All practice questions

Question 481Question →

Let the function $f$ be given by $f(x) = 3x - 5$ . If a second function $g$ is defined in terms of $f$ as $g(x) = 2f(x + 1) + 4$ , what is the value of $x$ for which $g(x) = 12$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(x) = 12

gives

2f(x + 1) + 4 = 12

. Subtracting 4 from both sides yields

2f(x + 1) = 8

. Dividing by 2 yields

f(x + 1) = 4

. Substituting

x+1

into the definition

f(x) = 3x - 5

gives

3(x + 1) - 5 = 4

. Simplifying the equation results in

3x + 3 - 5 = 4

, which is

3x - 2 = 4

. Solving for

x

gives

3x = 6

, so

x = 2

.

Step-by-Step Solution

1

Set the equation

g(x) = 12

using the definition of

g(x)

2f(x + 1) + 4 = 12

We are given that

g(x) = 12

and want to find the corresponding value of

x

.

2

Isolate the function term

f(x + 1)

f(x + 1) = 4

Subtract 4 from both sides of the equation to get

2f(x + 1) = 8

, then divide by 2.

3

Use the definition of

f(x)

to express

f(x + 1)

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

Substitute

x + 1

in place of

x

in the function

f(x) = 3x - 5

.

4

Set the expression for

f(x + 1)

equal to 4 and solve for

x

x = 2

Solve the linear equation

3x - 2 = 4

by adding 2 to both sides to get

3x = 6

, then dividing by 3.

Key Concept

Applying multiple transformations to a linear function and solving the resulting equation using function notation.

Question 482Question →

If the quadratic equation $x^2 - bx + 16 = 0$ , where $b$ is a positive constant, has two real solutions such that one solution is $4$ times the other, what is the value of $b$ ?

Show answer & explanation

Answer: 10

Answer

The value of

b

is

10

.

By writing the roots as

r

and

4r

, the quadratic equation can be represented as

(x-r)(x-4r) = x^2 - 5rx + 4r^2 = 0

. Comparing this to the given equation

x^2 - bx + 16 = 0

, we establish that

4r^2 = 16

and

b = 5r

. Solving for

r

gives

r^2 = 4

, which means

r = \pm 2

. Since

b

is a positive constant, we select

r = 2

, yielding

b = 5(2) = 10

. Alternatively, using Vieta's formulas, the product of the roots is

r \cdot 4r = 16 \implies 4r^2 = 16 \implies r = \pm 2

, and the sum of the roots is

r + 4r = b \implies 5r = b

. Since

b > 0

, we find

b = 10

.

Step-by-Step Solution

1

Define the roots in terms of a single variable and express the quadratic equation in factored form.

Let the roots be

r

and

4r

. The factored form is

(x - r)(x - 4r) = 0

.

This allows us to relate the given relationship between the roots to the coefficients of the quadratic equation.

2

Expand the factored quadratic expression.

x^2 - 5rx + 4r^2 = 0

This transforms the equation into the standard form

x^2 + Bx + C = 0

so we can match coefficients.

3

Compare the expanded equation with the given equation

x^2 - bx + 16 = 0

.

4r^2 = 16

and

b = 5r

Matching corresponding coefficients allows us to solve for the unknown variables.

4

Solve the constant term equation for

r

.

r = 2

or

r = -2

Finding the value of

r

is necessary to calculate the value of

b

.

5

Calculate the value of

b

using the constraint that

b

is positive.

b = 5(2) = 10

Using the negative root

r = -2

would yield a negative value for

b

(

b = -10

), which violates the constraint that

b

must be a positive constant.

Key Concept

Relating the roots of a quadratic equation to its coefficients

Estimated Time:1m 30s

Question 483Question →

The polynomial function $p$ is defined by $p(x) = x^3 - 4x^2 - kx + 36$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ has an $x$ -intercept at $(3, 0)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

An

x

-intercept at

(3, 0)

indicates that when

x = 3

,

p(x) = 0

. Substituting

x = 3

and

p(3) = 0

into the equation

p(x) = x^3 - 4x^2 - kx + 36

results in

3^3 - 4(3)^2 - k(3) + 36 = 0

. Simplifying the terms gives

27 - 36 - 3k + 36 = 0

. The

-36

and

+36

terms cancel, leaving

27 - 3k = 0

. Solving for

k

gives

3k = 27

, which results in

k = 9

.

