All practice questions

Question 1821Question →

The quadratic function $f$ is defined by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In the $xy$ -plane, the graph of $f$ is a parabola with vertex $(3, -5)$ that passes through the point $(0, 4)$ . What is the value of $a + b + c$ ?

11

-9

-1

-7

Show answer & explanation

Answer: -1

Answer

-1

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Substituting the given vertex

(3, -5)

gives

f(x) = a(x - 3)^2 - 5

. Since the graph passes through the point

(0, 4)

, substituting

x = 0

and

f(x) = 4

yields the equation

4 = a(0 - 3)^2 - 5

, which simplifies to

9a = 9

, or

a = 1

. The function is therefore defined by

f(x) = (x - 3)^2 - 5

. The expression

a + b + c

represents the sum of the coefficients of the quadratic function in standard form

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

. Evaluating the function at

x = 1

gives

f(1) = a(1)^2 + b(1) + c = a + b + c

. Substituting

x = 1

into our vertex form equation yields

f(1) = (1 - 3)^2 - 5 = (-2)^2 - 5 = 4 - 5 = -1

. Therefore, the value of

a + b + c

is

-1

.

Step-by-Step Solution

1

Write the quadratic function in vertex form using the given vertex

(3, -5)

.

f(x) = a(x - 3)^2 - 5

, where

a

is a constant.

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the given point

(0, 4)

into the vertex form equation to solve for

a

.

4 = a(0 - 3)^2 - 5

, which simplifies to

4 = 9a - 5

, and solving for

a

gives

a = 1

.

Since the graph passes through

(0, 4)

, substituting these coordinates into the function's equation must yield a true statement.

3

Find the value of

a + b + c

by evaluating

f(1)

.

f(1) = 1(1 - 3)^2 - 5 = 1(-2)^2 - 5 = 4 - 5 = -1

.

For any quadratic function

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

, evaluating the function at

x = 1

gives

f(1) = a(1)^2 + b(1) + c = a + b + c

.

Key Concept

Quadratic Functions and Graphs

Question 1822Question →

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 $r$ ?

Show answer & explanation

Answer: 3.8

Answer

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

.

Step-by-Step Solution

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.

Key Concept

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

Question 1823Question →

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

Show answer & explanation

Answer: 120

Answer

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.

Step-by-Step Solution

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}

.

Key Concept

Unit Conversions

Question 1824Question →

Which form of the noun 'archaeologist' grammatically completes the passage?

Fill in the blanks below

During a 2021 excavation in Egypt's Valley of the Kings, a team of archaeologists uncovered a series of previously undocumented tomb shafts. Because the humidity inside the chambers could damage the fragile wall paintings, the primary objective was to stabilize the microclimate before proceeding with any excavation.

Show answer & explanation

Answer

archaeologists'

The correct answer is 'archaeologists''. The context establishes that there are multiple archaeologists ('a team of archaeologists') and that the objective belongs to them ('primary objective'). To form the possessive of a plural noun ending in 's', an apostrophe is added to the end of the word, resulting in 'archaeologists''.

Step-by-Step Solution

1

Determine the number (singular or plural) of the noun based on context clues.

The noun refers back to 'a team of archaeologists,' which indicates multiple individuals. Therefore, the plural form is required.

The phrase 'a team of archaeologists' establishes a plural subject.

2

Determine the case (possessive or non-possessive) of the noun based on its grammatical role.

The noun modifies the noun phrase 'primary objective,' indicating ownership or association. Therefore, the possessive case is required.

A noun that shows ownership over another noun must be in the possessive form.

3

Apply standard spelling and punctuation rules for plural possessive nouns.

For a regular plural noun ending in 's' (archaeologists), the possessive is formed by adding an apostrophe after the final 's', yielding 'archaeologists''.

Plural possessive nouns ending in 's' receive only an apostrophe at the end, without an additional 's'.

Key Concept

Plural and possessive nouns require correct placement of the apostrophe depending on the number of the noun and its relationship to other words in the sentence.

Question 1825Question →

A retailer purchased a shipment of coats for a wholesale price of $x$ dollars each. The retailer marked up the wholesale price by $40\%$ to determine the retail price. During a winter sale, the retailer offered a discount of $20\%$ off the retail price of the coats. If the sale price of a coat was $84$ dollars, what was the wholesale price, $x$ , of the coat?

