All practice questions

Question 1901Question →

The graph of a linear equation in the $xy$ -plane has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, 3k)$ , where $k$ is a positive constant. If the line passes through the point $(2, 12)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of

k

is

6

.

The correct answer is

6

. The slope of the line can be found using the two intercepts

(k, 0)

and

(0, 3k)

:

m = \frac{3k - 0}{0 - k} = -3

. The equation of the line in slope-intercept form is

y = -3x + 3k

. Since the line passes through the point

(2, 12)

, we substitute

x = 2

and

y = 12

into the equation to get

12 = -3(2) + 3k

. Simplifying the equation yields

12 = -6 + 3k

, which gives

18 = 3k

. Dividing both sides by

3

results in

k = 6

.

Step-by-Step Solution

1

Find the slope of the line using the

x

-intercept

(k, 0)

and the

y

-intercept

(0, 3k)

.

The slope of the line is

-3

.

The slope formula is

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

. Substituting the points

(k, 0)

and

(0, 3k)

gives

m = \frac{3k - 0}{0 - k} = \frac{3k}{-k} = -3

.

2

Write the equation of the line in slope-intercept form.

The equation of the line is

y = -3x + 3k

.

Using the slope-intercept form

y = mx + b

, we substitute the slope

m = -3

and the

y

-intercept value

b = 3k

from the point

(0, 3k)

.

3

Substitute the given point

(2, 12)

into the equation and solve for

k

.

The value of

k

is

6

.

Substituting

x = 2

and

y = 12

into

y = -3x + 3k

yields

12 = -3(2) + 3k

. This simplifies to

12 = -6 + 3k

. Adding

6

to both sides gives

18 = 3k

, and dividing by

3

gives

k = 6

.

Key Concept

Linear equations in two variables, finding equations from intercepts, and constant determination via point substitution.

Question 1902Question →

The table below shows the daily high temperatures, in degrees Celsius, recorded in a city over a $9$ -day period.

Daily High Temperature (°C)	Number of Days
$18$	$3$
$19$	$1$
$20$	$2$
$21$	$3$

Based on the table, what was the median daily high temperature, in degrees Celsius, for the $9$ -day period?

Show answer & explanation

Answer: 20

Answer

The median daily high temperature for the 9-day period is

20

°C.

To find the median daily high temperature, we first sum the frequencies to find the total number of days:

3 + 1 + 2 + 3 = 9

. Because

9

is an odd number, the median is the middle value, which is the

5

th value when the data points are ordered from least to greatest. The first

3

values are

18

, the

4

th value is

19

, and the

5

th and

6

th values are

20

. Therefore, the

5

th value in the ordered list is

20

.

Step-by-Step Solution

1

Determine the total number of observations (days) in the data set.

9

days

To find the position of the median value, we must first calculate the total number of data points by adding the frequencies:

3 + 1 + 2 + 3 = 9

.

2

Find the position of the median in the ordered data set.

5

th position

For an odd number of observations

n

, the median is located at the

\frac{n+1}{2}

position. With

n = 9

, this corresponds to the

\frac{9+1}{2} = 5

th value.

3

Identify the temperature at the

5

th position using the frequency table.

20

Listing the temperatures in ascending order gives:

18, 18, 18

(positions 1–3),

19

(position 4), and

20, 20

(positions 5–6). The value at the

5

th position is

20

.

Key Concept

Finding the median of a frequency distribution

Question 1903Question →

A library administrator tracked a sample of $120$ books checked out last week, categorizing them by genre (fiction or nonfiction) and format (print or audiobook). The results are summarized in the table below.

Format	Fiction	Nonfiction
Print	$45$	$35$
Audiobook	$15$	$25$

Of the books in the sample that were checked out as audiobooks, what fraction were fiction?

Show answer & explanation

Answer: 0.375

Answer

The fraction of audiobooks that were fiction is 3/8 (which can also be gridded as the decimal .375).

To find the fraction of audiobooks that were fiction, find the total number of audiobooks in the sample by adding the values in the audiobook row:

15 + 25 = 40

. Next, find the number of fiction audiobooks, which is

15

. The fraction of audiobooks that were fiction is the ratio of these two values:

\frac{15}{40}

, which simplifies to

\frac{3}{8}

(or

0.375

in decimal form).

Step-by-Step Solution

1

Calculate the total number of audiobooks in the sample.

The total number of audiobooks is

15 + 25 = 40

.

The question asks for the fraction of books *that were checked out as audiobooks*, which means the denominator of our fraction must be the total number of audiobooks.

2

Identify the number of fiction audiobooks in the sample.