Step-by-Step Solution

1

Relate the

x

-intercept to the root of the polynomial function.

p(3) = 0

Since the graph of

y = p(x)

has an

x

-intercept at

(3, 0)

, the value of the function at

x = 3

must be

0

.

2

Substitute

x = 3

into the polynomial expression.

3^3 - 4(3)^2 - k(3) + 36 = 0

By setting the expression equal to

0

, we can solve for the unknown constant

k

.

3

Simplify the expression and solve for

k

.

k = 9

27 - 36 - 3k + 36 = 0

simplifies to

27 - 3k = 0

, which yields

3k = 27

and thus

k = 9

.

Key Concept

Relationship between x-intercepts of a graph and the roots of the polynomial function.

Question 484Question →

The graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, is a parabola in the $xy$ -plane that passes through the points $(-3, 22)$ and $(9, 22)$ . If the minimum value of $f(x)$ is $4$ , what is the value of $f(1)$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

The correct answer is 6. The axis of symmetry of the parabola is halfway between the points with equivalent y-values: x = 3. Using the minimum value of 4, the vertex is identified as (3, 4). Writing the equation in vertex form as f(x) = a(x - 3)^2 + 4 and substituting (9, 22) yields a = 0.5. Evaluating the function f(x) = 0.5(x - 3)^2 + 4 at x = 1 yields 6.

Step-by-Step Solution

1

Determine the axis of symmetry of the parabola.

The axis of symmetry is

x = 3

.

Because the parabola passes through the points

(-3, 22)

and

(9, 22)

, which have the same

y

-coordinate, the axis of symmetry must lie halfway between their

x

-coordinates:

x = \frac{-3 + 9}{2} = 3

.

2

Write the quadratic function in vertex form.

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

Since the function has a minimum value of

4

, the vertex of the upward-opening parabola is at

(3, 4)

.

3

Solve for the leading coefficient

a

.

a = 0.5

Substitute the point

(9, 22)

into the vertex form equation:

22 = a(9 - 3)^2 + 4

, which simplifies to

18 = 36a

, so

a = 0.5

.

4

Evaluate

f(1)

.

f(1) = 6

Substitute

x = 1

into the completed function

f(x) = 0.5(x - 3)^2 + 4

to get

f(1) = 0.5(1 - 3)^2 + 4 = 0.5(4) + 4 = 6

.

Key Concept

Using symmetry, vertex form, and given points to determine a quadratic function's equation and evaluate it.

Question 485Question →

A specialized laser cutter cuts steel sheet metal at a constant rate of $18$ inches per minute. At this rate, how many seconds does it take the laser cutter to cut a length of $6$ feet of steel? ( $1\text{ foot} = 12\text{ inches}$ )

Show answer & explanation

Answer: 240

Answer

The correct answer is 240.

To find the number of seconds it takes the laser cutter to cut

6

feet of steel, we first convert the length to inches:

6\text{ feet} \times 12\text{ inches/foot} = 72\text{ inches}

. Next, we divide this length by the cutting rate to find the duration in minutes:

72\text{ inches} \div 18\text{ inches/minute} = 4\text{ minutes}

. Finally, we convert the minutes into seconds: $4\text{ minutes} \times 60\text{ seconds/minute} = 240\text{ seconds}.

Step-by-Step Solution

1

Convert the total cutting distance from feet to inches.

72\text{ inches}

Since the rate is given in inches per minute, we must convert the length from feet to inches to align the units. Knowing that

1\text{ foot} = 12\text{ inches}

, we multiply

6

by

12

.

2

Calculate the time in minutes to cut

72\text{ inches}

at a rate of

18\text{ inches/minute}

.

4\text{ minutes}

Dividing the total distance (

72\text{ inches}

) by the cutting rate (

18\text{ inches/minute}

) yields the time in minutes.

3

Convert the time from minutes to seconds.

240\text{ seconds}

Since

1\text{ minute} = 60\text{ seconds}

, multiplying

4

minutes by

60

gives the final time in seconds.

Key Concept

Performing multi-step unit conversions using rate and dimensional analysis.