63

70

75

80

Show answer & explanation

Answer: $75$

Answer

75

The correct answer is

75

. The retail price is determined by increasing the wholesale price,

x

, by

40\%

, which can be represented as

1.40x

. The sale price is determined by decreasing the retail price by

20\%

, which is

0.80(1.40x) = 1.12x

. Since the sale price of the coat was

84

dollars, we set up the equation

1.12x = 84

. Solving for

x

yields

x = \frac{84}{1.12} = 75

.

Step-by-Step Solution

1

Represent the retail price in terms of the wholesale price

x

.

Retail price =

1.40x

A markup of

40\%

on

x

is equivalent to multiplying

x

by

1 + 0.40 = 1.40

.

2

Represent the sale price in terms of the retail price.

Sale price =

0.80(1.40x) = 1.12x

A discount of

20\%

off the retail price is equivalent to multiplying the retail price by

1 - 0.20 = 0.80

.

3

Set up the equation with the given sale price and solve for

x

.

1.12x = 84 \implies x = \frac{84}{1.12} = 75

The sale price of the coat is given as

84

dollars, so we set the algebraic representation equal to

84

and solve for

x

.

Key Concept

Applying consecutive percentage increases and decreases to find initial values.

Question 1826Question →

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 $p(6)$ ?

Show answer & explanation

Answer: 10

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Identifying the algebraic form of a polynomial from its

x

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

Question 1827Question →

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?

Show answer & explanation

Answer: 80

Answer

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.

Step-by-Step Solution

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.

Key Concept

Solving systems of linear equations in context

Question 1828Question →

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 $g(-1)$ ?

Show answer & explanation

Answer: 5

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Question 1829Question →

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 $p$ ?

Show answer & explanation

Answer: 20

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Compounding percent changes algebraically using variable markups and discounts.

Question 1830Question →

In the quadratic equation $x^2 - 8x + k = 0$ , $k$ is a constant. If the difference between the two real solutions to the equation is $2$ , what is the value of $k$ ?

15

-15

12

8

Show answer & explanation

Answer: 15

Answer

15

The correct answer is 15. The sum of the solutions to the quadratic equation

x^2 - 8x + k = 0

is given by

-\frac{-8}{1} = 8

. We are given that the difference between the two solutions is

2

. Let the two solutions be

x_1

and

x_2

. We can set up the system of equations:

x_1 + x_2 = 8

and

x_1 - x_2 = 2

. Adding the equations gives

2x_1 = 10

, so

x_1 = 5

. Thus,

x_2 = 3

. The product of the solutions is equal to the constant term

k

. Therefore,

k = x_1 \times x_2 = 5 \times 3 = 15

. Alternatively, using the difference of roots formula,

\frac{\sqrt{b^2 - 4ac}}{a} = 2 \Rightarrow \frac{\sqrt{(-8)^2 - 4(1)(k)}}{1} = 2 \Rightarrow \sqrt{64 - 4k} = 2

. Squaring both sides gives

64 - 4k = 4 \Rightarrow 4k = 60 \Rightarrow k = 15

.

Step-by-Step Solution

1

Find the sum of the two solutions of the quadratic equation.

The sum of the solutions is

8

.

For a quadratic equation in the form

x^2 + bx + c = 0

, the sum of the solutions is equal to

-b

. Here,

-(-8) = 8

.

2

Set up a system of linear equations using the sum and the given difference of the solutions to find the individual solutions.

The solutions are

5

and

3

.

Let the solutions be

x_1

and

x_2

. We have the system:

x_1 + x_2 = 8

and

x_1 - x_2 = 2

. Adding these two equations gives

2x_1 = 10

, so

x_1 = 5

. Substituting

5

back into the first equation gives

5 + x_2 = 8

, so

x_2 = 3

.

3

Use the product of the solutions to determine the value of

k

.

The value of

k

is

15

.

For a quadratic equation in the form

x^2 + bx + c = 0

, the product of the solutions is equal to

c

, which is

k

. Therefore,

k = x_1 \times x_2 = 5 \times 3 = 15

.

Key Concept

Relationship between the roots and coefficients of a quadratic equation (Vieta's Formulas).

Question 1831Question →

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 $b$ ?

Show answer & explanation

Answer: 19

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Determining the equation of a line given two points.

