All practice questions

Question 1941Question →

Several values of $x$ and the corresponding values of $f(x)$ for the quadratic function $f$ are shown in the table below.

$x$	$f(x)$
$-4$	$18$
$-2$	$6$
$0$	$2$
$2$	$6$
$4$	$18$

The function $g$ is defined by $g(x) = -2(f(x - 3) + 4)$ . If the vertex of the graph of $y = f(x)$ corresponds to the point $(p, q)$ on the graph of $y = g(x)$ , what is the value of $p + q$ ?

-15

-9

-5

3

Show answer & explanation

Answer: $-9$

Answer

-9

The vertex of the quadratic function

f(x)

is identified from the symmetric table values as

(0, 2)

. Under the transformation

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

, the input

x - 3

shifts the vertex horizontally to the right by 3 units, so the new x-coordinate is

p = 3

. Evaluating

g(3)

gives

q = -2(f(0) + 4) = -2(2 + 4) = -2(6) = -12

. The sum of these coordinates is

p + q = 3 + (-12) = -9

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x)

from the table.

Vertex of

f(x)

is

(0, 2)

.

Since

f

is a quadratic function and the table shows symmetry about

x = 0

(with

f(-2) = f(2) = 6

and

f(-4) = f(4) = 18

), the vertex must be at the point where

x = 0

, which gives

f(0) = 2

.

2

Determine the x-coordinate

p

of the corresponding point on the graph of

g(x)

.

p = 3

.

The function

g

is defined as

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

. The expression

f(x - 3)

indicates a horizontal translation of the graph of

f

to the right by 3 units. Therefore, the x-coordinate of the vertex shifts from

0

to

0 + 3 = 3

.

3

Determine the y-coordinate

q

of the corresponding point on the graph of

g(x)

by evaluating

g(3)

.

q = -12

.

Substitute

x = 3

into the definition of

g(x)

:

g(3) = -2(f(3 - 3) + 4) = -2(f(0) + 4)

. Since

f(0) = 2

, this simplifies to

-2(2 + 4) = -2(6) = -12

.

4

Calculate the value of

p + q

.

p + q = -9

.

Adding the coordinates

p = 3

and

q = -12

yields

3 + (-12) = -9

.

Key Concept

Function Notation and Transformations

Question 1942Question →

A technician is monitoring the pressure of a gas inside a container during an experiment. The pressure $P$ , in kilopascals ( $\text{kPa}$ ), and the time elapsed $t$ , in minutes, are related by a linear equation. At $t = 4$ minutes, the pressure is $112\text{ kPa}$ . At $t = 12$ minutes, the pressure is $136\text{ kPa}$ . If the pressure continues to increase at this constant rate, what will the pressure be, in $\text{kPa}$ , at $t = 15$ minutes?

Show answer & explanation

Answer: 145

Answer

145

The relationship between pressure

P

and time

t

is linear, which can be modeled by the equation

P = mt + b

, where

m

is the rate of change (slope) and

b

is the pressure at

t = 0

. Using the points

(4, 112)

and

(12, 136)

, the slope is calculated as

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

. Substituting the point

(4, 112)

and

m = 3

into the equation

P = mt + b

yields

112 = 3(4) + b

, which simplifies to

112 = 12 + b

, so

b = 100

. The linear equation is

P = 3t + 100

. Substituting

t = 15

into this equation gives

P = 3(15) + 100 = 45 + 100 = 145

.

Step-by-Step Solution

1

Calculate the rate of change (slope,

m

) using the two given coordinate points

(4, 112)

and

(12, 136)

.

m = \frac{136 - 112}{12 - 4} = \frac{24}{8} = 3

To find the constant rate at which the pressure is increasing per minute.

2

Set up the linear equation using the point-slope form

P - P_1 = m(t - t_1)

with the point

(4, 112)

.

P - 112 = 3(t - 4) \Rightarrow P = 3t + 100

To establish the linear relationship between pressure and time.