There are

15

fiction audiobooks.

The question asks what fraction of the audiobooks were *fiction*, so the numerator must be the number of books that are both audiobooks and fiction.

3

Write and simplify the fraction representing the conditional probability.

\frac{15}{40} = \frac{3}{8}

Divide the target number of books (15) by the total number of audiobooks (40). Simplify by dividing the numerator and the denominator by their greatest common divisor, which is 5.

Key Concept

Conditional Probability from a Two-Way Table

Estimated Time:1m 15s

Question 1904Question →

Complete the passage below by typing the grammatically correct form of the noun 'brother' in the blank.

Fill in the blanks below

In the late nineteenth century, brothers Wilbur and Orville Wright meticulously studied the aerodynamic behavior of soaring birds to design their first successful glider. The innovative wing-warping system, which allowed the aircraft to roll and turn smoothly, was directly inspired by the way buzzards tilt their wings to maintain balance in turbulent air currents.

Show answer & explanation

Answer

brothers'

The sentence refers to the wing-warping system designed by the two brothers, Wilbur and Orville Wright. Because there are two brothers, the noun must be plural ('brothers'). Because the system belongs to them, the noun must also be possessive. For regular plural nouns ending in -s, the possessive is formed by adding an apostrophe after the -s, resulting in 'brothers'.

Step-by-Step Solution

1

Identify whether the noun should be singular or plural based on the context.

The text refers to 'brothers Wilbur and Orville Wright' and uses the plural pronoun 'their'. This indicates the noun must be plural ('brothers').

Identifying the number of the noun establishes the correct base form.

2

Determine if the noun must be possessive.

The noun describes whose 'innovative wing-warping system' it is, meaning it must be in the possessive case.

Identifying possession indicates whether an apostrophe is needed.

3

Apply the possessive punctuation rule for a plural noun ending in -s.

For regular plural nouns ending in -s, the possessive is formed by adding an apostrophe after the -s, resulting in 'brothers'.

This matches standard English conventions for possessive nouns.

Key Concept

Plural and possessive nouns

Estimated Time:1m 0s

Question 1905Question →

A researcher monitored the mass of a cooling block of metal over time as it underwent sublimation in a vacuum chamber. The table below shows the mass of the metal block, $M$ , in grams, for several values of time, $t$ , in hours after the sublimation process began.

Time, $t$ (hours)	Mass, $M$ (grams)
$2$	$112.4$
$5$	$104.6$
$8$	$96.8$

If the relationship between $M$ and $t$ is linear, which of the following is the best interpretation of the slope of the graph of this relationship in the $tM$ -plane?

The mass of the metal block decreases by

2.6

grams each hour.

The initial mass of the metal block before sublimation began was

117.6

grams.

The mass of the metal block decreases by

7.8

grams each hour.

The mass of the metal block after

1

hour of sublimation is

115.0

grams.

Show answer & explanation

Answer: The mass of the metal block decreases by $2.6$ grams each hour.

Answer

The mass of the metal block decreases by

2.6

grams each hour.

The correct answer is the option stating that the mass of the metal block decreases by

2.6

grams each hour. The slope of a linear model is the rate of change, which is computed as the change in the dependent variable divided by the change in the independent variable. Here,

\frac{104.6 - 112.4}{5 - 2} = -2.6

, representing a change of

-2.6

grams per hour.

Step-by-Step Solution

1

Calculate the slope of the linear relationship using two points from the table.

Using the points

(2, 112.4)

and

(5, 104.6)

, the slope is

\frac{104.6 - 112.4}{5 - 2} = \frac{-7.8}{3} = -2.6

.

The slope represents the constant rate of change of the dependent variable (

M

) per unit change in the independent variable (

t

).

2

Interpret the calculated slope value in the context of the problem.

A slope of

-2.6

means that the mass

M

decreases by

2.6

grams for every

1

hour increase in time

t

.

The negative sign indicates a decrease, and the rate is expressed in units of the dependent variable per unit of the independent variable (grams per hour).

Key Concept

Interpreting the slope of a linear relationship in context.

Estimated Time:1m 30s

Question 1906Question →

A function $f$ has exactly two local extrema: a local maximum at the point $(-2, 5)$ and a local minimum at the point $(2, -3)$ . A second function $g$ is defined by $g(x) = 1 - 3f(2x - 4)$ . If the local minimum of the graph of $y = g(x)$ occurs at the point $(h, k)$ , what is the value of $h + k$ ?