Question 1832Question →

A polynomial function $p$ of degree 3 has $x$ -intercepts at $(-2, 0)$ with multiplicity 2, and $(3, 0)$ with multiplicity 1. In the $xy$ -plane, the graph of $y = p(x)$ intersects the $y$ -axis at $(0, 24)$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

8

16

36

-36

Show answer & explanation

Answer: $36$

Answer

The remainder when the polynomial function is divided by

x - 1

is 36.

The correct answer is 36. A polynomial with a root at

x = -2

of multiplicity 2 and a root at

x = 3

of multiplicity 1 has the form

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

. Since the

y

-intercept is

(0, 24)

, we solve

p(0) = a(2)^2(-3) = 24

to find

a = -2

. Thus, the polynomial is

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

. According to the Remainder Theorem, dividing

p(x)

by

x - 1

leaves a remainder of

p(1)

. Substituting

x = 1

yields

p(1) = -2(3)^2(-2) = 36

.

Step-by-Step Solution

1

Write the general form of the cubic polynomial using its roots and multiplicities.

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

Since there is an

x

-intercept at

x = -2

with multiplicity 2,

(x + 2)^2

is a factor. Since there is an

x

-intercept at

x = 3

with multiplicity 1,

(x - 3)

is a factor. Here,

a

is a constant coefficient.

2

Determine the value of the constant coefficient

a

using the

y

-intercept.

a = -2

, so

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

The graph intersects the

y

-axis at

(0, 24)

, meaning

p(0) = 24

. Substituting

x = 0

gives

p(0) = a(0 + 2)^2(0 - 3) = -12a

. Setting

-12a = 24

yields

a = -2

.

3

Apply the Remainder Theorem to find the required remainder.

The remainder is equal to

p(1)

.

By the Remainder Theorem, the remainder when a polynomial

p(x)

is divided by

x - c

is

p(c)

. Here, the divisor is

x - 1

, so we evaluate the polynomial at

x = 1

.

4

Calculate the value of

p(1)

.

p(1) = 36

Substituting

x = 1

into

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

gives

p(1) = -2(1 + 2)^2(1 - 3) = -2(9)(-2) = 36

.

Key Concept

Identifying a polynomial from its roots and multiplicities, solving for its leading coefficient using a given point, and applying the Remainder Theorem.

Question 1833Question →

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 $y = g(x)$ ?

Show answer & explanation

Answer: 13

Answer

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

.

Step-by-Step Solution

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.

Key Concept

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

Question 1834Question →

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 $(h, k)$ . The function $g$ is defined by $g(x) = f(x - 3) + 4$ . The graph of $g$ passes through the origin $(0,0)$ , and its vertex lies on the line $y = x$ in the first quadrant. What is the value of $f(0)$ ?

-1

-7

-3

-21

Show answer & explanation

Answer: -7

Answer

-7

The correct answer is

-7

. Representing

f(x)

in vertex form as

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

tells us that translating the function 3 units right and 4 units up shifts the vertex from

(h, k)

to

(h+3, k+4)

. Since this vertex lies on the line

y=x

, we have

h+3 = k+4

, or

k = h-1

. Since the vertex of

g

lies in the first quadrant, its coordinates must be positive, meaning

h > -3

and

k > -4

. Using the fact that the graph of

g

passes through the origin, we have

g(0) = 0 \implies f(-3) + 4 = 0 \implies f(-3) = -4

. Substituting

x = -3

into the vertex form of

f(x)

gives

-(-3-h)^2 + k = -4 \implies (h+3)^2 - k = 4

. Substituting

k = h-1

yields

(h+3)^2 - (h-1) = 4 \implies h^2 + 5h + 6 = 0

, which factors into

(h+2)(h+3) = 0

. The solution

h = -3

is discarded because it places the vertex of

g

at

(0,0)

, which is not in the first quadrant. Therefore,

h = -2

and

k = -3

. Calculating

f(0)

gives

-h^2 + k = -(-2)^2 + (-3) = -7

.

Step-by-Step Solution

1

Determine the vertex form of

f(x)

and the vertex of

g(x)

.

Since

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

has its vertex at

(h, k)

, its vertex form is

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

. The function

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

represents a horizontal shift of

f

by 3 units to the right and a vertical shift by 4 units up. Thus, the vertex of

g

is

(h + 3, k + 4)

.

Understanding translations allows us to write the coordinates of the new vertex in terms of the original vertex variables.