3

Substitute

t = 15

into the linear equation to find the pressure at

15

minutes.

P = 3(15) + 100 = 145

To determine the pressure at the requested time of

15

minutes.

Key Concept

Finding and applying a linear equation in two variables from two points.

Question 1943Question →

A forestry service uses a remote weather station powered by a solar battery system. During a period of heavy cloud cover, the remaining charge in the battery, $C$ , in watt-hours, can be modeled by the linear equation $C = 360 - 4.5h$ , where $h$ represents the number of hours since the cloud cover began. Based on the model, what is the best interpretation of the $h$ -intercept of the relationship?

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The initial charge of the battery, which is 360 watt-hours, when the cloud cover began.

The rate, which is 4.5 watt-hours per hour, at which the battery charge decreases.

The number of hours, which is 90, after the cloud cover began when the battery is completely discharged.

Show answer & explanation

Answer: The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

Answer

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The

h

-intercept of the relationship is the value of

h

when

C = 0

. Substituting

0

for

C

in the equation

C = 360 - 4.5h

gives

0 = 360 - 4.5h

. Solving for

h

yields

4.5h = 360

, or

h = 80

. In context, this represents the number of hours after the cloud cover began when the battery is completely discharged.

Step-by-Step Solution

1

Identify the meaning of the

h

-intercept in the context of the linear equation.

The

h

-intercept occurs when the dependent variable,

C

(remaining charge), is equal to 0.

By definition, the horizontal intercept of a function occurs where the vertical coordinate is zero.

2

Substitute

C = 0

into the equation and solve for

h

.

0 = 360 - 4.5h \implies 4.5h = 360 \implies h = 80

.

This algebraic isolation finds the specific value of

h

when the charge is completely depleted.

3

Interpret the resulting value of

h

in the context of the problem.

At

h = 80

hours, the remaining battery charge is 0 watt-hours, meaning the battery is fully discharged.

Connecting the mathematical coordinate

(80, 0)

back to the units of hours and watt-hours provides the physical interpretation.

Key Concept

Interpreting Linear Relationships in Context

Question 1944Question →

The table below shows some values of the exponential function $f$ , where $f(t) = p \cdot q^t$ for constants $p$ and $q$ .

$t$	$f(t)$
$0$	$80$
$2$	$180$
$4$	$405$

What is the value of $q$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5

The correct answer is 1.5. Since the value of the function at

t = 0

is

80

, the initial value coefficient is

80

. At

t = 2

, the value is

180

, which gives the equation

80 \cdot q^2 = 180

. Solving for

q^2

yields

q^2 = 2.25

, and taking the positive square root gives

q = 1.5

.

Step-by-Step Solution

1

Set up the general exponential equation using the initial value

f(0) = p \cdot q^0 = 80 \implies p = 80

The initial value at

t = 0

directly gives the coefficient

p

because

q^0 = 1

.

2

Substitute another point from the table to solve for the base

q

f(2) = 80 \cdot q^2 = 180 \implies q^2 = 2.25

Using the point

(2, 180)

allows us to write an equation with one variable,

q

.

3

Solve for

q

by taking the square root

q = 1.5

Since the base of an exponential function must be positive, we take the positive square root of 2.25.

Key Concept

Determining the base of an exponential function from a table of values

Question 1945Question →

To calculate the remaining budget, $B$ , in dollars, for a community project, a coordinator uses the formula $B = 1,200 - 3(2x + 50)$ , where $x$ is the number of volunteer shifts scheduled. If the coordinator wants the remaining budget to be at most $450, which of the following inequalities represents all possible values of$ x$?

x \le 100

x \ge 150

x \ge 10

x \ge 100

Show answer & explanation

Answer: $x \ge 100$

Answer

The correct answer is

x \ge 100

.