Question 486Question →

The function $f$ is defined by $f(x) = 2x^2 - 12x + c$ , where $c$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(h, 5)$ , where $h$ is a constant. What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

For a quadratic function in standard form

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

, the x-coordinate of the vertex is given by

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

. In this function,

a = 2

and

b = -12

, so

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

. The vertex coordinates are

(3, 5)

, which means

f(3) = 5

. Substituting

x = 3

into the function gives

2(3)^2 - 12(3) + c = 5

. Simplifying the terms gives

18 - 36 + c = 5

, which becomes

-18 + c = 5

. Adding

18

to both sides yields

c = 23

.

Step-by-Step Solution

1

Identify the x-coordinate formula for the vertex of a quadratic function

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

.

The x-coordinate

h

is given by

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

.

To find the axis of symmetry and the horizontal position of the vertex.

2

Substitute

a = 2

and

b = -12

from the given equation

f(x) = 2x^2 - 12x + c

into the vertex formula.

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

To calculate the specific x-coordinate of the vertex for this function.

3

Substitute the vertex coordinates

(3, 5)

into the function

f(x)

.

f(3) = 2(3)^2 - 12(3) + c = 5

Since the vertex lies on the graph of the function, its coordinates must satisfy the function's equation.

4

Simplify the equation and solve for the constant

c

.

18 - 36 + c = 5 \implies -18 + c = 5 \implies c = 23

To determine the final value of the constant

c

.

Key Concept

Vertex of a quadratic function

Question 487Question →

A fiber-optic internet connection downloads data at a constant rate of $40$ megabits per second. There are $8$ megabits in $1$ megabyte and $1,000$ megabytes in $1$ gigabyte. At this rate, how many gigabytes of data can the connection download in $1$ hour?

Show answer & explanation

Answer: 18

Answer

The connection can download 18 gigabytes of data in 1 hour.

First, convert the download rate from megabits per second to megabytes per second:

40 \div 8 = 5

megabytes per second. Next, find the number of seconds in

1

hour:

60 \text{ minutes} \times 60 \text{ seconds} = 3,600

seconds. Multiply the rate in megabytes per second by the number of seconds in an hour to find the total megabytes downloaded:

5 \times 3,600 = 18,000

megabytes. Finally, convert megabytes to gigabytes:

18,000 \div 1,000 = 18

gigabytes.

Step-by-Step Solution

1

Convert the download speed from megabits per second to megabytes per second.

5

megabytes per second

Since

8\text{ megabits} = 1\text{ megabyte}

, divide

40

megabits per second by

8

.

2

Calculate the total number of seconds in

1

hour.

3,600

seconds

Since there are

60

minutes in

1

hour and

60

seconds in

1

minute, multiply

60

by

60

.

3

Determine the total megabytes downloaded in

1

hour.

18,000

megabytes

Multiply the rate of

5

megabytes per second by the elapsed time of

3,600

seconds.

4

Convert the total megabytes to gigabytes.

18

gigabytes

Since

1,000\text{ megabytes} = 1\text{ gigabyte}

, divide

18,000

megabytes by

1,000

.

Key Concept

Unit Conversions

Question 488Question →

The graph of the quadratic function $f(x) = x^2 - 4x + 7$ is translated $3$ units to the right and $2$ units down in the $xy$ -plane to form the graph of the function $g(x) = x^2 + px + q$ , where $p$ and $q$ are constants. What is the value of $q$ ?

Show answer & explanation

Answer: 26

Answer

The value of

q

is

26

.

To find the constant term

q

of the translated quadratic function

g(x) = x^2 + px + q

, we first determine the vertex of the original function

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

. Completing the square gives

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

, which identifies the vertex of the parabola as

(2, 3)

. Translating the graph

3

units to the right and

2

units down shifts the vertex to

(2 + 3, 3 - 2) = (5, 1)

. Because the translation does not affect the shape of the parabola, the leading coefficient remains

1

. The vertex form of the new function is

g(x) = (x - 5)^2 + 1

. Expanding this expression yields

g(x) = x^2 - 10x + 25 + 1 = x^2 - 10x + 26

. Comparing this to

g(x) = x^2 + px + q

, we find that

q = 26

.

Step-by-Step Solution

1

Convert the original function

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

into vertex form,

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

, to identify its vertex

(h, k)

.

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

, which represents a parabola with vertex

(2, 3)

.

Finding the vertex of the original function allows us to apply the translation directly to the vertex coordinates.

2

Apply the translation of

3

units to the right and

2

units down to the coordinates of the vertex

(2, 3)

.

The new vertex is

(2 + 3, 3 - 2) = (5, 1)

.

Translating a graph shifts its vertex by the corresponding horizontal and vertical amounts.

