All practice questions

Question 501Question →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(3, -8)$ and passes through the point $(1, 0)$ . If the graph intersects the $y$ -axis at $(0, c)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.

By writing the function in vertex form

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

and substituting the vertex

(3, -8)

, we obtain

f(x) = a(x - 3)^2 - 8

. Substituting the point

(1, 0)

into the equation gives

0 = a(1 - 3)^2 - 8

, which simplifies to

4a = 8

and yields

a = 2

. With the function fully defined as

f(x) = 2(x - 3)^2 - 8

, the

y

-intercept is found by calculating

f(0) = 2(0 - 3)^2 - 8 = 2(9) - 8 = 10

. Thus,

c = 10

.

Step-by-Step Solution

1

Express the quadratic function in vertex form using the given vertex

(3, -8)

.

f(x) = a(x - 3)^2 - 8

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 point

(1, 0)

into the function to solve for the leading coefficient

a

.

a = 2

Since the graph passes through

(1, 0)

, substituting

x = 1

and

f(x) = 0

must satisfy the equation.

3

Substitute

x = 0

into the completed function

f(x) = 2(x - 3)^2 - 8

to find the value of

c

.

c = 10

The intersection with the

y

-axis occurs where the input

x

is 0.

Key Concept

Writing and evaluating quadratic functions using the vertex form

Question 502Question →

A researcher wants to estimate the total number of households in a city of $40,000$ households that own at least one hybrid vehicle. The researcher randomly selects a sample of households to contact for a survey. The survey has a response rate of $40\%$ . Of the households that responded, $15\%$ reported owning at least one hybrid vehicle. A follow-up study of a random sample of the non-respondents indicates that the proportion of non-respondents who own at least one hybrid vehicle is half the proportion of the respondents who do. Based on the results of the survey and the follow-up study, what is the estimated total number of households in the city that own at least one hybrid vehicle?

Show answer & explanation

Answer: 4200

Answer

The estimated total number of households in the city that own at least one hybrid vehicle is 4,200.

The correct answer of 4,200 is found by dividing the population of 40,000 households into two strata: respondents (40%, or 16,000 households) and non-respondents (60%, or 24,000 households). The number of hybrid-owning households among respondents is estimated as 15% of 16,000, which is 2,400. The proportion of hybrid owners among non-respondents is half of the respondents' rate, or 7.5%. The number of hybrid-owning households among non-respondents is estimated as 7.5% of 24,000, which is 1,800. Summing these estimates yields 2,400 + 1,800 = 4,200.

Step-by-Step Solution

1

Calculate the size of the respondent and non-respondent strata in the population.

Respondent stratum:

16,000

households; Non-respondent stratum:

24,000

households.

Since the survey's response rate is

40\%

, we estimate that

40\%

of the

40,000

households are potential respondents, and the remaining

60\%

(

24,000

households) are potential non-respondents.

2

Determine the proportion of hybrid vehicle owners in each stratum.

Respondent proportion:

15\%

(or

0.15

); Non-respondent proportion:

7.5\%

(or

0.075

).

The respondent proportion is given as

15\%

. The follow-up study indicates the non-respondent proportion is half of the respondent proportion, which is

\frac{1}{2} \times 15\% = 7.5\%

.

3

Calculate the estimated number of households owning a hybrid vehicle in each stratum and sum them.

2,400 + 1,800 = 4,200

households.

Multiply each stratum size by its respective proportion:

16,000 \times 0.15 = 2,400

and

24,000 \times 0.075 = 1,800

. Adding these gives the total estimate of

4,200

.

Key Concept

Generalizing from stratified random samples to estimate population parameters while accounting for non-response bias.

Question 503Question →

The table below shows selected values for a polynomial function $f$ of degree 3.

$x$	$f(x)$
$-2$	$0$
$1$	$0$
$3$	$0$
$4$	$18$

What is the value of $f(0)$ ?

Show answer & explanation

Answer: 6

Answer

The value of

f(0)

is 6.

The table shows the roots of the cubic function

f(x)

are

x = -2

,

x = 1

, and

x = 3

. This allows the function to be written in factored form as

f(x) = a(x + 2)(x - 1)(x - 3)

. Using the point

(4, 18)

from the table, we solve for

a

:

18 = a(4 + 2)(4 - 1)(4 - 3) \Rightarrow 18 = 18a \Rightarrow a = 1

. The function is

f(x) = (x + 2)(x - 1)(x - 3)

. Substituting

x = 0

gives

f(0) = (2)(-1)(-3) = 6

.

Step-by-Step Solution

1

Identify the roots of the polynomial from the table.

The roots of the function are

x = -2

,

x = 1

, and

x = 3

.

The table shows that

f(x) = 0

at these values, indicating they are the

x

-intercepts or roots.