The correct answer is the inequality stating that

x

is greater than or equal to

100

. Distributing

-3

to both terms inside the parentheses gives

1,200 - 6x - 150 \le 450

. Combining the constant terms on the left side results in

1,050 - 6x \le 450

. Subtracting

1,050

from both sides gives

-6x \le -600

. Dividing both sides by

-6

and reversing the inequality sign because of the division by a negative number yields

x \ge 100

.

Step-by-Step Solution

1

Substitute the formula for the remaining budget

B

into the inequality

B \le 450

.

1,200 - 3(2x + 50) \le 450

To find the possible values of

x

when the budget is at most

450

.

2

Distribute

-3

to both terms inside the parentheses.

1,200 - 6x - 150 \le 450

To simplify the expression by removing the parentheses.

3

Combine the constant terms

1,200

and

-150

on the left side.

1,050 - 6x \le 450

To simplify the inequality further before isolating the variable.

4

Subtract

1,050

from both sides of the inequality.

-6x \le -600

To isolate the variable term on one side of the inequality.

5

Divide both sides of the inequality by

-6

and reverse the inequality sign.

x \ge 100

Dividing by a negative number requires reversing the direction of the inequality sign to maintain the truth value.

Key Concept

Solving linear inequalities in one variable, including the distributive property and sign reversal when dividing by a negative number.

Question 1946Question →

A scientist is monitoring the population of a bacterial culture in a petri dish. The population of the culture grows exponentially, doubling every $3$ hours. If the population of the culture after $9$ hours is $1,600$ , what was the initial population of the culture?

Show answer & explanation

Answer: 200

Answer

The initial population of the culture was

200

.

Since the bacterial culture doubles every

3

hours, the population undergoes

9 \div 3 = 3

doubling periods over a span of

9

hours. An initial population

P_0

that doubles

3

times will grow to

P_0 \times 2^3 = 8P_0

. Setting this expression equal to the final population of

1,600

gives

8P_0 = 1,600

. Dividing both sides of this equation by

8

yields

P_0 = 200

. Therefore, the initial population of the culture was

200

.

Step-by-Step Solution

1

Determine the number of doubling periods that occurred in

9

hours.

3

doubling periods

Since the population doubles every

3

hours, dividing the total time of

9

hours by the doubling time of

3

hours gives

9 \div 3 = 3

periods.

2

Set up an equation representing the exponential growth.

P_0 \times 2^3 = 1,600

An initial population

P_0

doubling

3

times grows by a factor of

2^3

, which is equal to

8

times the initial amount.

3

Solve the equation

8P_0 = 1,600

for the initial population

P_0

.

P_0 = 200

Dividing both sides of the equation by

8

isolates

P_0

and gives the initial value.

Key Concept

Exponential growth models and solving for the initial value.

Question 1947Question →

In the quadratic equation $x^2 + kx + 36 = 0$ , $k$ is a constant. If the equation has two distinct real solutions, which of the following could be the value of $k$ ?

-12

-15

-10

8

Show answer & explanation

Answer: $-15$

Answer

The value

-15

is a possible value for

k

because it is the only option that satisfies the inequality

k < -12

or

k > 12

, which guarantees a positive discriminant and thus two distinct real solutions.

A quadratic equation has two distinct real solutions when its discriminant is positive. For

x^2 + kx + 36 = 0

, the discriminant is

k^2 - 4(1)(36) = k^2 - 144

. The inequality

k^2 - 144 > 0

simplifies to

k^2 > 144

, which means

k > 12

or

k < -12

. Among the choices, the value

-15

is the only one that satisfies this condition.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have two distinct real solutions.

The discriminant,

b^2 - 4ac

, must be strictly greater than

0

.

A positive discriminant ensures the quadratic formula yields two distinct real solutions.

2

Substitute the coefficients of the given equation

x^2 + kx + 36 = 0

into the discriminant formula.

Here,

a = 1

,

b = k

, and

c = 36

. The discriminant is

k^2 - 4(1)(36) = k^2 - 144

.