2

Use the line

y = x

and quadrant constraints to find a relation between

h

and

k

.

Since the vertex of

g

lies on the line

y = x

, we set its coordinates equal:

h + 3 = k + 4

, which simplifies to

k = h - 1

. Additionally, because the vertex is in the first quadrant, we must have

h + 3 > 0

and

k + 4 > 0

.

The geometric placement of the vertex on the line

y = x

constrains its coordinate values.

3

Apply the condition that the graph of

g

passes through the origin.

g(0) = 0 \implies f(-3) + 4 = 0 \implies f(-3) = -4

. Substituting

x = -3

into the vertex form of

f(x)

gives

-(-3 - h)^2 + k = -4

, which simplifies to

(h + 3)^2 - k = 4

.

The point

(0,0)

lying on the graph of

g

provides an equation to solve for the vertex parameters.

4

Substitute

k = h - 1

into the quadratic equation and solve for

h

.

(h + 3)^2 - (h - 1) = 4 \implies h^2 + 6h + 9 - h + 1 = 4 \implies h^2 + 5h + 6 = 0

. Factoring gives

(h + 2)(h + 3) = 0

, so

h = -2

or

h = -3

.

Substituting the linear relation into the quadratic equation isolates the variable

h

.

5

Filter the solutions using the first quadrant constraint and calculate

f(0)

.

If

h = -3

, then

k = -4

, giving the vertex of

g

at

(0, 0)

, which is not in the first quadrant. If

h = -2

, then

k = -3

, giving the vertex of

g

at

(1, 1)

, which is in the first quadrant. Thus,

h = -2

and

k = -3

. The value of

f(0)

is

f(0) = -(0 - h)^2 + k = -h^2 + k = -(-2)^2 + (-3) = -4 - 3 = -7

.

The quadrant condition uniquely determines the correct vertex coordinates, allowing us to find the y-intercept of the original function.

Key Concept

Quadratic Functions and Graphs

Question 1835Question →

For which values of $x$ is the inequality $-4(2x - 5) \geq 3x + 9$ true?

x \geq 1

x \leq -1

x \leq 1

x \leq -\frac{29}{11}

Show answer & explanation

Answer: $x \leq 1$

Answer

The inequality is true for all values of

x

such that

x

is less than or equal to

1

.

The correct inequality representing the solution set is the one that shows the variable is less than or equal to one.

Step-by-Step Solution

1

Distribute the factor of

-4

to the terms inside the parentheses on the left side of the inequality.

-8x + 20 \geq 3x + 9

Applying the distributive property gives

-4 \times 2x = -8x

and

-4 \times -5 = 20

.

2

Subtract

3x

from both sides of the inequality to collect all terms with the variable

x

on the left side.

-11x + 20 \geq 9

Grouping the variable terms helps isolate the variable.

3

Subtract

20

from both sides of the inequality to isolate the variable term.

-11x \geq -11

Subtracting twenty from both sides moves the constant terms to the right side of the inequality.

4

Divide both sides by

-11

and reverse the direction of the inequality sign.

x \leq 1

Dividing or multiplying an inequality by a negative number requires flipping the inequality symbol.

Key Concept

Linear Inequalities in One Variable

Question 1836Question →

Unlike almost all other vertebrates, fish of the family Channichthyidae, or Antarctic icefish, do not possess hemoglobin to transport oxygen in their blood. Instead, they survive in the freezing, oxygen-rich waters of the Southern Ocean by relying on oxygen dissolved directly in their blood plasma ______ biological anomaly is facilitated by their unusually large hearts and wide blood vessels, which pump blood at high rates.

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

, this

this

; this

, although this

Show answer & explanation

Answer: ; this

Answer

The correct answer is the option containing a semicolon followed by the pronoun 'this' ('; this').

The correct answer is the option that reads '; this'. The sentence contains two independent clauses: 'Instead, they survive in the freezing, oxygen-rich waters of the Southern Ocean by relying on oxygen dissolved directly in their blood plasma' and 'this biological anomaly is facilitated by their unusually large hearts and wide blood vessels, which pump blood at high rates.' A semicolon is a grammatically correct way to link two independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Analyze the structure of the two clauses separated by the blank.