-17

-15

-13

-5

Show answer & explanation

Answer: -13

Answer

-13

To locate the local minimum of the transformed function

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

, we analyze the vertical reflection and scaling. The negative coefficient in

-3f(2x - 4)

reflects the graph vertically, which means the local maximum of

f(x)

at the point

(-2, 5)

becomes the local minimum of

g(x)

. To find the corresponding

x

-coordinate

h

, we solve

2h - 4 = -2

, which gives

h = 1

. To find the corresponding

y

-coordinate

k

, we evaluate

g(1) = 1 - 3f(-2) = 1 - 3(5) = -14

. Thus, the local minimum occurs at

(1, -14)

, and the sum of these coordinates is

1 + (-14) = -13

.

Step-by-Step Solution

1

Analyze how the vertical reflection in

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

affects the extrema.

Due to the negative coefficient in

-3f(2x - 4)

, the graph is reflected vertically. Therefore, the local maximum of

f(x)

at

(-2, 5)

transforms into the local minimum of

g(x)

, while the local minimum of

f(x)

transforms into the local maximum of

g(x)

.

A vertical reflection inverts the relative heights of the outputs, converting peaks to valleys and vice versa.

2

Determine the horizontal transformation to find the

x

-coordinate

h

of the new local minimum.

Set the input of

f

in the definition of

g(x)

equal to the

x

-coordinate of the maximum of

f(x)

, which is

-2

:

2h - 4 = -2

. Solving this equation gives

2h = 2

, which yields

h = 1

.

The horizontal shift and compression require solving for the new input variable that produces the same argument for the inner function.

3

Determine the vertical transformation to find the

y

-coordinate

k

of the new local minimum.

Substitute

h = 1

into

g(x)

to find the output value:

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

. Since the maximum value of

f(x)

is

f(-2) = 5

, we compute

k = 1 - 3(5) = 1 - 15 = -14

.

The vertical transformations (stretch, reflection, and shift) are applied directly to the function output.

4

Calculate the sum of the coordinates

h + k

.

Compute

h + k = 1 + (-14) = -13

.

The question asks for the sum of the coordinates of the local minimum of

g(x)

.

Key Concept

Analyzing function transformations including horizontal compression, horizontal translation, vertical stretch, reflection, and vertical translation to determine the coordinates of key features (local extrema) of a transformed function.

Estimated Time:3m 0s

Question 1907Question →

In the quadratic equation $2x^2 - bx + 18 = 0$ , $b$ is a positive constant. If one of the solutions to the equation is $4$ times the other solution, what is the value of $b$ ?

Show answer & explanation

Answer: 15

Answer

15

By Vieta's formulas, the product of the roots of the quadratic equation

2x^2 - bx + 18 = 0

is

\frac{18}{2} = 9

. Letting the roots be

r_1

and

r_2

with

r_1 = 4r_2

, we have

4r_2^2 = 9

, which yields

r_2 = \frac{3}{2}

(since

b > 0

implies the roots must be positive). Thus,

r_1 = 6

. The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and by Vieta's formulas, this sum equals

\frac{b}{2}

. Solving for

b

gives

15

.

Step-by-Step Solution

1

Set up the relationships for the product and sum of the roots using Vieta's formulas.

r_1 \cdot r_2 = 9

and

r_1 + r_2 = \frac{b}{2}

Vieta's formulas relate the coefficients of a quadratic equation to the sum and product of its roots.

2

Substitute the given condition that one root is

4

times the other (

r_1 = 4r_2

) into the product equation.

4r_2^2 = 9

This reduces the product equation to a single variable equation in terms of

r_2

.

3

Solve for

r_2

and determine its sign based on the constraint that

b

is positive.

r_2 = \frac{3}{2}

Since

b > 0

, the sum of the roots

5r_2 = \frac{b}{2}

must be positive, which requires

r_2 > 0

.

4

Calculate the second root

r_1

and then use the sum of the roots to find

b

.

b = 15

The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and since

r_1 + r_2 = \frac{b}{2}

, we have

\frac{b}{2} = \frac{15}{2}

.

Key Concept

Relationship between roots and coefficients of a quadratic equation

Question 1908Question →

In the quadratic equation $x^2 - px + q = 0$ , $p$ and $q$ are positive constants. If the equation has exactly one real solution, what is the value of $\frac{p^2}{q}$ ?

-4

4

2

16

Show answer & explanation

Answer: $4$

Answer

The correct answer is

4

.