This expresses the discriminant in terms of the constant

k

.

3

Solve the inequality

k^2 - 144 > 0

for

k

.

k^2 > 144 \Rightarrow |k| > 12

, which means

k > 12

or

k < -12

.

This gives the range of values for

k

that produce two distinct real solutions.

4

Compare the given choices to the solution range.

Only

-15

satisfies the condition because

-15 < -12

.

This determines which of the options is a valid value for

k

.

Key Concept

Determining the number of real solutions of a quadratic equation using the discriminant.

Alternative Method

Instead of solving the inequality algebraically, substitute each answer choice for

k

into the equation and compute the discriminant

b^2 - 4ac

directly to identify which option yields a positive value.

Estimated Time:1m 30s

Question 1948Question →

A cargo ship travels at a constant speed of $24\text{ knots}$ . The ship's engine consumes fuel at a constant rate of $23\text{ liters per minute}$ . The fuel has a density of $0.80\text{ grams per milliliter}$ . Given that $1\text{ knot} = 1.15\text{ miles per hour}$ and $1\text{ metric ton} = 1,000\text{ kilograms}$ , what is the ship's fuel consumption rate in metric tons per mile?

0.04

0.053

0.0625

0.40

Show answer & explanation

Answer: 0.04

Answer

0.04

The correct answer is obtained by first converting the ship's speed to miles per hour (

24 \times 1.15 = 27.6\text{ mph}

). Next, the fuel usage is converted to liters per hour (

23 \times 60 = 1,380\text{ L/hr}

). Using the density conversion where

0.80\text{ g/mL}

is equal to

0.80\text{ kg/L}

, the fuel consumption is converted to mass (

1,380 \times 0.80 = 1,104\text{ kg/hr}

), which is equivalent to

1.104

metric tons per hour. Finally, dividing the fuel rate by the speed gives the rate per mile:

1.104 / 27.6 = 0.04

metric tons per mile.

Step-by-Step Solution

1

Convert the speed of the cargo ship from knots to miles per hour.

24\text{ knots} \times 1.15\text{ miles per hour per knot} = 27.6\text{ miles per hour}

To match the final unit of miles, we convert knots to miles per hour.

2

Convert the fuel consumption rate from liters per minute to liters per hour.

23\text{ liters/minute} \times 60\text{ minutes/hour} = 1,380\text{ liters/hour}

Since speed is in miles per hour, we convert the fuel consumption to a per-hour rate.

3

Convert the volume consumption rate to a mass consumption rate using the density.

1,380\text{ liters/hour} \times 0.80\text{ kilograms/liter} = 1,104\text{ kilograms/hour}

Density of

0.80\text{ g/mL}

is equivalent to

0.80\text{ kg/L}

because there are

1,000

grams in a kilogram and

1,000

milliliters in a liter.

4

Convert the mass rate from kilograms per hour to metric tons per hour.

\frac{1,104\text{ kilograms/hour}}{1,000\text{ kilograms/metric ton}} = 1.104\text{ metric tons/hour}

To match the final unit of metric tons, we divide the kilograms by

1,000

.

5

Calculate the fuel consumption rate in metric tons per mile by dividing the hourly fuel consumption by the hourly speed.

\frac{1.104\text{ metric tons/hour}}{27.6\text{ miles/hour}} = 0.04\text{ metric tons/mile}

Dividing the rate of fuel consumed per hour by the distance traveled per hour gives the fuel consumed per mile.

Key Concept

Unit conversions with compound rates and density scaling

Estimated Time:2m 30s

Question 1949Question →

By analyzing the annual growth rings of ancient conifers, dendrochronologists can reconstruct historical climate patterns with remarkable accuracy. Wide rings indicate periods of abundant rainfall and optimal growth _______ narrow rings reveal times of severe drought and environmental stress. This high-resolution ecological data helps researchers understand how ecosystems historically responded to rapid temperature shifts.