2

Write the general factored form of a degree 3 polynomial with these roots.

f(x) = a(x + 2)(x - 1)(x - 3)

A cubic polynomial with roots

r_1

,

r_2

, and

r_3

can be expressed as

f(x) = a(x - r_1)(x - r_2)(x - r_3)

for a constant coefficient

a

.

3

Find the value of the constant coefficient

a

using a known point from the table.

a = 1

Substituting the table values

x = 4

and

f(4) = 18

yields

18 = a(4 + 2)(4 - 1)(4 - 3)

, which simplifies to

18 = 18a

, so

a = 1

.

4

Calculate the value of

f(0)

using the fully defined function.

f(0) = 6

Substituting

x = 0

into the equation

f(x) = (x + 2)(x - 1)(x - 3)

gives

f(0) = (0 + 2)(0 - 1)(0 - 3) = 2 \times (-1) \times (-3) = 6

.

Key Concept

Using polynomial roots and extra points to define a polynomial function and evaluate it.

Question 504Question →

For the function $f(x) = 4^x - 3$ , a new function $g$ is created by reflecting the graph of $y = f(x)$ across the $x$ -axis, followed by a vertical translation of $c$ units upward. The graph of $y = g(x)$ contains the point $(2, -5)$ . What is the value of the constant $c$ ?

Show answer & explanation

Answer: 8

Answer

8

Reflecting the graph of

f(x) = 4^x - 3

across the

x

-axis results in the function

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

. Translating this graph vertically upward by

c

units defines

g(x) = -4^x + 3 + c

. Substituting the given point

(2, -5)

yields

-5 = -4^2 + 3 + c

, which simplifies to

-5 = -13 + c

. Adding

13

to both sides gives

c = 8

.

Step-by-Step Solution

1

Reflect the function

f(x)

across the

x

-axis

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

Reflecting a graph across the

x

-axis is represented by negating the output of the function, which transforms

y = f(x)

to

y = -f(x)

.

2

Apply the vertical translation upward by

c

units to define

g(x)

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

Translating a function vertically upward by

c

units adds

c

to the function's expression.

3

Substitute the point

(2, -5)

into

g(x)

and solve for

c

c = 8

Since the graph of

g

passes through

(2, -5)

, substitute

x = 2

and

g(x) = -5

into the equation:

-5 = -4^2 + 3 + c

. This simplifies to

-5 = -16 + 3 + c

, then

-5 = -13 + c

, which gives

c = 8

.

Key Concept

Function transformations, including vertical reflections and translations

Question 505Question →

An online streaming service tracks the number of subscribers in two regions, Region X and Region Y. At the beginning of 2020 ( $t = 0$ ), Region X had $12,000$ subscribers, and the number of subscribers in Region X increased by a constant amount of $2,500$ each year. Region Y had $8,000$ subscribers at the beginning of 2020, and the number of subscribers in Region Y increased by a constant $15\%$ each year. What is the positive difference between the number of subscribers in Region X and the number of subscribers in Region Y at the beginning of 2022 ( $t = 2$ )?

Show answer & explanation

Answer: 6420

Answer

The positive difference between the subscriber counts in Region X and Region Y at the beginning of 2022 is

6,420

.

To find the positive difference, we first calculate the subscriber counts for both regions at

t = 2

. Region X grows linearly at a constant rate of

2,500

subscribers per year, so its population at

t = 2

is

12,000 + 2,500(2) = 17,000

. Region Y grows exponentially at a constant rate of

15\%

per year, so its population at

t = 2

is

8,000 \times (1.15)^2 = 10,580

. The positive difference between the subscriber counts is

17,000 - 10,580 = 6,420

.

Step-by-Step Solution

1

Calculate the subscriber count for Region X at

t = 2

.

17,000

Since Region X grows linearly by a constant amount of

2,500

subscribers per year, the number of subscribers at

t = 2

is given by

12,000 + 2,500(2) = 17,000

.

2

Calculate the subscriber count for Region Y at

t = 2

.

10,580

Since Region Y grows exponentially by a constant rate of

15\%

per year, the number of subscribers at

t = 2

is given by

8,000 \times (1.15)^2 = 10,580

.

3

Calculate the positive difference between the two subscriber counts.

6,420

Subtract the smaller subscriber count from the larger subscriber count:

17,000 - 10,580 = 6,420

.

Key Concept

Distinguishing between linear growth (constant rate of change per unit of time) and exponential growth (constant percent rate of change per unit of time) to solve real-world problems.

Question 506Question →

A set of 5 positive integers has a mean of 12, a median of 10, and a unique mode of 8. What is the maximum possible value of the largest integer in this set?

Show answer & explanation

Answer: 23

Answer

The maximum possible value of the largest integer in the set is 23.