3

Write the equation of the translated function

g(x)

in vertex form using the new vertex

(5, 1)

and the original leading coefficient

a = 1

.

g(x) = (x - 5)^2 + 1

A translation does not change the shape or vertical stretch of the parabola, so the coefficient of

x^2

remains

1

.

4

Expand the vertex form of

g(x)

into standard form,

g(x) = x^2 + px + q

, to determine the constant term

q

.

g(x) = x^2 - 10x + 26

, which means

q = 26

.

Expanding the equation allows us to compare it directly with the standard form of

g(x)

and identify the value of the constant term.

Key Concept

Vertex form and translations of quadratic functions

Alternative Method

Alternatively, the translation can be applied directly to the variable

x

in the function equation. Translating a function

f(x)

by

3

units to the right and

2

units down yields

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

. Substituting

x - 3

into the original function gives:

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

. Simplifying this expression:

g(x) = (x^2 - 6x + 9) - (4x - 12) + 5 = x^2 - 10x + 26

. This directly shows that the constant term

q

is

26

.

Estimated Time:1m 30s

Question 489Question →

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the difference between the two solutions to the equation is $5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 13

Answer

The value of the positive constant

k

is

13

.

For the quadratic equation

x^2 - kx + 36 = 0

with solutions

r_1

and

r_2

, the sum of the solutions is

r_1 + r_2 = k

and the product of the solutions is

r_1 \cdot r_2 = 36

. Given that the difference between the two solutions is

5

, we can write

|r_1 - r_2| = 5

. Squaring both sides yields

(r_1 - r_2)^2 = 25

. Using the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we substitute the known values to obtain

25 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is positive,

k = 13

. Alternatively, we can find two numbers whose product is

36

and whose difference is

5

. These numbers are

9

and

4

, because

9 \cdot 4 = 36

and

9 - 4 = 5

. The sum of these solutions is

9 + 4 = 13

, which matches the coefficient of the linear term,

k

.

Step-by-Step Solution

1

Relate the roots of the quadratic equation

x^2 - kx + 36 = 0

to its coefficients using Vieta's formulas.

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 \cdot r_2 = 36

.

This sets up the system of equations representing the roots.

2

Express the given root difference of

5

mathematically and square it.

|r_1 - r_2| = 5 \implies (r_1 - r_2)^2 = 25

.

Squaring the difference allows us to use standard algebraic identities.

3

Apply the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

to substitute the sum and product expressions.

25 = k^2 - 4(36)

This converts the relationship between the roots into an equation with the single variable

k

.

4

Solve the equation for

k

, selecting the positive value.

25 = k^2 - 144 \implies k^2 = 169 \implies k = 13

(since

k > 0

).

To determine the final value of

k

satisfying the constraint that

k

is a positive constant.

Key Concept

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

Question 490Question →

A quadratic function $f$ has its vertex at $(4, 12)$ and a $y$ -intercept at $(0, -4)$ in the $xy$ -plane. The function $g$ is defined by $g(x) = f(x + 2) + k$ , where $k$ is a constant. If the $y$ -intercept of the graph of $g$ is $(0, 15)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

The quadratic function

f(x)

is determined to be

f(x) = -(x - 4)^2 + 12

by substituting the vertex

(4, 12)

and the

y

-intercept

(0, -4)

into the vertex form. The transformation

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

translates the function horizontally left by 2 units and vertically by

k

units, resulting in

g(x) = -(x - 2)^2 + 12 + k

. Using the

y

-intercept of

g

, which is

(0, 15)

, we substitute

x = 0

to get

15 = -(0 - 2)^2 + 12 + k

, simplifying to

15 = 8 + k

, which yields

k = 7

.

Step-by-Step Solution

1

Write the vertex form of

f(x)

f(x) = a(x - 4)^2 + 12

The vertex form of a quadratic function with vertex

(h, k_{vertex})

is given by

f(x) = a(x - h)^2 + k_{vertex}

.

2

Determine the value of the coefficient

a

a = -1

, so

f(x) = -(x - 4)^2 + 12

Substitute the coordinates of the

y

-intercept

(0, -4)

into the vertex form equation to solve for

a

.

3

Express the transformed function

g(x)

in terms of

x

and

k

g(x) = -(x - 2)^2 + 12 + k

Apply the translation rules: substituting

x + 2

for

x

shifts the graph left by 2 units, and adding

k

shifts the graph vertically by

k

units.