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

, but

,

but

, so

Show answer & explanation

Answer: , but

Answer

The option containing ', but' is the correct answer because it properly uses a comma and the coordinating conjunction 'but' to connect the two independent clauses in a contrasting relationship.

The correct option successfully links the two independent clauses with a comma followed by the coordinating conjunction 'but', which is grammatically correct and logically conveys the contrast between wide and narrow conifer growth rings.

Step-by-Step Solution

1

Identify the clause structure around the blank.

The text before the blank ('Wide rings indicate periods of abundant rainfall and optimal growth') and the text after the blank ('narrow rings reveal times of severe drought and environmental stress') are both independent clauses.

Determining if the clauses are independent or dependent dictates the punctuation and conjunctions needed to link them.

2

Determine the logical relationship between the two independent clauses.

The relationship is contrast: 'wide rings' and 'optimal growth' are compared to 'narrow rings' and 'severe drought'.

This helps select the appropriate coordinating conjunction or transition word.

3

Evaluate the choices based on grammatical correctness and logical meaning.

Using a comma and the contrast coordinator 'but' satisfies both grammatical rules for joining independent clauses and the logical need for a contrast transition.

Other choices either create grammatical errors (like a comma splice or run-on) or use a coordinator that fails to convey the correct logical relationship.

Key Concept

Clause Boundaries and Linking

Question 1950Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(3, -4)$ and passes through the point $(1, 8)$ . The function $g$ is defined by $g(x) = f(x - h) + k$ , where $h$ and $k$ are constants. If the graph of $g$ has its vertex at $(0, 0)$ , what is the value of $g(4)$ ?

16

48

12

24

Show answer & explanation

Answer: 48

Answer

48

The quadratic function

f(x)

can be written in vertex form as

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

. Substituting the point

(1, 8)

gives

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

, which simplifies to

12 = 4a

, so

a = 3

. This means

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

. The function

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

has its vertex at

(0, 0)

, which is a translation of the vertex of

f(x)

from

(3, -4)

to

(0, 0)

. This shifts the function to

g(x) = 3x^2

. Evaluating

g(4)

yields

3(4)^2 = 3(16) = 48

.

Step-by-Step Solution

1

Write the vertex form of the quadratic function

f(x)

.

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

The vertex of the parabola is given as

(3, -4)

, so we can substitute these coordinates into the vertex form

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

.

2

Substitute the point

(1, 8)

into the equation to solve for the constant

a

.

8 = a(1 - 3)^2 - 4 \implies 12 = 4a \implies a = 3

Since the graph passes through

(1, 8)

, these coordinates must satisfy the equation of the function.

3

Determine the equation of the transformed function

g(x)

.

g(x) = 3x^2

The vertex of

f(x)

is at

(3, -4)

. Shifting this vertex to

(0, 0)

removes the horizontal and vertical offsets, simplifying the function to

g(x) = ax^2

with the same leading coefficient

a = 3

.

4

Evaluate

g(4)

.

g(4) = 3(4)^2 = 3(16) = 48

Substitute

x = 4

into the equation for

g(x)

to find the required value.

Key Concept

Quadratic Functions and Graphs

Question 1951Question →

A quality control analyst inspected a batch of 180 microchips from two production lines, Line A and Line B. The microchips were classified as Defective, Acceptable, or Premium. Some of the data from the inspection are shown in the table below.

Production Line	Defective	Acceptable	Premium
Line A	10	50	$x$
Line B	15	$y$	35
Total	25	55	180

If a microchip is selected at random from those classified as Defective or Premium, what is the probability that the chip was produced by Line A?

Show answer & explanation

Answer: 0.375

Answer

The correct answer is 3/8 (or 0.375).