To find the maximum possible value of the largest integer in a set of 5 positive integers with a mean of 12, a median of 10, and a unique mode of 8: First, calculate the total sum of the integers, which is

12 \times 5 = 60

. Let the sorted integers be

x_1 \le x_2 \le x_3 \le x_4 \le x_5

. The median is the middle term, so

x_3 = 10

. Since 8 is the unique mode and is less than the median, it must appear at least twice in the first two slots, so

x_1 = 8

and

x_2 = 8

. The sum of the remaining two integers is

60 - (8 + 8 + 10) = 34

. To maximize the largest integer

x_5

, we must minimize

x_4

. Since the integers are sorted,

x_4 \ge x_3 = 10

. However, if

x_4 = 10

, then 10 would appear twice, making it a second mode alongside 8, which violates the unique mode condition. Thus, the smallest possible integer value for

x_4

is 11, which gives a maximum possible value of

34 - 11 = 23

for the largest integer.

Step-by-Step Solution

1

Calculate the sum of the five integers.

The sum of the five integers is 60.

Since the mean of 5 numbers is 12, their sum is

12 \times 5 = 60

.

2

Set up the ordered list of integers and identify the median.

For integers

x_1 \le x_2 \le x_3 \le x_4 \le x_5

, the median is

x_3 = 10

.

The median of 5 numbers in ordered sequence is the third number.

3

Determine the values of the first two integers using the mode constraint.

x_1 = 8

and

x_2 = 8

.

The mode must be unique and equal to 8. Since 8 is less than the median 10, 8 must occupy the first two spots to appear more than once.

4

Write the sum equation for the remaining unknown integers.

x_4 + x_5 = 34

.

Since the total sum is 60, we subtract the known values:

60 - (8 + 8 + 10) = 34

.

5

Minimize the fourth integer to maximize the fifth integer while keeping the unique mode of 8.

The minimum value for

x_4

is 11, which gives the maximum value for

x_5

as 23.

We must have

x_4 \ge 10

due to ordering. If

x_4 = 10

, 10 would be a mode, so the minimum valid integer for

x_4

is 11.

Key Concept

Using measures of center (mean, median) and measures of frequency (mode) to determine constraints on individual data values in a data distribution.

Alternative Method

Instead of setting up inequalities, one can test integers starting from the maximum mathematical limit if there were no mode constraint (which is 24, since

x_4 \ge 10

, giving

60 - 26 - 10 = 24

). Testing 24 yields the set {8, 8, 10, 10, 24}, which has two modes. The next highest value to test is 23, which yields the valid set {8, 8, 10, 11, 23}.

Estimated Time:1m 30s

Question 507Question →

In similar triangles $JKL$ and $WXY$ , the ratio of the length of side $JK$ to the length of its corresponding side $WX$ is $1$ to $4$ . If the length of side $WX$ is $x + 9$ and the length of side $JK$ is $x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Because triangles

JKL

and

WXY

are similar, the ratio of corresponding side lengths is constant. Since the ratio of

JK

to

WX

is

1

to

4

, the relationship can be written as

\frac{x}{x + 9} = \frac{1}{4}

. Cross-multiplying yields

4x = x + 9

. Subtracting

x

from both sides gives

3x = 9

, and dividing by

3

gives the final answer

x = 3

.

Step-by-Step Solution

1

Set up a proportion using the ratio of corresponding side lengths.

\frac{x}{x + 9} = \frac{1}{4}

The ratio of corresponding side lengths in similar triangles is constant.

2

Cross-multiply to eliminate fractions.

4x = x + 9

Multiplying both sides by both denominators simplifies the equation.

3

Solve for the variable

x

.

x = 3

Subtracting

x

from both sides gives

3x = 9

, and dividing by

3

gives

x = 3

.

Key Concept

The corresponding side lengths of similar triangles are proportional.

Question 508Question →

A solid right circular cylinder has a radius of $r$ centimeters and a height of $6$ centimeters. A solid sphere has a radius of $r$ centimeters. If the sum of the volumes of the cylinder and the sphere is equal to the volume of a right circular cylinder with a radius of $2r$ centimeters and a height of $2$ centimeters, what is the value of $r$ ?

Show answer & explanation

Answer: 1.5

Answer

The correct answer is 1.5 (or 3/2).

The correct answer is 1.5. By using the formulas for the volume of a cylinder (

V = \pi r^2 h

) and a sphere (

V = \frac{4}{3}\pi r^3

), the total volume of the first cylinder and the sphere is

6\pi r^2 + \frac{4}{3}\pi r^3

. The second cylinder has a volume of

\pi(2r)^2(2) = 8\pi r^2

. Setting them equal gives

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

. Subtracting

6\pi r^2

from both sides gives

\frac{4}{3}\pi r^3 = 2\pi r^2

. Dividing both sides by

\pi r^2

(since

r > 0

) simplifies to

\frac{4}{3}r = 2

, which solves to

r = 1.5

.