The correct answer is

4

. For any quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

\Delta = b^2 - 4ac

must equal

0

. In the equation

x^2 - px + q = 0

, we identify

a = 1

,

b = -p

, and

c = q

. Substituting these values into the discriminant formula gives

(-p)^2 - 4(1)(q) = 0

, which simplifies to

p^2 - 4q = 0

. Rearranging this equation gives

p^2 = 4q

. Since

q

is a positive constant, we can divide both sides of the equation by

q

to isolate the ratio, yielding

\frac{p^2}{q} = 4

.

Step-by-Step Solution

1

Recall the condition for a quadratic equation of the form

ax^2 + bx + c = 0

to have exactly one real solution.

The discriminant,

\Delta = b^2 - 4ac

, must be equal to

0

.

A quadratic equation has exactly one real solution if and only if its discriminant is zero, meaning the parabola is tangent to the x-axis.

2

Identify the coefficients of the given quadratic equation

x^2 - px + q = 0

and substitute them into the discriminant formula.

Here,

a = 1

,

b = -p

, and

c = q

. Thus,

\Delta = (-p)^2 - 4(1)(q) = p^2 - 4q = 0

.

To apply the one-solution condition specifically to the coefficients of the given equation.

3

Solve the equation

p^2 - 4q = 0

for the ratio

\frac{p^2}{q}

.

p^2 = 4q \implies \frac{p^2}{q} = 4

.

Isolate the target expression by dividing both sides of the equation by

q

, which is valid since

q

is a positive constant and thus non-zero.

Key Concept

Evaluating the discriminant of a quadratic equation to determine the number of real solutions.

Estimated Time:1m 30s

Question 1909Question →

A digital music playlist contains only rock songs and pop songs. At the beginning of the month, $60\%$ of the songs in the playlist were rock songs. During the month, the number of rock songs in the playlist increased by $10\%$ , and the number of pop songs in the playlist increased by $35\%$ . If there were $120$ rock songs in the playlist at the beginning of the month, what percent of the songs in the playlist at the end of the month were pop songs?

Show answer & explanation

Answer: 45

Answer

The percent of the songs in the playlist at the end of the month that were pop songs is 45.

To find the final percentage of pop songs, we first determine the initial number of songs. Since

120

rock songs represent

60\%

of the playlist, the initial total is

120 / 0.60 = 200

songs. This means there are

200 - 120 = 80

pop songs initially. During the month, the rock songs increase by

10\%

, yielding

120 \times 1.10 = 132

songs. The pop songs increase by

35\%

, yielding

80 \times 1.35 = 108

songs. The final total number of songs is

132 + 108 = 240

. The final percentage of pop songs is

\frac{108}{240} \times 100 = 45\%

. Thus, the correct value to grid in is 45.

Step-by-Step Solution

1

Determine the initial total number of songs and initial pop songs.

The initial total number of songs is 200, and the initial number of pop songs is 80.

Since 120 rock songs represent 60% of the playlist, the initial total is 120 / 0.60 = 200. The remaining songs are pop songs: 200 - 120 = 80.

2

Calculate the updated number of rock songs and pop songs after the percentage increases.

The final rock song count is 132, and the final pop song count is 108.

The number of rock songs increased by 10%, which is 120 * 1.10 = 132. The number of pop songs increased by 35%, which is 80 * 1.35 = 108.

3

Calculate the final total number of songs and the final percentage of pop songs.

The final total is 240 songs, and the final pop song percentage is 45%.

The final total is 132 + 108 = 240. The final pop song percentage is (108 / 240) * 100 = 45%.

Key Concept

Calculating absolute quantities from initial percentages, applying percentage increases to separate groups, and finding a new conditional percentage based on the final total.

Question 1910Question →

An industrial printing press uses ink at a constant rate of $0.05$ milliliters per square centimeter of printed paper. The press prints a continuous roll of paper that is $40$ centimeters wide at a constant speed of $2$ meters per second. At this rate, how many liters of ink does the press use during $1$ hour of continuous printing?

Show answer & explanation

Answer: 1440

Answer

1440

The correct answer is 1440. First, convert the speed of the paper from meters per second to centimeters per second:

2 \text{ m/s} \times 100 \text{ cm/m} = 200 \text{ cm/s}

. Next, find the rate at which the paper surface area is printed:

40 \text{ cm} \times 200 \text{ cm/s} = 8,000 \text{ cm}^2/\text{s}

. Multiply this by the ink usage rate to find milliliters of ink per second:

8,000 \text{ cm}^2/\text{s} \times 0.05 \text{ mL/cm}^2 = 400 \text{ mL/s}

. Since there are

3,600

seconds in

1

hour, the press uses

400 \text{ mL/s} \times 3,600 \text{ s} = 1,440,000 \text{ mL}

of ink per hour. Finally, convert milliliters to liters:

1,440,000 \text{ mL} \div 1,000 \text{ mL/L} = 1,440

liters.