To find the probability that a randomly selected microchip was produced by Line A given that it is classified as Defective or Premium, we restrict the sample space to only the Defective and Premium microchips. From the table, the total number of Defective microchips is 25, and the total number of Premium microchips is 55, giving a combined group size of 25 + 55 = 80 microchips. Next, we determine how many of these 80 microchips were produced by Line A. Line A produced 10 Defective microchips and x Premium microchips. Since the total number of Premium microchips is 55 and Line B produced 35, Line A produced x = 55 - 35 = 20 Premium microchips. Thus, the number of microchips produced by Line A that are Defective or Premium is 10 + 20 = 30. The conditional probability is the number of favorable outcomes divided by the restricted total outcomes, which is 30/80 = 3/8 (or 0.375).

Step-by-Step Solution

1

Find the value of

x

(Line A Premium chips) using the total number of Premium chips.

x = 20

Since the total number of Premium chips is 55 and Line B produced 35, Line A must have produced

55 - 35 = 20

Premium chips.

2

Calculate the total number of chips produced by Line A.

Line A Total = 80

Sum the Defective, Acceptable, and Premium chips produced by Line A:

10 + 50 + 20 = 80

.

3

Calculate the total number of chips produced by Line B.

Line B Total = 100

Subtract the total number of Line A chips from the grand total of 180 chips:

180 - 80 = 100

.

4

Find the value of

y

(Line B Acceptable chips).

y = 50

Subtract the Defective (15) and Premium (35) chips of Line B from its total (100):

100 - 15 - 35 = 50

.

5

Identify the total number of chips in the conditioning category 'Defective or Premium'.

Total Defective or Premium = 80

Sum the total number of Defective chips (25) and Premium chips (55):

25 + 55 = 80

.

6

Identify the number of chips produced by Line A that are either Defective or Premium.

Favorable chips = 30

Sum the Defective chips from Line A (10) and the Premium chips from Line A (

x = 20

):

10 + 20 = 30

.

7

Calculate the probability by dividing the favorable outcomes by the total outcomes of the conditioning category.

30/80 = 3/8 = 0.375

The probability of selecting a Line A chip from the Defective or Premium group is the ratio of favorable chips to total chips in that group.

Key Concept

Conditional Probability from Two-Way Tables

Estimated Time:2m 30s

Question 1952Question →

A commercial printing press has a reservoir of yellow ink. The volume of yellow ink in the reservoir, $V$ , in milliliters, after printing $p$ pages of a color brochure is modeled by the equation $V = 1,200 - 0.15p$ . According to the model, what is the decrease, in milliliters, in the volume of yellow ink in the reservoir for every 100 pages printed?

Show answer & explanation

Answer: 15

Answer

The volume of yellow ink in the reservoir decreases by 15 milliliters for every 100 pages printed.

In the equation

V = 1,200 - 0.15p

, the coefficient of

p

is

-0.15

. This represents the rate of change of the volume of yellow ink in the reservoir with respect to the number of pages printed. Specifically, it means the volume decreases by

0.15

milliliters for each additional page printed. To find the decrease in volume for every 100 pages printed, multiply the rate per page by 100:

0.15 \times 100 = 15

milliliters.

Step-by-Step Solution

1

Identify the rate of change per page from the linear equation.

The rate of change is 0.15 milliliters per page.

In the linear equation

V = 1,200 - 0.15p

, the coefficient of the independent variable

p

represents the change in the dependent variable

V

for each unit increase in

p

.

2

Calculate the decrease in ink volume for 100 pages.

15 milliliters

Since the volume decreases by 0.15 milliliters for each page printed, printing 100 pages results in a total decrease of

0.15 \times 100 = 15

milliliters.

Key Concept

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

Question 1953Question →

If $3^{2x - 1} = 81 \cdot 9^{y + 2}$ , which of the following equations correctly expresses $x$ in terms of $y$ ?

x = \frac{1}{2}y + \frac{7}{2}

x = y + \frac{7}{2}

x = y + \frac{9}{2}

x = 4y + \frac{17}{2}

Show answer & explanation

Answer: $x = y + \frac{9}{2}$

Answer

x = y + \frac{9}{2}

The correct equation is found by expressing

81

as

3^4

and

9^{y+2}

as

3^{2y+4}

. Applying the product rule for exponents, the right side becomes

3^{2y+8}

. Since the bases are the same, equating the exponents gives

2x - 1 = 2y + 8

. Solving for

x

yields

x = y + \frac{9}{2}

.