Step-by-Step Solution

1

Express the volume of the first cylinder using the formula

V = \pi r^2 h

.

V_1 = 6\pi r^2

To represent the cylinder's volume in terms of its radius.

2

Express the volume of the sphere using the formula

V = \frac{4}{3}\pi r^3

.

V_2 = \frac{4}{3}\pi r^3

To represent the sphere's volume in terms of its radius.

3

Express the volume of the second cylinder, which has a radius of

2r

and a height of

2

.

V_3 = \pi (2r)^2(2) = 8\pi r^2

To represent the larger cylinder's volume in terms of

r

.

4

Set up the equation where the sum of the first cylinder and sphere volumes equals the second cylinder volume.

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

To relate the volumes as given in the problem statement.

5

Simplify the equation by subtracting

6\pi r^2

from both sides.

\frac{4}{3}\pi r^3 = 2\pi r^2

To isolate the cubic term on one side.

6

Divide both sides of the equation by

\pi r^2

since

r > 0

.

\frac{4}{3}r = 2

To reduce the polynomial equation to a linear equation.

7

Multiply both sides by

\frac{3}{4}

to solve for

r

.

r = 1.5

To find the final numerical value of the radius.

Key Concept

Volume of cylinders and spheres

Question 509Question →

The graph of the quadratic equation $y = 2x^2 - 12x + c$ , where $c$ is a constant, is a parabola in the $xy$ -plane. If the $y$ -coordinate of the vertex of this parabola is $-5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 13

Answer

13

To find the value of the constant

c

, we calculate the coordinates of the vertex of the parabola. The x-coordinate of the vertex for a quadratic function in standard form

y = ax^2 + bx + c

is given by

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

. Substituting

a = 2

and

b = -12

gives

x = -\frac{-12}{2(2)} = 3

. Evaluating the quadratic equation at

x = 3

gives the y-coordinate of the vertex:

y = 2(3)^2 - 12(3) + c = 18 - 36 + c = c - 18

. Since we are given that the y-coordinate of the vertex is

-5

, we set

c - 18 = -5

and solve to find

c = 13

.

Step-by-Step Solution

1

Calculate the x-coordinate of the vertex using the vertex formula.

x = 3

The axis of symmetry and the x-coordinate of the vertex are located at

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

.

2

Substitute the x-coordinate of the vertex into the equation to express the y-coordinate in terms of

c

.

y = c - 18

Evaluating the quadratic function at the vertex's x-coordinate gives the minimum or maximum value of the function.

3

Equate the expression for the y-coordinate to the given vertex y-coordinate of

-5

and solve for

c

.

c = 13

Setting the calculated y-coordinate expression equal to the given value allows us to isolate and solve for the unknown constant.

Key Concept

Determining the vertex of a quadratic function from its standard form and solving for a constant coefficient.

Alternative Method

Alternatively, we can complete the square to write the quadratic equation in vertex form,

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

. Factoring the leading coefficient from the variable terms gives

y = 2(x^2 - 6x) + c

. To complete the square inside the parentheses, add and subtract

9

:

y = 2(x^2 - 6x + 9 - 9) + c = 2((x-3)^2 - 9) + c = 2(x-3)^2 - 18 + c

. In this vertex form, the y-coordinate of the vertex is

k = c - 18

. Since the vertex y-coordinate is

-5

, we set

c - 18 = -5

to get

c = 13

.

Estimated Time:1m 30s

Question 510Question →

A sociologist is studying the print newspaper subscription habits of residents in a city. The table below shows the total number of households, the sample size, and the number of households in each sample that report subscribing to a print newspaper for the city's two districts.

District	Total households	Sample size	Subscribing households in sample
West	$10,000$	$200$	$35$
East	$14,000$	$250$	$60$

Based on the results of these samples, what is the estimated total number of households in the city that subscribe to a print newspaper?

Show answer & explanation

Answer: 5110

Answer

5110

To find the total estimated number of households in the city that subscribe to a print newspaper, we must estimate the subscriptions for each district separately and then sum them. For District West, the proportion of households subscribing is

\frac{35}{200} = 0.175

. Multiplying this by the total number of households in District West (

10,000

) gives an estimate of

1,750

subscribing households. For District East, the proportion of households subscribing is

\frac{60}{250} = 0.24

. Multiplying this by the total number of households in District East (

14,000

) gives an estimate of

3,360

subscribing households. Adding these two estimates together (

1,750 + 3,360

) results in a total estimate of

5,110

households.

Step-by-Step Solution

1

Calculate the estimated number of print newspaper subscriptions in District West.

1,750 households

The proportion of subscribing households in the District West sample is applied to the total number of households in District West.