4

Solve for the constant

k

k = 7

Use the

y

-intercept of

g(x)

, which is

(0, 15)

, so

g(0) = 15

. Setting

15 = -(0 - 2)^2 + 12 + k

simplifies to

15 = 8 + k

, which yields

k = 7

.

Key Concept

Quadratic function vertex form and transformations

Question 491Question →

A shipping company records the weights, in pounds, of $10$ packages in a delivery batch. The table below shows the distribution of the weights.

Weight (pounds)	Frequency
$4$	$3$
$6$	$3$
$8$	$3$
$26$	$1$

If the heaviest package is removed from the batch, by how much will the mean weight of the remaining packages decrease, in pounds?

Show answer & explanation

Answer: 2

Answer

The mean weight of the remaining packages will decrease by

2

pounds.

The original mean weight of the

10

packages is calculated by finding the total weight and dividing it by

10

. The total weight is

(3 \times 4) + (3 \times 6) + (3 \times 8) + (1 \times 26) = 12 + 18 + 24 + 26 = 80

pounds, so the initial mean is

\frac{80}{10} = 8

pounds. After removing the heaviest package of

26

pounds, the remaining

9

packages have a total weight of

80 - 26 = 54

pounds. The new mean weight is

\frac{54}{9} = 6

pounds. The decrease in the mean weight is

8 - 6 = 2

pounds.

Step-by-Step Solution

1

Calculate the total weight of the

10

packages.

80

pounds

By summing the products of each weight and its corresponding frequency.

2

Calculate the initial mean weight.

8

pounds

By dividing the total weight of

80

pounds by the total count of

10

packages.

3

Calculate the sum of the weights of the remaining packages after removing the heaviest package.

54

pounds

By subtracting the weight of the heaviest package (

26

pounds) from the original total weight (

80

pounds).

4

Calculate the new mean weight of the remaining

9

packages.

6

pounds

By dividing the remaining weight (

54

pounds) by the new count of packages (

9

).

5

Find the decrease in the mean weight.

2

pounds

By subtracting the new mean weight (

6

pounds) from the original mean weight (

8

pounds).

Key Concept

Effect of outlier removal on the mean of a data distribution

Question 492Question →

An electric scooter travels at a constant speed of $15$ miles per hour. Given that $1\text{ mile} = 1.6$ kilometers, how many seconds does it take the scooter to travel $100$ meters?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15. The scooter takes 15 seconds to travel 100 meters.

To find the time in seconds to travel 100 meters, we first convert the speed of 15 miles per hour to kilometers per hour by multiplying by 1.6, yielding 24 kilometers per hour. We then convert this speed to meters per second by multiplying by 1,000 (to convert kilometers to meters) and dividing by 3,600 (to convert hours to seconds), which simplifies to 20/3 meters per second. Finally, dividing the distance of 100 meters by this speed of 20/3 meters per second gives 15 seconds.

Step-by-Step Solution

1

Convert the speed of the scooter from miles per hour to kilometers per hour.

The speed is

24

kilometers per hour.

Since

1\text{ mile} = 1.6

kilometers, we multiply

15

by

1.6

to find the speed in kilometers per hour.

2

Convert the speed from kilometers per hour to meters per second.

The speed is

\frac{20}{3}

meters per second.

There are

1,000

meters in

1

kilometer and

3,600

seconds in

1

hour. Thus, the speed is

\frac{24 \times 1,000}{3,600} = \frac{20}{3}

meters per second.

3

Calculate the time required to travel

100

meters.

The time is

15

seconds.

Using the formula

\text{time} = \frac{\text{distance}}{\text{speed}}

, we divide

100

meters by the speed of

\frac{20}{3}

meters per second:

100 \div \frac{20}{3} = 15

seconds.

Key Concept

Converting compound units and applying rate formulas to solve for time.

Question 493Question →

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(4, -3)$ . If the graph passes through the point $(1, 15)$ , what is the value of $f(2)$ ?

Show answer & explanation

Answer: 5

Answer

The value of

f(2)

is

5

.

By using the vertex form of a quadratic function,

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

with vertex

(4, -3)

, the function can be written as

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

. Substituting the point

(1, 15)

gives

15 = a(1 - 4)^2 - 3

, which simplifies to

18 = 9a

, leading to

a = 2

. Substituting

a = 2

back into the function gives

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

. Finally, evaluating at

x = 2

gives

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

.

