

n

(number of rides) to

0

gives the starting value of the dependent variable

E

(net earnings).

3

Interpret the negative value in terms of daily operations.

An initial net earning of

-25

dollars means the driver starts with a cost or loss of

25

dollars.

A negative starting value indicates a cost or fee that must be paid before any revenue is generated.

Anahtar Kavram

Interpreting the y-intercept of a linear model in context.

Soru 1654Soru →

The table below shows several values of $x$ and the corresponding values for the linear functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
1	7	17
2	9	14
4	13	8

If $(x, y)$ is the solution to the system of equations formed by $y = f(x)$ and $y = g(x)$ , what is the value of $2x + y$ ?

5

17

15

25

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The correct value of the expression is 17.

The correct answer is the value obtained by finding the linear equations for both functions, solving the resulting system of equations, and substituting the coordinates into the expression. The equation for the first function is

y = 2x + 5

and the equation for the second function is

y = -3x + 20

. Equating the two expressions yields

2x + 5 = -3x + 20

, which simplifies to

5x = 15

or

x = 3

. Substituting

x = 3

back into either equation yields

y = 11

. Evaluating the expression

2x + y

with these values gives

2(3) + 11 = 17

.

Adım Adım Çözüm

1

Determine the linear equation for

f(x)

using the points in the table.

The slope of

f(x)

is

\frac{9 - 7}{2 - 1} = 2

. Using the point-slope form with

(1, 7)

, the equation is

y - 7 = 2(x - 1)

, which simplifies to

y = 2x + 5

.

Finding the equation of the first line is necessary to set up the system of linear equations.

2

Determine the linear equation for

g(x)

using the points in the table.

The slope of

g(x)

is

\frac{14 - 17}{2 - 1} = -3

. Using the point-slope form with

(1, 17)

, the equation is

y - 17 = -3(x - 1)

, which simplifies to

y = -3x + 20

.

Finding the equation of the second line completes the system of equations.

3

Find the intersection point of

y = f(x)

and

y = g(x)

by setting the two equations equal to each other.

2x + 5 = -3x + 20 \Rightarrow 5x = 15 \Rightarrow x = 3

. Substituting

x = 3

into

y = 2x + 5

gives

y = 2(3) + 5 = 11

.

Solving the system of equations yields the values of

x

and

y

at the intersection point.

4

Calculate the value of

2x + y

using the solution

(3, 11)

.

2(3) + 11 = 6 + 11 = 17

.

This evaluates the specific linear combination requested in the question.

Anahtar Kavram

Solving systems of linear equations by translating tabular data into linear equations and finding their point of intersection.

Soru 1655Soru →

A parabola in the $xy$ -plane has a vertex at $(3, -4)$ and passes through the point $(1, 0)$ . A second parabola is created by reflecting the original parabola across the $x$ -axis, then translating the resulting graph $4$ units to the left and $2$ units down. If this second parabola represents the graph of the function $g$ , which of the following equations defines $g(x)$ ?

g(x) = -(x - 7)^2 + 2

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

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

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

Cevabı ve açıklamayı göster

Cevap: $g(x) = -(x + 1)^2 + 2$

Cevap

The equation

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

The correct equation is

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

. Since the vertex of the original parabola

f(x)

is

(3, -4)

and it passes through

(1, 0)

, its vertex form equation is

f(x) = (x - 3)^2 - 4

. Reflecting

f(x)

across the

x

-axis negates the entire function, resulting in

-(x - 3)^2 + 4

. Shifting this function

4

units to the left replaces

x

with

x + 4

, yielding

-(x + 1)^2 + 4

. Finally, shifting it

2

units down subtracts

2

from the entire expression, giving

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

.

Adım Adım Çözüm

1

Determine the equation of the original parabola

f(x)

using its vertex form

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

.

f(x) = (x - 3)^2 - 4

Since the vertex is

(3, -4)

, the vertex form is

f(x) = a(x - 3)^2 - 4

. Substituting the point

(1, 0)

gives

0 = a(1 - 3)^2 - 4 \implies 4a = 4 \implies a = 1

.

2

Reflect the graph of

f(x)

across the

x

-axis.

-f(x) = -(x - 3)^2 + 4

A reflection across the

x

-axis negates the entire function, so

y = -f(x)

.