2

Calculate the estimated number of print newspaper subscriptions in District East.

3,360 households

The proportion of subscribing households in the District East sample is applied to the total number of households in District East.

3

Find the total estimated number of subscribing households in the city.

5,110 households

Adding the estimates from both districts yields the overall population estimate.

Key Concept

Estimating population parameters from stratified random samples

Question 511Question →

A hybrid vehicle uses $2$ gallons of fuel to travel $80$ miles in the city, and $2.5$ gallons of fuel to travel $120$ miles on the highway. A driver takes a trip that includes $40$ miles of city driving and $180$ miles of highway driving. What is the total number of gallons of fuel the vehicle is expected to use for the entire trip?

Show answer & explanation

Answer: 4.75

Answer

4.75

The correct answer is 4.75 (or 19/4). The city fuel economy is 40 miles per gallon (80 miles / 2 gallons), and the highway fuel economy is 48 miles per gallon (120 miles / 2.5 gallons). For the city portion of 40 miles, the vehicle uses 1 gallon of fuel (40 miles / 40 mpg). For the highway portion of 180 miles, the vehicle uses 3.75 gallons of fuel (180 miles / 48 mpg). The total fuel used for the entire trip is 1 + 3.75 = 4.75 gallons.

Step-by-Step Solution

1

Calculate the fuel economy rate in miles per gallon for city driving.

40

miles per gallon

Divide

80

miles by

2

gallons to find how many miles can be traveled per gallon of fuel in the city.

2

Calculate the fuel economy rate in miles per gallon for highway driving.

48

miles per gallon

Divide

120

miles by

2.5

gallons to find how many miles can be traveled per gallon of fuel on the highway.

3

Find the amount of fuel used for the city portion of the trip.

1

gallon

Divide the city distance of the trip (

40

miles) by the city fuel rate (

40

miles per gallon).

4

Find the amount of fuel used for the highway portion of the trip.

3.75

gallons

Divide the highway distance of the trip (

180

miles) by the highway fuel rate (

48

miles per gallon).

5

Calculate the total fuel used for the entire trip.

4.75

gallons

Sum the city fuel used (

1

gallon) and the highway fuel used (

3.75

gallons).

Key Concept

Calculating and applying distinct rates to different portions of a trip.

Question 512Question →

In the quadratic equation $x^2 - kx + 5 = 0$ , $k$ is a constant. If the solutions to the equation are $x = 3 \pm \sqrt{a}$ , where $a$ is a positive constant, what is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. By relating the roots of a quadratic equation to its coefficients, the sum of the roots is

(3 + \sqrt{a}) + (3 - \sqrt{a}) = 6

, which equals the linear coefficient term

k

in

x^2 - kx + 5 = 0

. Using

k = 6

, we can rewrite the equation as

x^2 - 6x + 5 = 0

. Completing the square yields

(x - 3)^2 - 9 + 5 = 0

, which simplifies to

(x - 3)^2 = 4

, or

x = 3 \pm \sqrt{4}

. Comparing this to the given expression

x = 3 \pm \sqrt{a}

gives

a = 4

. Alternatively, using the product of roots,

(3 + \sqrt{a})(3 - \sqrt{a}) = 9 - a

. Since the product of roots is the constant term 5, we have

9 - a = 5

, which directly yields

a = 4

.

Step-by-Step Solution

1

Find the sum of the given solutions to determine the value of the constant

k

.

k = 6

The sum of the solutions

(3 + \sqrt{a}) + (3 - \sqrt{a})

is

6

. For a quadratic equation in the form

x^2 - kx + c = 0

, the sum of the solutions is equal to

k

.

2

Substitute

k = 6

back into the quadratic equation.

x^2 - 6x + 5 = 0

This gives the complete quadratic equation with numerical coefficients.

3

Solve the quadratic equation by completing the square to express the solutions in the form

3 \pm \sqrt{a}

.

x = 3 \pm \sqrt{4}

Completing the square gives

(x - 3)^2 - 9 + 5 = 0

, which simplifies to

(x - 3)^2 = 4

, so

x - 3 = \pm \sqrt{4}

.

4

Compare the obtained solutions to the given form

x = 3 \pm \sqrt{a}

to identify the value of

a

.

a = 4

Comparing

x = 3 \pm \sqrt{4}

with

x = 3 \pm \sqrt{a}

yields

a = 4

.

Key Concept

Relationship between roots and coefficients of a quadratic equation, or solving quadratic equations by completing the square.