Step-by-Step Solution

1

Write the quadratic function in vertex form and substitute the vertex

(4, -3)

.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the point

(1, 15)

into the equation and solve for the constant

a

.

a = 2

Since the graph passes through

(1, 15)

, substituting

x = 1

and

f(x) = 15

allows us to solve for

a

.

3

Substitute

x = 2

into the completed function

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

to find

f(2)

.

f(2) = 5

Evaluating the function at

x = 2

yields the required value.

Key Concept

Determining a quadratic function's equation from its vertex and a point, then evaluating it.

Question 494Question →

A polynomial function $f$ has the form $f(x) = a(x - 2)(x + 3)(x - 5)$ , where $a$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a $y$ -intercept of $(0, 60)$ . What is the value of $f(1)$ ?

Show answer & explanation

Answer: 32

Answer

32

The

y

-intercept of the graph is given as

(0, 60)

, which means that when

x = 0

,

f(0) = 60

. Substituting

x = 0

into the function gives

f(0) = a(0 - 2)(0 + 3)(0 - 5) = 30a

. Equating this to 60 gives

30a = 60

, so

a = 2

. Therefore, the function is

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

. To find the value of

f(1)

, substitute

x = 1

into this expression:

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

.

Step-by-Step Solution

1

Identify the relation between the

y

-intercept and the function's value.

f(0) = 60

The

y

-intercept of a graph

y = f(x)

is the point where

x = 0

.

2

Substitute

x = 0

into the definition of

f(x)

and set it equal to 60.

a(0 - 2)(0 + 3)(0 - 5) = 60 \implies 30a = 60

This allows us to solve for the unknown constant coefficient

a

.

3

Solve the linear equation for

a

.

a = 2

Dividing both sides of the equation by 30 isolates

a

.

4

Evaluate the complete function

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

at

x = 1

.

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

This yields the requested value of

f(1)

.

Key Concept

Using the factors and a known point (such as the y-intercept) of a polynomial function to determine its algebraic expression and evaluate it.

Question 495Question →

A coffee shop owner records the number of customers served during each hour of an 8-hour shift. The recorded numbers of customers are $11$ , $13$ , $14$ , $15$ , $17$ , $18$ , $20$ , and $x$ . If the mean number of customers served per hour is equal to the median number of customers served per hour for this shift, and $x > 18$ , what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

20

The correct value is 20. When

x > 18

, the 4th and 5th values of the sorted dataset are 15 and 17, giving a median of 16. Setting the mean, which is (108 + x)/8, equal to 16 yields x = 20, which is consistent with the condition

x > 18

.

Step-by-Step Solution

1

Determine the median of the dataset given the constraint

x > 18

.

The median of the dataset is

16

.

When the 8 values are sorted in ascending order, the first six values must be 11, 13, 14, 15, 17, and 18. The remaining two values, 20 and

x

, will occupy the 7th and 8th positions (in either order). Since there are 8 values, the median is the average of the 4th and 5th values:

\frac{15 + 17}{2} = 16

.

2

Set up the equation for the mean of the dataset.

The mean is represented by the expression

\frac{108 + x}{8}

.

The mean is the sum of all 8 values divided by 8:

\frac{11 + 13 + 14 + 15 + 17 + 18 + 20 + x}{8} = \frac{108 + x}{8}

.

3

Equate the mean to the median and solve for

x

.

x = 20

Setting the mean equal to the median gives

\frac{108 + x}{8} = 16

. Multiplying by 8 yields

108 + x = 128

. Subtracting 108 from both sides results in

x = 20

, which satisfies the condition

x > 18

.

Key Concept

Analyzing the mean and median of a dataset containing a variable constraint

Question 496Question →

In the quadratic equation $2x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $x = 6$ , what is the other solution?

Show answer & explanation

Answer: 1.5

Answer

1.5

The sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

. For the equation

2x^2 - 15x + c = 0

, the sum of the roots is

-\frac{-15}{2} = 7.5

. Since one of the solutions is

6

, the other solution is

7.5 - 6 = 1.5

. Alternatively, substituting

x = 6

into the equation yields

2(6)^2 - 15(6) + c = 0 \implies 72 - 90 + c = 0 \implies c = 18

. The equation becomes

2x^2 - 15x + 18 = 0

, which factors as

(2x - 3)(x - 6) = 0

. Setting the factor

2x - 3 = 0

gives the other solution,

x = 1.5

(or

\frac{3}{2}

).