Step-by-Step Solution

1

Rewrite all parts of the equation using a common base of

3

.

Since

81 = 3^4

and

9 = 3^2

, the term

9^{y+2}

becomes

(3^2)^{y+2} = 3^{2y+4}

. The equation can be rewritten as

3^{2x - 1} = 3^4 \cdot 3^{2y + 4}

.

Expressing all exponential terms with the same base allows the use of exponent rules to simplify the equation.

2

Simplify the product on the right side of the equation using the product rule for exponents,

a^m \cdot a^n = a^{m+n}

.

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

, which simplifies to

3^{2x - 1} = 3^{2y + 8}

.

Adding the exponents of terms with a common base simplifies the right side into a single exponential expression.

3

Set the exponents equal to each other.

2x - 1 = 2y + 8

If two exponential expressions with the same positive base (other than

1

) are equal, their exponents must be equal.

4

Solve for

x

in terms of

y

.

Add

1

to both sides to get

2x = 2y + 9

, then divide by

2

to obtain

x = y + \frac{9}{2}

.

This isolates the variable

x

to express it as a function of

y

.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent laws.

Question 1954Question →

A manufacturing company uses two types of machines, Type P and Type Q, to complete a production run of custom components. The rate of production of a Type P machine is to that of a Type Q machine in the ratio $3:2$ . A team consisting of $4$ Type P machines and $3$ Type Q machines can complete the entire production run in $10$ hours when working together. The company starts the run with this team, but after $4$ hours, $2$ of the Type P machines break down and are removed from service. To complete the remainder of the production run on schedule, additional Type Q machines are added to the remaining machines. What is the total number of Type Q machines that must be added?

2

3

5

6

Show answer & explanation

Answer: 3

Answer

3 additional Type Q machines must be added.

To complete the production run on schedule, we express the rate of Type P and Type Q machines as

3r

and

2r

respectively. The initial team of

4

Type P and

3

Type Q machines produces at a combined rate of

4(3r) + 3(2r) = 18r

components per hour. The total work for the

10

-hour project is

18r \times 10 = 180r

components. In the first

4

hours, the team completes

18r \times 4 = 72r

components, leaving

180r - 72r = 108r

components to be completed in the remaining

6

hours. This requires a combined rate of

108r / 6 = 18r

components per hour. After

2

Type P machines break down, the remaining machines (

2

Type P and

3

Type Q) provide a rate of

2(3r) + 3(2r) = 12r

components per hour. To reach the required rate of

18r

, an additional rate of

18r - 12r = 6r

is needed. Since each Type Q machine has a rate of

2r

, the number of additional Type Q machines needed is

6r / 2r = 3

.

Step-by-Step Solution

1

Define the individual production rates of Type P and Type Q machines using a common variable

r

.

Let the rate of a Type P machine be

3r

components per hour, and the rate of a Type Q machine be

2r

components per hour.

This represents the given

3:2

ratio of their production rates.

2

Calculate the combined rate of the initial team and the total work required for the production run.

Combined initial rate

= 4(3r) + 3(2r) = 18r

components per hour. Total work

W = 18r \times 10 \text{ hours} = 180r

components.

The team consists of

4

Type P and

3

Type Q machines working for

10

hours.

3

Determine the work completed in the first

4

hours and the remaining work.

Completed work

= 18r \times 4 = 72r

components. Remaining work

= 180r - 72r = 108r

components.

The team worked for

4

hours before the breakdown occurred.

4

Find the required rate to complete the remaining work in the remaining time.