Question 513Question →

The quadratic function $f$ is defined by $f(x) = (x - 3)(x - 7)$ . The function $g$ is defined by $g(x) = f(x - 2) + k$ , where $k$ is a constant. In the $xy$ -plane, the vertex of the graph of $y = g(x)$ is $(7, 5)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

9

To find the vertex of the function

f(x) = (x - 3)(x - 7)

, we calculate the midpoint of the

x

-intercepts,

x = 5

, and evaluate the function at this value to get

f(5) = -4

, giving the vertex

(5, -4)

. The graph of

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

shifts the graph of

f

right by

2

units and up by

k

units, resulting in a vertex of

(7, -4 + k)

. Since the vertex of

g(x)

is

(7, 5)

, we have

-4 + k = 5

, which solves to

k = 9

.

Step-by-Step Solution

1

Find the vertex of the quadratic function

f(x) = (x - 3)(x - 7)

.

The vertex of

f(x)

is

(5, -4)

.

The

x

-coordinate of the vertex of a quadratic function in factored form

f(x) = a(x - r_1)(x - r_2)

lies midway between the roots

r_1

and

r_2

. Here,

x = \frac{3 + 7}{2} = 5

. Evaluating the function at

x = 5

gives the

y

-coordinate:

f(5) = (5 - 3)(5 - 7) = -4

.

2

Apply the transformations to find the coordinates of the vertex of

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

.

The vertex of

g(x)

is

(7, -4 + k)

.

The transformation

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

shifts the graph of

f(x)

horizontally by

h

units to the right and vertically by

k

units up. For

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

, the vertex of

f

,

(5, -4)

, is shifted

2

units to the right and

k

units up, resulting in the new vertex

(5 + 2, -4 + k) = (7, -4 + k)

.

3

Equate the

y

-coordinate of the vertex of

g(x)

to the given value and solve for

k

.

k = 9

We are given that the vertex of the graph of

y = g(x)

is

(7, 5)

. Setting the

y

-coordinate of the transformed vertex equal to

5

gives

-4 + k = 5

. Adding

4

to both sides yields

k = 9

.

Key Concept

Function Transformations of Quadratic Graphs

Question 514Question →

In the $xy$ -plane, a square has vertices at $(0,0)$ , $(10,0)$ , $(10,10)$ , and $(0,10)$ . A line that passes through the points $(0,2)$ and $(8,10)$ divides the square into two regions. What is the area of the larger region?

Show answer & explanation

Answer: 68

Answer

The area of the larger region is 68.

The total area of the square is

10^2 = 100

. The line segment between

(0,2)

on the left boundary and

(8,10)

on the top boundary forms a right triangle with the top-left vertex of the square

(0,10)

. The legs of this right triangle have lengths

10 - 2 = 8

and

8 - 0 = 8

, so its area is

\frac{1}{2} \times 8 \times 8 = 32

. The area of the other region is

100 - 32 = 68

. The larger area is therefore 68.

Step-by-Step Solution

1

Calculate the area of the square

100

The square has vertices at

(0,0)

,

(10,0)

,

(10,10)

, and

(0,10)

, which gives a side length of 10. The area of a square is

\text{side}^2

.

2

Determine the dimensions of the smaller triangular region formed by the line

Legs of length 8 and 8

The line intersects the left edge at

(0,2)

and the top edge at

(8,10)

. The corner of the square is at

(0,10)

. The distance from

(0,2)

to

(0,10)

is 8, and the distance from

(8,10)

to

(0,10)

is 8.

3

Calculate the area of the smaller triangular region

32

The area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8 = 32

.

4

Subtract the area of the smaller region from the total area of the square to find the area of the larger region

68

The area of the larger region is the total area of the square minus the area of the smaller region:

100 - 32 = 68

.

Key Concept

Calculating the area of a region by partitioning a geometric shape or using subtraction of areas.

Question 515Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . If the length of side $AB$ is 6 and the length of side $AC$ is 10, what is the length of side $BC$ ?

Show answer & explanation

Answer: 8

Answer

The length of side

BC

is 8.

By applying the Pythagorean theorem to right triangle

ABC

, we have

AB^2 + BC^2 = AC^2

. Substituting the given values gives

6^2 + BC^2 = 10^2

, or

36 + BC^2 = 100

. Subtracting 36 from both sides gives

BC^2 = 64

, and taking the square root yields

BC = 8

.

Step-by-Step Solution

1

Identify the hypotenuse and the given leg in the right triangle.

The hypotenuse is

AC = 10

and the given leg is

AB = 6

.

The side opposite the

90^\circ

angle (angle

B

) is always the hypotenuse.

2

Apply the Pythagorean theorem.

AB^2 + BC^2 = AC^2

, which becomes

6^2 + BC^2 = 10^2

.

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

3

Solve for the unknown leg

BC

.

36 + BC^2 = 100 \implies BC^2 = 64 \implies BC = 8

.

Simplify the squares, subtract 36 from both sides, and find the principal square root of 64.