Step-by-Step Solution

1

Convert the speed of the paper from meters per second to centimeters per second.

200 \text{ cm/s}

To align the speed unit with the paper width unit (

40 \text{ cm}

).

2

Calculate the surface area of paper printed per second.

8,000 \text{ cm}^2/\text{s}

Multiply the width of the paper (

40 \text{ cm}

) by the converted speed (

200 \text{ cm/s}

).

3

Calculate the volume of ink used per second in milliliters.

400 \text{ mL/s}

Multiply the area printed per second (

8,000 \text{ cm}^2/\text{s}

) by the ink consumption rate (

0.05 \text{ mL/cm}^2

).

4

Convert the ink volume rate from per second to per hour.

1,440,000 \text{ mL/h}

Multiply the rate per second by

3,600

, since there are

3,600

seconds in

1

hour.

5

Convert the final volume from milliliters to liters.

1440

Divide the total milliliters (

1,440,000

) by

1,000

, since

1

liter is equal to

1,000

milliliters.

Key Concept

Multi-step dimensional analysis and compound unit conversion (length, area, volume, and time).

Question 1911Question →

A printer prints $24$ pages in $3$ minutes. At this constant rate, how many pages can the printer print in $10$ minutes?

1.25

72

80

240

Show answer & explanation

Answer: 80

Answer

80

The correct answer is 80. The printer's rate of printing is determined by dividing 24 pages by 3 minutes, which is 8 pages per minute. Multiplying this rate of 8 pages per minute by 10 minutes gives a total of 80 pages printed.

Step-by-Step Solution

1

Calculate the unit rate of pages printed per minute.

24 \div 3 = 8

pages per minute

Dividing the total pages by the total minutes gives the constant rate of printing.

2

Multiply the unit rate by the new amount of time.

8 \times 10 = 80

pages

Multiplying the pages per minute by 10 minutes gives the total number of pages printed in that time.

Key Concept

Calculating and applying a unit rate to solve a proportion.

Question 1912Question →

A town's population was $12,000$ in the year $2010$ . Since $2010$ , the population of the town has increased by a constant amount of $350$ people each year. Based on this information, what was the population of the town in the year $2018$ ?

Show answer & explanation

Answer: 14800

Answer

14,800

The population starts at

12,000

in

2010

and grows linearly by

350

people per year. The number of years from

2010

to

2018

is

2018 - 2010 = 8

years. The total increase in population is

350 \times 8 = 2,800

people. Adding this to the initial population of

12,000

gives a total population of

12,000 + 2,800 = 14,800

in the year

2018

.

Step-by-Step Solution

1

Calculate the number of years that passed between

2010

and

2018

.

8

years

To determine the total population growth, we need the elapsed time in years.

2

Calculate the total population growth over this

8

-year period.

2,800

people

Since the growth is linear, the total growth is the constant annual rate of change multiplied by the number of years.

3

Add the total population growth to the initial population from

2010

.

14,800

The population in

2018

is the initial population plus the total growth over the

8

-year period.

Key Concept

Linear growth represents a quantity that increases by a constant amount per unit of time.

Question 1913Question →

An online retailer offers free shipping on orders of $75 or more. A customer has placed a shirt costing$ 22.50 and a pair of pants costing $34.80 in their shopping cart. They want to buy some pairs of socks that cost$ 4.50 per pair to qualify for free shipping. What is the minimum number of pairs of socks the customer must add to their cart to qualify for free shipping?

Show answer & explanation

Answer: 4

Answer

The minimum number of pairs of socks the customer must add to the cart is 4.

The customer needs to spend at least

75 to qualify for free shipping. The current subtotal of the cart is

22.50 +

34.80 =

57.30. The remaining amount needed to reach

75 is

75.00 -

57.30 =

17.70. Let

s

represent the number of pairs of socks purchased at

4.50 per pair. This can be represented by the inequality

4.50s \geq 17.70

. Dividing both sides of the inequality by

4.50

gives

s \geq 3.93$. Since the customer cannot purchase a fraction of a pair of socks, the minimum number of pairs of socks they must purchase is 4.

Step-by-Step Solution

1

Calculate the total cost of the items already in the shopping cart.

22.50 +

34.80 = $57.30

To determine how much has already been spent before adding socks.

2

Set up an inequality to represent the total cost of the order including

s

pairs of socks meeting the free shipping threshold of $75.

57.30 + 4.50s \geq 75

The total cost of the shirt, pants, and socks must be greater than or equal to $75.