The first clause ('Instead, they survive... blood plasma') is an independent clause because it has a subject and a verb and can stand alone. The second clause ('this biological anomaly... high rates') is also an independent clause with its own subject ('this biological anomaly') and verb ('is facilitated').

Identifying clause types is necessary to determine the appropriate punctuation and conjunctions needed to link them.

2

Determine the grammatically correct way to connect two independent clauses.

Two independent clauses must be joined by a period, a semicolon, or a coordinating conjunction (like 'and') preceded by a comma.

Joining two independent clauses with only a comma creates a comma splice, and joining them with no punctuation at all creates a run-on sentence.

3

Evaluate the choices to find the one that correctly connects the clauses without introducing grammatical or logical errors.

The choice containing '; this' uses a semicolon to separate the two independent clauses, which is grammatically correct. The choice with ', this' is a comma splice, the choice with 'this' is a run-on sentence, and the choice with ', although this' incorrectly uses a subordinating conjunction of contrast when the relationship between the clauses is explanatory.

The correct option must satisfy standard English grammar rules and preserve the logical connection between the ideas.

Key Concept

Clause Boundaries and Linking

Question 1837Question →

If $x$ satisfies the equation below, what is the value of $x - 3$ ?

x - \sqrt{2x + 7} = 4

-2

6

9

-4

Show answer & explanation

Answer: 6

Answer

6

The correct answer is the value

6

. Isolating the radical in

x - \sqrt{2x + 7} = 4

gives

x - 4 = \sqrt{2x + 7}

. Squaring both sides yields

x^2 - 8x + 16 = 2x + 7

, which simplifies to the quadratic equation

x^2 - 10x + 9 = 0

. Factoring this equation gives

(x - 9)(x - 1) = 0

, yielding potential solutions of

x = 9

and

x = 1

. Substituting these potential solutions back into the original equation reveals that

x = 9

is a valid solution because

9 - \sqrt{25} = 4

, whereas

x = 1

is extraneous because

1 - \sqrt{9} = -2 \neq 4

. The value of the expression

x - 3

is

9 - 3 = 6

.

Step-by-Step Solution

1

Isolate the radical term in the equation.

x - 4 = \sqrt{2x + 7}

Before squaring both sides of a radical equation, it is best to isolate the radical term on one side.

2

Square both sides of the equation to eliminate the radical.

(x - 4)^2 = 2x + 7

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

3

Expand the squared binomial and move all terms to one side to set the quadratic equation to zero.

x^2 - 10x + 9 = 0

(x - 4)^2

expands to

x^2 - 8x + 16

. Subtracting

2x

and

7

from both sides gives the standard quadratic form

ax^2 + bx + c = 0

.

4

Factor the quadratic equation.

(x - 9)(x - 1) = 0

Finding two numbers that multiply to

9

and add to

-10

gives

-9

and

-1

. Thus, the potential solutions are

x = 9

and

x = 1

.

5

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

x = 9

is valid, while

x = 1

is extraneous.

Checking

x = 9

:

9 - \sqrt{2(9)+7} = 9 - 5 = 4

(true). Checking

x = 1

:

1 - \sqrt{2(1)+7} = 1 - 3 = -2 \neq 4

(false). Thus,

x = 9

is the only real solution.

6

Calculate the value of the requested expression

x - 3

.

9 - 3 = 6

Since the only valid solution is

x = 9

, we substitute

9

into the expression

x - 3

to find the final answer.

Key Concept

Radical equations require isolating the radical, squaring both sides, solving the resulting equation, and checking for extraneous solutions that do not satisfy the original equation.

Estimated Time:1m 30s

Question 1838Question →

A student is editing a passage about early botanical photography. Complete the passage by typing the correct form of the noun 'botanist' in the blank. Based on the grammatical context of the passage, what is the correct plural possessive form of the noun 'botanist' that should be written in the blank?

Fill in the blanks below

During the nineteenth century, standard academic practice required botanists to preserve plant specimens through delicate drying and pressing techniques. However, the introduction of cyanotype photography offered a more durable and efficient alternative. By capturing precise silhouettes of leaves and flowers on light-sensitive paper, the new process greatly simplified efforts to document diverse flora.

Show answer & explanation

Answer

botanists'

The sentence requires a plural possessive noun because the context refers to the collective efforts of multiple botanists introduced in the first sentence. The plural noun is 'botanists', and its possessive form is created by adding an apostrophe after the 's', resulting in 'botanists''.