Key Concept

The Pythagorean Theorem

Question 516Question →

The number of active participants in an online educational forum has been growing exponentially since the forum was launched. At the launch of the forum ( $t = 0$ ), there were $5,000$ active participants. Two months after the launch ( $t = 2$ ), there were $7,200$ active participants. If the number of active participants continues to grow exponentially at this constant monthly percent rate, what is the number of active participants in the forum four months after the launch ( $t = 4$ )?

Show answer & explanation

Answer: 10368

Answer

The correct number of active participants in the forum four months after the launch is 10,368.

To find the number of participants at

t = 4

, we identify that the growth is exponential, meaning the population increases by a constant factor over equal time intervals. Since the population grew from

5,000

to

7,200

over a 2-month interval, the 2-month growth factor is

\frac{7200}{5000} = 1.44

. Because the interval from

t = 2

to

t = 4

is also exactly 2 months, the population will increase by the same factor of

1.44

again. Thus, the population at

t = 4

is

7200 \times 1.44 = 10368

.

Step-by-Step Solution

1

Set up the general exponential growth model equation.

P(t) = P_0 \cdot b^t

, where

P(t)

is the number of participants at time

t

months,

P_0

is the initial number of participants, and

b

is the monthly growth factor.

Establishing the model is necessary to use the given data to solve for unknowns.

2

Solve for the growth factor squared (

b^2

) using the given data points.

7200 = 5000 \cdot b^2 \implies b^2 = \frac{7200}{5000} = 1.44

. Taking the square root gives a monthly growth factor of

b = 1.2

.

Determining the growth factor allows us to project the population for future times.

3

Calculate the active participant population at

t = 4

using the growth factor.

P(4) = P(2) \cdot b^2 = 7200 \cdot 1.44 = 10368

(or

P(4) = 5000 \cdot (1.2)^4 = 10368

).

Finding the number of participants four months after launch answers the question.

Key Concept

Linear and Exponential Growth

Question 517Question →

A scientist measures the concentration of a chemical solution across five different trials. The concentrations recorded for the first four trials are $12.1$ grams per liter ( $\text{g/L}$ ), $12.2\text{ g/L}$ , $12.4\text{ g/L}$ , and $12.5\text{ g/L}$ . The concentration for the fifth trial is $x\text{ g/L}$ , where $x < 12.0$ . If the mean of the concentrations from all five trials is equal to their median, what is the value of $x$ ?

Show answer & explanation

Answer: 11.8

Answer

11.8

The correct answer is

11.8

. Since

x < 12.0

, sorting the five trial concentrations in ascending order gives

x

,

12.1

,

12.2

,

12.4

, and

12.5

. The median is the middle value of this ordered list, which is

12.2

. The mean is the sum of the five values divided by

5

, which is

\frac{x + 12.1 + 12.2 + 12.4 + 12.5}{5} = \frac{x + 49.2}{5}

. Setting the mean equal to the median gives the equation

\frac{x + 49.2}{5} = 12.2

. Multiplying both sides by

5

yields

x + 49.2 = 61

. Subtracting

49.2

from both sides gives

x = 11.8

.

Step-by-Step Solution

1

Order the data set in ascending order using the condition

x < 12.0

.

x, 12.1, 12.2, 12.4, 12.5

To find the median, the values must be listed in order. Since

x

is less than

12.0

, it must be the smallest value in the data set.

2

Identify the median of the five trials.

12.2

For a set of

5

ordered values, the median is the third value.

3

Write an equation representing the mean of the five trials set equal to the median.

\frac{x + 12.1 + 12.2 + 12.4 + 12.5}{5} = 12.2

The problem states that the mean of the five concentrations is equal to their median.

4

Solve the equation for

x

.

x = 11.8

Multiply both sides of the equation by

5

to get

x + 49.2 = 61.0

, then subtract

49.2

from both sides.

Key Concept

Calculating the mean and identifying the median of a data set, and solving for an unknown variable under inequality constraints.

Question 518Question →

A right circular cylinder and a right circular cone have equal volumes. The radius of the cylinder's base is $3$ times the radius of the cone's base. If the height of the cone is $k$ times the height of the cylinder, what is the value of $k$ ?

Show answer & explanation

Answer: 27

Answer

27

The volume of a cylinder is

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and the volume of a cone is

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

. Let the base radius of the cone be

r

and the height of the cylinder be

h

. According to the problem, the base radius of the cylinder is

3r

and the height of the cone is

kh

. Setting the volumes equal to each other gives

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

. Squaring the term in parentheses simplifies the equation to

9 \pi r^2 h = \frac{k}{3} \pi r^2 h

. Dividing both sides by the common term

\pi r^2 h

yields

9 = \frac{k}{3}

. Multiplying both sides by 3 results in

k = 27

.