3

Subtract $57.30 from both sides of the inequality to find the minimum amount that needs to be spent on socks.

4.50s \geq 17.70

To isolate the term representing the cost of the socks.

4

Divide both sides of the inequality by $4.50 to find the minimum number of pairs of socks.

s \geq 3.93

To isolate the variable representing the number of pairs of socks.

5

Determine the smallest integer value of

s

that satisfies the inequality.

4

The number of pairs of socks must be a whole number, and 4 is the smallest integer greater than or equal to 3.93.

Key Concept

Solving linear inequalities in one variable to find the minimum integer solution in a real-world context.

Estimated Time:1m 30s

Question 1914Question →

Discovered accidentally during high-temperature materials science experiments in 2009, YInMn Blue was the first new blue pigment identified in more than two hundred ________ its unique crystal structure, which absorbs red and green wavelengths of light while reflecting blue, makes the compound exceptionally stable and resistant to fading. Consequently, the pigment has found widespread use in both fine arts and industrial coatings.

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

years;

years,

years

years, but because

Show answer & explanation

Answer: years;

Answer

The correct choice is the option that ends with a semicolon, which properly connects the two independent clauses.

The clause preceding the blank and the clause following the blank are both independent clauses because they can each stand alone as a complete sentence. A semicolon is used to connect two closely related independent clauses when no coordinating conjunction is present.

Step-by-Step Solution

1

Analyze the structure of the clauses before and after the blank.

The clause before the blank ('Discovered accidentally... two hundred years') and the clause after the blank ('its unique crystal structure... resistant to fading') are both independent clauses.

Identifying clause types is necessary to determine the correct punctuation or conjunction needed to link them.

2

Evaluate the choices to find a grammatically correct way to link two independent clauses.

A semicolon is the only option that correctly links the two independent clauses without causing a comma splice, run-on, or fragment.

Standard English grammar requires a semicolon, period, colon, or comma plus coordinating conjunction to separate independent clauses.

Key Concept

Clause Boundaries and Linking

Question 1915Question →

A quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ contains the points $(-1, 3)$ and $(7, 3)$ . The function $g$ is defined by $g(x) = f(x + 2) - 4$ , and its graph has a vertex at $(p, q)$ . If $p + q = 5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of k is 8.

The axis of symmetry of the quadratic function f is located at the midpoint of the x-values of any two points on the graph that have the same y-value. Since the graph of f contains (-1, 3) and (7, 3), the x-coordinate of its vertex, h, is equal to the midpoint (-1 + 7) / 2 = 3. Thus, the vertex of f is (3, k). The definition of g(x) = f(x + 2) - 4 indicates a horizontal translation of 2 units to the left and a vertical translation of 4 units down. Applying these translations to the vertex (3, k) of f gives the vertex of g as (3 - 2, k - 4) = (1, k - 4). Since the vertex of g is defined as (p, q), we have p = 1 and q = k - 4. Substituting these values into the equation p + q = 5 yields 1 + (k - 4) = 5, which simplifies to k - 3 = 5, or k = 8.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the function f

h = 3

Because the points (-1, 3) and (7, 3) have the same y-coordinate, the axis of symmetry (and thus the x-coordinate of the vertex) must lie exactly halfway between x = -1 and x = 7.

2

Determine the vertex of the function g(x)

(p, q) = (1, k - 4)

The function g(x) = f(x + 2) - 4 represents a translation of the graph of f by 2 units to the left and 4 units down. Since the vertex of f is (3, k), the vertex of g is (3 - 2, k - 4).

3

Solve for k using the given sum p + q = 5

k = 8

Substitute p = 1 and q = k - 4 into the equation p + q = 5 to get 1 + k - 4 = 5, which simplifies to k = 8.

Key Concept

Quadratic functions and graphs, including symmetry and translations.

Question 1916Question →

In the $xy$ -plane, the graph of the cubic function $f(x) = x^3 - 7x^2 + kx - 12$ , where $k$ is a constant, is tangent to the $x$ -axis at one point and intersects the $x$ -axis at another point. If all roots of $f(x)$ are real numbers, which of the following could be the value of $k$ ?

16

-16

7

12

Show answer & explanation

Answer: 16

Answer

16

The correct answer is the value that makes the cubic function have a double root at

2

and a single root at

3

. Setting

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

and expanding it yields

x^3 - 7x^2 + 16x - 12

. Comparing this to

f(x) = x^3 - 7x^2 + kx - 12

shows that

k = 16

.