Step-by-Step Solution

1

Find the sum of the roots of the quadratic equation using Vieta's formulas.

The sum of the roots is

-\frac{-15}{2} = 7.5

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of its roots is equal to

-\frac{b}{a}

.

2

Set up an equation using the known root to find the unknown root.

6 + x_2 = 7.5

, where

x_2

is the other root.

Since one root is

6

, adding it to the second root must equal the sum of the roots.

3

Solve for the second root.

x_2 = 1.5

.

Subtracting

6

from both sides of the equation yields

1.5

.

Key Concept

Sum of roots of a quadratic equation (Vieta's Formulas)

Estimated Time:1m 30s

Question 497Question →

The function $f$ is defined by $f(x) = -x^2 + 6x - 1$ . The function $g$ is defined by $g(x) = f(x + 2) - 5$ . If the maximum value of $g(x)$ in the $xy$ -plane occurs at the point $(h, k)$ , what is the value of $h + k$ ?

Show answer & explanation

Answer: 4

Answer

4

The vertex of the original quadratic function

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

is located at

(3, 8)

. The transformation

g(x) = f(x + 2) - 5

translates the graph horizontally to the left by

2

units and vertically down by

5

units. This moves the vertex from

(3, 8)

to

(3 - 2, 8 - 5) = (1, 3)

. Therefore,

h = 1

and

k = 3

, and their sum

h + k

equals

4

.

Step-by-Step Solution

1

Find the vertex of the function

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

.

The vertex of

f(x)

is at

(3, 8)

.

By writing

f(x)

in vertex form,

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

, we find that the maximum value of

f(x)

occurs at

(3, 8)

.

2

Determine the vertex

(h, k)

of the transformed function

g(x) = f(x + 2) - 5

.

(h, k) = (1, 3)

The horizontal shift of

f(x + 2)

translates the graph to the left by

2

units, changing the

x

-coordinate from

3

to

3 - 2 = 1

. The vertical shift of

- 5

translates the graph down by

5

units, changing the

y

-coordinate from

8

to

8 - 5 = 3

.

3

Calculate the value of

h + k

.

4

Adding the coordinates of the transformed vertex yields

h + k = 1 + 3 = 4

.

Key Concept

Vertex form of a quadratic function and translation of functions.

Question 498Question →

An environmental study monitors the populations of two fish species in a lake. At the start of the study ( $t = 0$ ), Species A has a population of $1,200$ and its population decreases by $40$ each year. Species B has a population of $800$ and its population increases by $5\%$ each year. To the nearest whole number, what is the positive difference in the populations of the two species at year $t = 3$ ?

Show answer & explanation

Answer: 154

Answer

The positive difference in the populations of the two species at year

t = 3

is 154.

At

t = 3

, the population of Species A, which decreases linearly by 40 each year, is

1,200 - 40(3) = 1,080

. The population of Species B, which grows exponentially at

5\%

each year, is

800(1.05)^3 = 926.1

. The positive difference between these populations is

1,080 - 926.1 = 153.9

, which rounds to 154.

Step-by-Step Solution

1

Calculate the population of Species A at

t = 3

using a linear decay model.

A(3) = 1,080

Since Species A decreases by a constant number of 40 fish each year, its population follows the linear model

A(t) = 1,200 - 40t

. Substituting

t = 3

gives

1,200 - 120 = 1,080

.

2

Calculate the population of Species B at

t = 3

using an exponential growth model.

B(3) = 926.1

Since Species B increases by a constant percent of

5\%

each year, its population follows the exponential model

B(t) = 800(1.05)^t

. Substituting

t = 3

gives

800(1.05)^3 = 800(1.157625) = 926.1

.

3

Find the positive difference between the two populations and round to the nearest whole number.

154

The positive difference is

|A(3) - B(3)| = |1,080 - 926.1| = 153.9

. Rounding to the nearest whole number yields 154.

Key Concept

Distinguishing between linear models (constant rate of change per unit time) and exponential models (constant percent rate of change per unit time) to make projections.

Question 499Question →

In the quadratic equation $3x^2 - 18x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to this equation is $26$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

The value of

c

is

15

.