Step-by-Step Solution

1

State the standard volume formulas for a right circular cylinder and a right circular cone.

V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}

and

V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}

To establish the mathematical equations governing the volume of each solid.

2

Define variables for the cone's radius (

r

) and the cylinder's height (

h

), then write the cylinder's radius and the cone's height using the given relationships.

r_{\text{cylinder}} = 3r

,

r_{\text{cone}} = r

,

h_{\text{cylinder}} = h

, and

h_{\text{cone}} = kh

To express all variables in terms of

r

,

h

, and the constant

k

so they can be compared directly.

3

Substitute the expressions into the volume formulas and set the two volumes equal to each other.

\pi (3r)^2 h = \frac{1}{3} \pi r^2 (kh)

The problem states that the cylinder and the cone have equal volumes.

4

Simplify the equation by squaring the cylinder's radius and dividing both sides by the common factors.

9 \pi r^2 h = \frac{k}{3} \pi r^2 h \implies 9 = \frac{k}{3} \implies k = 27

To isolate and solve for the constant

k

.

Key Concept

Relating the volumes of geometric solids through algebraic substitution and dimension scaling.

Question 519Question →

The polynomial function $p$ is defined by $p(x) = x^4 - 3x^3 + ax^2 - 7x + 10$ , where $a$ is a constant. If $x - 2$ is a factor of $p(x)$ , what is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

3

According to the Factor Theorem, if

x - 2

is a factor of the polynomial function

p(x)

, then evaluating the polynomial at

x = 2

must yield a value of

0

. Substituting

x = 2

into the equation gives

p(2) = 2^4 - 3(2)^3 + a(2)^2 - 7(2) + 10 = 0

. Simplifying the terms results in

16 - 24 + 4a - 14 + 10 = 0

, which further simplifies to

4a - 12 = 0

. Adding

12

to both sides and dividing by

4

yields

a = 3

.

Step-by-Step Solution

1

Apply the Factor Theorem to set up the equation.

p(2) = 0

By the Factor Theorem,

x - c

is a factor of

p(x)

if and only if

p(c) = 0

.

2

Substitute

x = 2

into the polynomial expression.

2^4 - 3(2)^3 + a(2)^2 - 7(2) + 10 = 0

Evaluating

p(2)

yields a linear equation in terms of the unknown constant

a

.

3

Simplify the arithmetic terms in the equation.

4a - 12 = 0

Simplifying the constant values:

16 - 24 - 14 + 10 = -12

.

4

Solve the linear equation for

a

.

a = 3

Isolating the variable term gives

4a = 12

, and dividing by

4

yields

a = 3

.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

x - c

if and only if

p(c) = 0

.

Question 520Question →

A local library recorded the number of books checked out by 9 patrons on a certain day. The numbers of books checked out were:

2, 3, 3, 5, 6, 7, 8, 9, 12

If a 10th patron who checked out $x$ books is included, the mean of the numbers of books checked out by the 10 patrons is equal to the median of the numbers of books checked out by the 10 patrons. If $x$ is a positive integer, what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

10

To find the value of

x

, we first find the sum of the 9 given numbers:

2 + 3 + 3 + 5 + 6 + 7 + 8 + 9 + 12 = 55

. When a 10th number

x

is added, the mean of the 10 numbers is

\frac{55 + x}{10}

. Since

x

is a positive integer, we analyze the median of the 10 numbers depending on the value of

x

. If

x \ge 8

, the sorted list of the 10 numbers is

2, 3, 3, 5, 6, 7, 8, 9, 12, x

(with the positions of

9

,

12

, and

x

potentially rearranged depending on how large

x

is). In this case, the 5th and 6th numbers in the sorted list are

6

and

7

. Thus, the median is

\frac{6 + 7}{2} = 6.5

. Setting the mean equal to the median gives

\frac{55 + x}{10} = 6.5

, which simplifies to

55 + x = 65

, or

x = 10

. Since

10

is indeed a positive integer and is

\ge 8

,

10

is the correct answer.

Step-by-Step Solution

1

Find the sum of the original 9 data points.

Sum = 55

This is needed to write the formula for the mean of the 10 data points: Mean = (55 + x)/10.

2

Determine the median of the 10 data points when x is a large positive integer.

Median = 6.5

For x >= 8, the ordered dataset is 2, 3, 3, 5, 6, 7, 8, 9, 12, x. The median of a 10-element dataset is the average of the 5th and 6th elements (6 and 7), which is 6.5.

3

Equate the mean and median expressions and solve for x.

x = 10

Setting (55 + x)/10 = 6.5 gives 55 + x = 65, which results in x = 10. Since 10 is a positive integer and is greater than or equal to 8, it is the correct value.

Key Concept

Calculating and comparing the mean and median of a dataset after adding a new value.

Answer

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Common Mistakes

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