Step-by-Step Solution

1

Determine whether the noun 'botanist' should be singular or plural based on the context.

The first sentence refers to 'botanists' in the plural, establishing that the passage discusses the collective work of multiple scientists.

This identifies the correct grammatical number for the noun.

2

Determine whether the noun requires a possessive form to modify the noun 'efforts'.

The blank precedes the noun 'efforts', indicating that the efforts belong to the botanists. Thus, a possessive form is required.

This identifies the correct grammatical case for the noun.

3

Form the plural possessive of 'botanist'.

The plural form 'botanists' ends in 's', so the possessive is formed by adding an apostrophe after the 's', yielding 'botanists''.

This applies the correct English punctuation rule for plural possessive nouns.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1839Question →

A 3D printer extrudes plastic filament at a constant rate of $8$ millimeters per second. What is this rate, in meters per hour?

0.48

2.88

28.8

28,800

Show answer & explanation

Answer: $28.8$

Answer

The rate is

28.8

meters per hour.

To convert the rate from millimeters per second to meters per hour, we apply the unit conversion factors. We convert millimeters to meters by dividing by

1000

, and convert seconds to hours by multiplying by

3600

. This gives:

8\text{ mm/s} \times \frac{1\text{ m}}{1000\text{ mm}} \times \frac{3600\text{ s}}{1\text{ hour}} = 8 \times 3.6 = 28.8\text{ meters per hour}

.

Step-by-Step Solution

1

Convert the length unit from millimeters to meters.

Since

1\text{ meter} = 1000\text{ millimeters}

, a rate of

8\text{ millimeters per second}

is equal to

\frac{8}{1000} = 0.008\text{ meters per second}

.

To convert from a smaller unit (millimeters) to a larger unit (meters), divide by the conversion factor of

1000

.

2

Convert the time unit from seconds to hours.

Since

1\text{ hour} = 3600\text{ seconds}

, a rate of

0.008\text{ meters per second}

is equal to

0.008 \times 3600 = 28.8\text{ meters per hour}

.

To find the distance printed in one hour, multiply the distance printed in one second by the number of seconds in one hour (

3600

).

Key Concept

Unit Conversions

Alternative Method

We can combine the conversion steps into a single conversion factor:

\frac{3600\text{ seconds/hour}}{1000\text{ millimeters/meter}} = 3.6\text{ (meters \cdot seconds) / (millimeters \cdot hour)}

. Multiplying the rate of

8\text{ mm/s}

by

3.6

gives the rate directly as

28.8\text{ m/h}

.

Estimated Time:1m 0s

Question 1840Question →

An electric vehicle is connected to a charging station. The energy stored in the vehicle's battery pack, $E$ , in kilowatt-hours (kWh), can be modeled by the equation $E = 0.75m + 24.0$ , where $m$ is the number of minutes the vehicle has been charging. Which of the following is the best interpretation of the number $0.75$ in this context?

The initial energy stored in the battery pack, in kilowatt-hours, before the vehicle began charging

The increase in the charging time, in minutes, for each additional kilowatt-hour of energy stored in the battery pack

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The total energy stored in the battery pack, in kilowatt-hours, after the vehicle has been charging for

1

minute

Show answer & explanation

Answer: The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

Answer

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The coefficient of

m

in the linear equation

E = 0.75m + 24.0

is the slope of the line. In context, the slope represents the rate of change of the dependent variable (stored energy,

E

) per unit increase of the independent variable (time,

m

). Thus,

0.75

represents an increase of

0.75

kilowatt-hours in stored energy for each additional minute of charging.

Step-by-Step Solution

1

Identify the form and components of the given equation.

The equation

E = 0.75m + 24.0

is in the slope-intercept form

y = mx + b

, where

m = 0.75

is the slope and

b = 24.0

is the y-intercept.

To relate the numerical values in the equation to their mathematical interpretations.

2

Interpret the meaning of the slope in context.

The slope,

0.75

, represents the unit rate of change of the dependent variable (energy

E

in kWh) per unit change of the independent variable (time

m

in minutes).

To explain what the coefficient of

m

means in real-world terms.

3

Combine the units and direction of change.

Since

0.75

is positive, the energy increases by

0.75

kWh for every additional

1

minute of charging time.

To formulate the final contextual interpretation of the slope.

Key Concept

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

Answer

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