3

Translate the reflected graph

4

units to the left.

-(x + 1)^2 + 4

Translating a function

h(x)

to the left by

c

units is represented by

h(x + c)

. Replacing

x

with

x + 4

in

-(x - 3)^2 + 4

gives

-((x + 4) - 3)^2 + 4 = -(x + 1)^2 + 4

.

4

Translate the graph

2

units down to find the final function

g(x)

.

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

Translating a function down by

d

units is represented by subtracting

d

from the function. Subtracting

2

from

-(x + 1)^2 + 4

gives

-(x + 1)^2 + 2

.

Anahtar Kavram

Applying transformations (reflections and horizontal/vertical translations) to quadratic functions in vertex form.

Soru 1656Soru →

The quadratic function $f$ is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and has its vertex at $(2, 4)$ . The function $g$ is defined by $g(x) = f(x - 3) + k$ , where $k$ is a constant. The vertex of the graph of $y = g(x)$ in the $xy$ -plane lies on the line $y = 2x - 3$ . If the product of the $x$ -intercepts of the graph of $y = g(x)$ is $11$ , what is the $y$ -intercept of the graph of $y = f(x)$ ?

2

6

-4

-6

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The correct answer is 2. By writing the quadratic function

f(x)

in vertex form as

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

, we can apply the transformation rules to express

g(x) = f(x - 3) + k

as having a vertex at

(5, 4 + k)

. Since this vertex lies on the line

y = 2x - 3

, substituting

x = 5

gives

4 + k = 7

, meaning

g(x) = a(x - 5)^2 + 7

. The roots of

g(x) = 0

are

5 \pm \sqrt{-7/a}

, and their product is

25 + 7/a

. Setting this product equal to 11 yields

a = -1/2

. Substituting

a = -1/2

back into the expression for

f(0) = 4a + 4

gives 2.

Adım Adım Çözüm

1

Write the function

f(x)

in vertex form using its vertex

(2, 4)

.

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

Any quadratic function with a vertex at

(h, k)

can be written in the form

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

.

2

Determine the vertex of the transformed function

g(x) = f(x - 3) + k

.

The vertex of

g(x)

is

(5, 4 + k)

.

The transformation

f(x - 3)

shifts the graph 3 units to the right, changing the

x

-coordinate of the vertex from 2 to 5. The addition of

k

shifts the graph vertically by

k

units, changing the

y

-coordinate of the vertex from 4 to

4 + k

.

3

Use the condition that the vertex of

y = g(x)

lies on the line

y = 2x - 3

to find the value of

k

and the vertex coordinates.

4 + k = 7 \implies k = 3

. The vertex of

g(x)

is

(5, 7)

.

Since the vertex

(5, 4 + k)

lies on the line

y = 2x - 3

, substituting

x = 5

into the line equation gives

y = 2(5) - 3 = 7

.

4

Write

g(x)

in vertex form and express its roots (the

x

-intercepts) in terms of

a

.

g(x) = a(x - 5)^2 + 7

. The roots are

x = 5 \pm \sqrt{-\frac{7}{a}}

.

Setting

g(x) = 0

gives

a(x - 5)^2 + 7 = 0

, which simplifies to

(x - 5)^2 = -\frac{7}{a}

, yielding

x = 5 \pm \sqrt{-\frac{7}{a}}

.

5

Calculate the product of the roots and set it equal to 11 to solve for

a

.

25 + \frac{7}{a} = 11 \implies a = -\frac{1}{2}

.

The product of the roots is

(5 + \sqrt{-\frac{7}{a}})(5 - \sqrt{-\frac{7}{a}}) = 25 - (-\frac{7}{a}) = 25 + \frac{7}{a}

. Setting this equal to 11 gives

\frac{7}{a} = -14

, so

a = -\frac{1}{2}

.

6

Evaluate

f(0)

to find the

y

-intercept of the graph of

f(x)

.

f(0) = 4(-\frac{1}{2}) + 4 = 2

.

The

y

-intercept is the value of the function at

x = 0

. Substituting

x = 0

and

a = -\frac{1}{2}

into

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

yields

4a + 4 = 2

.