To find the value of

c

, we apply Vieta's formulas to the equation

3x^2 - 18x + c = 0

. The sum of the solutions is

x_1 + x_2 = -\frac{-18}{3} = 6

, and the product of the solutions is

x_1 x_2 = \frac{c}{3}

. Using the identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

, we substitute the given values:

26 = 6^2 - 2\left(\frac{c}{3}\right)

. This simplifies to

26 = 36 - \frac{2c}{3}

. Subtracting

36

from both sides yields

-10 = -\frac{2c}{3}

, which gives

2c = 30

, and thus

c = 15

. Alternatively, since the vertex of the corresponding parabola is at

x = -\frac{-18}{2(3)} = 3

, the two solutions can be represented symmetrically as

3 + d

and

3 - d

. The sum of their squares is

(3 + d)^2 + (3 - d)^2 = (9 + 6d + d^2) + (9 - 6d + d^2) = 18 + 2d^2

. Setting this equal to the given value of

26

yields

18 + 2d^2 = 26 \implies 2d^2 = 8 \implies d^2 = 4 \implies d = 2

. Therefore, the solutions are

3 + 2 = 5

and

3 - 2 = 1

. Substituting either solution back into the original equation, such as

x = 1

, yields

3(1)^2 - 18(1) + c = 0 \implies -15 + c = 0 \implies c = 15

.

Step-by-Step Solution

1

Find the sum and product of the solutions to the quadratic equation.

The sum of the solutions is

x_1 + x_2 = 6

, and their product is

x_1 x_2 = \frac{c}{3}

.

According to Vieta's formulas, for a quadratic equation

ax^2 + bx + c = 0

, the sum of the solutions is

-\frac{b}{a}

and the product of the solutions is

\frac{c}{a}

.

2

Relate the sum of the squares of the solutions to their sum and product using the algebraic identity.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 \implies 26 = (6)^2 - 2\left(\frac{c}{3}\right)

.

The algebraic identity

(x_1 + x_2)^2 = x_1^2 + 2x_1 x_2 + x_2^2

can be rearranged to express the sum of squares

x_1^2 + x_2^2

in terms of the sum and product.

3

Solve the resulting linear equation for the constant

c

.

26 = 36 - \frac{2c}{3} \implies \frac{2c}{3} = 10 \implies c = 15

.

Subtracting

36

from both sides gives

-10 = -\frac{2c}{3}

. Multiplying by

-3

gives

30 = 2c

, which yields

c = 15

.

Key Concept

Vieta's Formulas and Algebraic Identities

Question 500Question →

In triangle $XYZ$ , point $W$ lies on side $XY$ and point $V$ lies on side $XZ$ such that line segment $WV$ is parallel to side $YZ$ . If $XW = 4$ , $WY = 2$ , and $WV = 6$ , what is the length of side $YZ$ ?

Show answer & explanation

Answer: 9

Answer

9

Because segment

WV

is parallel to segment

YZ

, triangle

XWV

is similar to triangle

XYZ

by the Angle-Angle (AA) similarity theorem. The ratio of the corresponding side lengths is constant, which gives the proportion

\frac{YZ}{WV} = \frac{XY}{XW}

. The length of side

XY

is

XW + WY = 4 + 2 = 6

. Substituting the values into the proportion yields

\frac{YZ}{6} = \frac{6}{4}

. Solving for

YZ

gives

YZ = 9

.

Step-by-Step Solution

1

Calculate the length of side

XY

XY = 6

The length of side

XY

is the sum of segment lengths

XW

and

WY

:

XY = XW + WY = 4 + 2 = 6

.

2

Determine that triangle

XWV

is similar to triangle

XYZ

\triangle XWV \sim \triangle XYZ

Because segment

WV

is parallel to segment

YZ

, corresponding angles

\angle XWV

and

\angle XYZ

are congruent, and corresponding angles

\angle XVW

and

\angle XZY

are congruent. Since they also share

\angle X

, the two triangles are similar by the Angle-Angle (AA) similarity criterion.

3

Set up a proportion and solve for

YZ

YZ = 9

Corresponding sides of similar triangles are proportional:

\frac{YZ}{WV} = \frac{XY}{XW}

. Substituting the known values gives

\frac{YZ}{6} = \frac{6}{4}

. Multiplying both sides by 6 yields

YZ = \frac{36}{4} = 9

.

Key Concept

Finding side lengths in similar triangles using the Angle-Angle similarity theorem when a line is parallel to one side of a triangle.

Answer

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