Step-by-Step Solution

1

Express the cubic function in factored form based on its roots.

f(x) = (x - r_1)^2(x - r_2)

Since the graph is tangent to the

x

-axis at one point and intersects it at another, it has one real root of multiplicity 2 (let it be

r_1

) and one real root of multiplicity 1 (let it be

r_2

).

2

Expand the factored form to obtain coefficients in terms of

r_1

and

r_2

.

f(x) = x^3 - (2r_1 + r_2)x^2 + (r_1^2 + 2r_1 r_2)x - r_1^2 r_2

Comparing this expansion to the given expression

f(x) = x^3 - 7x^2 + kx - 12

allows us to set up a system of equations.

3

Set up and solve the system of equations for

r_1

and

r_2

.

2r_1 + r_2 = 7

,

r_1^2 r_2 = 12

, and

k = r_1^2 + 2r_1 r_2

. Substituting

r_2 = 7 - 2r_1

into the second equation gives

2r_1^3 - 7r_1^2 + 12 = 0

. Testing integer root candidates shows

r_1 = 2

is a solution, which yields

r_2 = 3

.

This determines the values of the roots

r_1

and

r_2

.

4

Calculate the value of

k

using the values of the roots.

k = 2^2 + 2(2)(3) = 16

Substituting

r_1 = 2

and

r_2 = 3

into the expression for

k

gives the target coefficient.

Key Concept

Analyzing the relationship between a polynomial's algebraic factors, roots, and its graphical features such as tangency and intercepts.

Question 1917Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -2(x - d)^2 + 8$ , where $d$ is a positive constant, intersects the $x$ -axis at the point $(2, 0)$ . The function $g$ is defined by $g(x) = f(x + 3) - 5$ . What is the $y$ -coordinate of the $y$ -intercept of the graph of $g$ in the $xy$ -plane?

-15

-95

1

11

Show answer & explanation

Answer: 1

Answer

The

y

-coordinate of the

y

-intercept of the graph of

g

is

1

.

To find the correct answer, we first substitute the point

(2,0)

into

f(x) = -2(x - d)^2 + 8

to obtain

0 = -2(2 - d)^2 + 8

. Simplifying this equation gives

(2 - d)^2 = 4

, which has solutions

d = 0

and

d = 4

. Since the problem states that

d

is a positive constant, we have

d = 4

, which means

f(x) = -2(x - 4)^2 + 8

. We then find the equation for the transformed function

g(x)

by replacing

x

with

x + 3

and subtracting

5

, which yields

g(x) = f(x + 3) - 5 = -2(x + 3 - 4)^2 + 8 - 5 = -2(x - 1)^2 + 3

. To find the

y

-coordinate of the

y

-intercept, we evaluate

g(x)

at

x = 0

, giving

g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1

.

Step-by-Step Solution

1

Use the given

x

-intercept

(2,0)

of

f(x)

to solve for the constant

d

.

d = 4

Substitute

x = 2

and

f(x) = 0

into the equation

f(x) = -2(x - d)^2 + 8

to get

0 = -2(2 - d)^2 + 8

. Dividing by

-2

and isolating the squared term gives

(2 - d)^2 = 4

. Taking the square root of both sides gives

2 - d = 2

or

2 - d = -2

, which yields

d = 0

or

d = 4

. Since

d

must be a positive constant, we must choose

d = 4

.

2

Determine the equation of the transformed function

g(x)

.

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

Substitute

d = 4

into

f(x)

to get

f(x) = -2(x - 4)^2 + 8

. Since

g(x) = f(x + 3) - 5

, substitute

x + 3

for

x

in

f(x)

and subtract

5

:

g(x) = -2((x + 3) - 4)^2 + 8 - 5

, which simplifies to

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

.

3

Find the

y

-coordinate of the

y

-intercept of the graph of

g

.

1

The

y

-intercept of a graph is the point where

x = 0

. Substitute

x = 0

into the equation for

g(x)

to find

g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1

.

Key Concept

Applying horizontal and vertical translations to quadratic equations and identifying graph features.

Question 1918Question →

An equation is shown below.

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

If

x

satisfies the equation above, what is the value of

x + 3

?

4

2

12

6

Show answer & explanation

Answer: 12

Answer

12

To solve the equation, square both sides to obtain

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

. Rearranging terms gives the quadratic equation

x^2 - 10x + 9 = 0

, which factors as

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

. This yields potential solutions of

x = 9

and

x = 1

. Testing these in the original equation shows that

x = 9

is a valid solution because

\sqrt{25} = 5

, but

x = 1

is extraneous because

\sqrt{9} \neq -3

. Thus, the only real solution is

x = 9

. Evaluating

x + 3

for this solution gives

9 + 3 = 12

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

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

, which expands to

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

.