Anahtar Kavram

Quadratic transformations and properties of roots in vertex form

Soru 1657Soru →

\begin{aligned} y &= 2x^2 - 5x - 1 \\ y &= 3x - 7 \end{aligned}

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

-11

5

-3

1

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Substituting

y = 3x - 7

into

y = 2x^2 - 5x - 1

yields the quadratic equation

3x - 7 = 2x^2 - 5x - 1

. Moving all terms to one side gives

2x^2 - 8x + 6 = 0

. Dividing by 2 simplifies this to

x^2 - 4x + 3 = 0

, which factors as

(x - 1)(x - 3) = 0

. This gives the solutions

x = 1

and

x = 3

. Substituting these into the linear equation yields the coordinate points

(1, -4)

and

(3, 2)

. Since the problem states that

y > 0

, we choose the solution

(3, 2)

. The value of

x + y

is

3 + 2 = 5

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the linear equation into the quadratic equation.

3x - 7 = 2x^2 - 5x - 1

Since both equations are solved for

y

, they can be set equal to each other to find the

x

-coordinates of the intersection points.

2

Rearrange the terms to set the quadratic equation equal to zero, and then divide by the common factor.

2x^2 - 8x + 6 = 0

, which simplifies to

x^2 - 4x + 3 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

allows us to factor and solve for

x

.

3

Factor the quadratic equation to find the values of

x

.

(x - 1)(x - 3) = 0

, so

x = 1

or

x = 3

Factoring allows us to find the roots of the equation directly.

4

Substitute the

x

values back into the linear equation to solve for the corresponding

y

values.

For

x = 1

,

y = 3(1) - 7 = -4

, giving the solution

(1, -4)

. For

x = 3

,

y = 3(3) - 7 = 2

, giving the solution

(3, 2)

.

Finding the

y

-coordinates completes the solutions

(x, y)

to the system.

5

Apply the given constraint

y > 0

and calculate

x + y

.

The solution

(3, 2)

satisfies

y > 0

since

2 > 0

. Thus,

x + y = 3 + 2 = 5

.

The problem asks for the sum of the coordinates of the solution that has a positive

y

-value.

Anahtar Kavram

Solving a system consisting of a linear equation and a quadratic equation by substitution and applying coordinate constraints.

Tahmini Süre:1m 30s

Soru 1658Soru →

A quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(0, 5)$ and $(4, 5)$ . If the minimum value of $f(x)$ for $0 \le x \le 3$ is $1$ , what is the value of $f(6)$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The value of

f(6)

is

17

.

The correct answer is

17

. Since the quadratic function passes through

(0, 5)

and

(4, 5)

, its axis of symmetry is the line

x = 2

, which means the vertex

x

-coordinate is

h = 2

. For the interval

0 \le x \le 3

, this vertex is within the bounds. An upward-opening parabola has its minimum value at its vertex, so the minimum value of

1

must be the

y

-coordinate of the vertex, giving

k = 1

. The function can then be written as

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

. Substituting

(0, 5)

into this equation gives

5 = a(0 - 2)^2 + 1

, which simplifies to

4a = 4

, meaning

a = 1

. The fully determined function is

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

. Evaluating this at

x = 6

yields

f(6) = (6 - 2)^2 + 1 = 17

.

Adım Adım Çözüm

1

Determine the axis of symmetry of the parabola.

h = 2

Since the points

(0, 5)

and

(4, 5)

have the same

y

-value, they are symmetric across the vertical line of symmetry of the parabola. The

x

-coordinate of the vertex

h

is the average of the

x

-coordinates of these two points:

h = \frac{0 + 4}{2} = 2

.

2

Determine the

y

-coordinate of the vertex

k

using the minimum value over the interval

[0, 3]

.

k = 1

The vertex

x

-coordinate

x = 2

lies within the interval

0 \le x \le 3

. For a quadratic function that opens upward, the minimum value occurs at the vertex. Thus, the minimum value on this interval is the vertex

y

-coordinate, so

k = 1

.

3

Substitute the point

(0, 5)

into the vertex form of the equation to solve for

a

.

a = 1

Using the vertex form

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

, substitute

x = 0

and

f(x) = 5

:

5 = a(0 - 2)^2 + 1 \implies 5 = 4a + 1 \implies 4a = 4 \implies a = 1

.