Squaring both sides is the standard method to solve radical equations.

2

Rearrange the terms to set the quadratic equation to zero.

x^2 - 10x + 9 = 0

.

This allows us to solve the quadratic equation by factoring.

3

Factor the quadratic equation.

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

, giving potential solutions of

x = 9

and

x = 1

.

Factoring helps find the roots of the quadratic equation.

4

Verify each potential solution in the original equation to check for extraneous roots.

x = 9

is valid because

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

is true (

5=5

).

x = 1

is extraneous because

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

is false (

3 \neq -3

).

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

5

Substitute the valid solution

x = 9

into the expression

x + 3

.

9 + 3 = 12

.

The question asks for the value of the expression

x + 3

rather than just

x

.

Key Concept

Solving radical equations and checking for extraneous solutions

Question 1919Question →

A sample of a radioactive isotope decays according to the function $A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$ , where $A_0$ is the initial mass of the sample, $t$ is the time in years, and $h$ is the half-life of the isotope in years. If the sample decays to $12.5\%$ of its initial mass in $45$ years, what is the value of $h$ ?

5.625

22.5

15

135

Show answer & explanation

Answer: 15

Answer

15

The correct answer is 15. The remaining fraction of the radioactive isotope is

12.5\%

, which can be written as

0.125

or

\frac{1}{8}

. Expressing this as a power of the decay base yields

\frac{1}{8} = \left(\frac{1}{2}\right)^3

. Setting the decay formula equal to this fraction gives

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \left(\frac{1}{2}\right)^3

. Equating the exponents gives

\frac{45}{h} = 3

, which yields

h = 15

.

Step-by-Step Solution

1

Set up the decay equation based on the given remaining percentage.

A(45) = 0.125 A_0

, which simplifies to

\frac{A(45)}{A_0} = 0.125 = \frac{1}{8}

.

This establishes the fraction of the substance remaining after

45

years.

2

Substitute the remaining fraction and the time

t = 45

into the exponential decay model.

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \frac{1}{8}

.

This sets up the equation in terms of the unknown half-life parameter

h

.

3

Express both sides of the equation with a common base of

\frac{1}{2}

.

\left(\frac{1}{2}\right)^{\frac{45}{h}} = \left(\frac{1}{2}\right)^3

.

Converting

\frac{1}{8}

to

\left(\frac{1}{2}\right)^3

allows us to equate the exponents directly.

4

Equate the exponents and solve for

h

.

\frac{45}{h} = 3 \implies 3h = 45 \implies h = 15

.

Since the bases are identical, their exponents must be equal to satisfy the equation.

Key Concept

Solving exponential equations by expressing both sides with a common base.

Estimated Time:1m 30s

Question 1920Question →

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - 10)$ , where $a$ is a positive constant. In the $xy$ -plane, the graph of $f$ has a vertex with a $y$ -coordinate of $-18$ . What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of

a

is 2.

The vertex of a parabola lies on the axis of symmetry, which is located midway between its x-intercepts. Since the function is

f(x) = a(x - 4)(x - 10)

, the x-intercepts are at

x = 4

and

x = 10

. The midpoint of these values is

\frac{4 + 10}{2} = 7

. Thus, the x-coordinate of the vertex is 7. Since the y-coordinate of the vertex is given as

-18

, the vertex is at

(7, -18)

. Substituting these coordinates into the function equation yields

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = a(3)(-3)

, or

-18 = -9a

. Solving for

a

gives

a = 2

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex using the symmetry of the parabola.

The x-coordinate of the vertex is 7.

The x-coordinate of the vertex lies on the axis of symmetry, which is the midpoint of the x-intercepts

x = 4

and

x = 10

.

2

Identify the coordinates of the vertex.

The vertex is at

(7, -18)

.

The y-coordinate of the vertex is given as

-18

, and its x-coordinate was calculated as 7.

3

Substitute the vertex coordinates into the function definition to solve for

a

.

a = 2

Substituting

(7, -18)

into

f(x) = a(x - 4)(x - 10)

gives

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = -9a

.

Key Concept

Finding the vertex of a quadratic function from its factored form and solving for a leading coefficient.

Alternative Method

Alternatively, the function

f(x) = a(x - 4)(x - 10)

can be expanded to standard form:

f(x) = a(x^2 - 14x + 40)

. The x-coordinate of the vertex can be found using the formula

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

, which gives

x = -\frac{-14a}{2a} = 7

. Then, substitute

x = 7

and

f(7) = -18

to solve for

a

.

Estimated Time:1m 30s

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