4

Evaluate

f(6)

using the complete quadratic function formula.

f(6) = 17

Substitute

x = 6

into the equation

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

:

f(6) = (6 - 2)^2 + 1 = 4^2 + 1 = 16 + 1 = 17

.

Anahtar Kavram

Using symmetry properties and interval extrema to determine the equation of a quadratic function in vertex form.

Soru 1659Soru →

\begin{aligned} y + 11 &= x^2 \\ y - 3x &= 7 \end{aligned}

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

Cevabı ve açıklamayı göster

Cevap: 31

Cevap

The correct answer is 31.

Solving the system of equations by expressing

y

in terms of

x

from the linear equation gives

y = 3x + 7

. Substituting this expression for

y

in the first equation yields

(3x + 7) + 11 = x^2

, which simplifies to

x^2 - 3x - 18 = 0

. Factoring the quadratic expression gives

(x - 6)(x + 3) = 0

, so

x = 6

or

x = -3

. Since the problem specifies that

x > 0

, the

x

-value of the solution must be

6

. Substituting

x = 6

back into the linear equation gives

y = 3(6) + 7 = 25

. The sum of

x

and

y

is

6 + 25 = 31

.

Adım Adım Çözüm

1

Express

y

in terms of

x

from the linear equation.

y = 3x + 7

This isolates the variable

y

to facilitate substitution.

2

Substitute the expression for

y

into the quadratic equation.

(3x + 7) + 11 = x^2

This eliminates the variable

y

, yielding a single equation in terms of

x

.

3

Rearrange the equation into standard quadratic form.

x^2 - 3x - 18 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve by factoring.

4

Factor the quadratic equation.

(x - 6)(x + 3) = 0

Finding factors helps determine the possible values of

x

.

5

Find the values of

x

and apply the positive constraint.

x = 6

The equation yields

x = 6

and

x = -3

. The condition

x > 0

restricts the solution to

x = 6

.

6

Calculate the corresponding value of

y

.

y = 25

Substituting

x = 6

into the linear equation gives

y = 3(6) + 7 = 25

.

7

Calculate the sum of

x

and

y

.

31

The problem asks for the value of

x + y

.

Anahtar Kavram

Solving systems of nonlinear equations algebraically using substitution and quadratic factoring.

Alternatif Yöntem

Alternatively, solve the linear equation for

x

to get

x = \frac{y - 7}{3}

and substitute this into the quadratic equation to solve for

y

first. This approach is more complex because it introduces fractional terms.

Tahmini Süre:1m 30s

Soru 1660Soru →

In the nineteenth century, mathematical physicist James Clerk Maxwell proposed a famous thought experiment involving a hypothetical entity _______ a microscopic 'demon' capable of sorting fast and slow molecules to decrease entropy. By separating the molecules based on their thermal energy without performing work, this imaginary creature appeared to violate the second law of thermodynamics.

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

entity:

entity;

entity, it was

entity; it being

Cevabı ve açıklamayı göster

Cevap: entity:

Cevap

The correct option is the one that inserts a colon after "entity".

The punctuation mark must connect the independent clause preceding the blank to the explanatory appositive phrase that follows. A colon is appropriate here because it introduces an explanation, illustration, or definition of a concept mentioned in the preceding independent clause.

Adım Adım Çözüm

1

Analyze the clause structure before the blank.

The clause "In the nineteenth century, mathematical physicist James Clerk Maxwell proposed a famous thought experiment involving a hypothetical entity" is independent.

It contains a complete subject and verb and can stand alone as a sentence.

2

Analyze the structure of the phrase after the blank.

The phrase "a microscopic 'demon' capable of sorting fast and slow molecules to decrease entropy" is a noun phrase that serves as an appositive explaining "entity".

It lacks a finite verb and cannot stand alone as an independent clause.

3

Select the appropriate punctuation mark to link these structures.

A colon is the correct punctuation to introduce explanatory information or an appositive following an independent clause.

A semicolon requires independent clauses on both sides, while a comma splice would occur if a comma were used to link two independent clauses.

Anahtar Kavram

Colons, Semicolons, and Dashes

Tahmini Süre:1m 0s

Cevap

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Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

Alternatif Yöntem

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