Remaining time

= 10 - 4 = 6

hours. Required rate

= 108r / 6 = 18r

components per hour.

The production run must be completed on the original schedule of

10

total hours.

5

Calculate the rate of the remaining machines and the rate deficit.

Current rate of remaining machines (

2

Type P and

3

Type Q)

= 2(3r) + 3(2r) = 12r

components per hour. Rate deficit

= 18r - 12r = 6r

components per hour.

Two Type P machines broke down, leaving

2

Type P and the original

3

Type Q machines.

6

Find the number of additional Type Q machines needed to cover the rate deficit.

Number of additional Type Q machines

= 6r / 2r = 3

.

Each Type Q machine contributes a rate of

2r

components per hour.

Key Concept

Work-rate problems with varying machine efficiencies and changing team compositions over time.

Question 1955Question →

A parabola in the $xy$ -plane has vertex $(3, 18)$ and passes through the origin. If the equation of the parabola is written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b$ ?

Show answer & explanation

Answer: 10

Answer

10

The vertex form of a quadratic function with vertex

(h, k)

is

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

. Substituting the given vertex

(3, 18)

yields the equation

y = a(x - 3)^2 + 18

. Since the parabola passes through the origin, we can substitute the point

(0, 0)

into the equation to find the value of

a

:

0 = a(0 - 3)^2 + 18 \implies 9a = -18 \implies a = -2

. Substituting

a = -2

back into the vertex form and expanding gives

y = -2(x - 3)^2 + 18 = -2(x^2 - 6x + 9) + 18 = -2x^2 + 12x

. Comparing this to the standard form

y = ax^2 + bx + c

, we identify

a = -2

and

b = 12

. The sum of these constants is

a + b = -2 + 12 = 10

.

Step-by-Step Solution

1

Write the equation of the parabola in vertex form.

y = a(x - 3)^2 + 18

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the coordinates of the origin

(0, 0)

to solve for

a

.

a = -2

The parabola passes through the origin

(0, 0)

, so these coordinates must satisfy the equation.

3

Expand the vertex form equation into standard form

y = ax^2 + bx + c

.

y = -2x^2 + 12x

Expanding the equation allows us to identify the coefficients

a

and

b

directly.

4

Identify the values of

a

and

b

, and calculate

a + b

.

a + b = 10

Comparing

y = -2x^2 + 12x

to

y = ax^2 + bx + c

gives

a = -2

and

b = 12

, so

a + b = -2 + 12 = 10

.

Key Concept

Vertex form of quadratic functions and conversion to standard form

Question 1956Question →

In the $xy$ -plane, the graph of the quadratic function $f$ has its vertex at the point $(4, -3)$ . The function $g$ is defined by $g(x) = f(x + 2) + 5$ . Which of the following ordered pairs represents the vertex of the graph of $g$ ?

(6, 2)

(2, -8)

(2, 2)

(6, -8)

Show answer & explanation

Answer: $(2, 2)$

Answer

(2, 2)

The vertex of

f(x)

is given as

(4, -3)

. The translation

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

shifts the graph

2

units to the left and

5

units upward. Applying these transformations to the vertex of

f(x)

yields the new vertex

(4 - 2, -3 + 5) = (2, 2)

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x)

The vertex of

f(x)

is

(4, -3)

This is given in the problem and serves as the starting point for the transformations.

2

Determine the effect of the horizontal translation

f(x + 2)

The vertex shifts

2

units to the left, changing the

x

-coordinate from

4

to

4 - 2 = 2

For any function

f(x)

, the graph of

f(x + c)

is shifted

c

units to the left when

c > 0

.

3

Determine the effect of the vertical translation

+ 5

The vertex shifts

5

units upward, changing the

y

-coordinate from

-3

to

-3 + 5 = 2

Adding a constant to a function shifts its graph vertically upward by that constant's value.

4

Combine the translated coordinates to find the vertex of

g(x)

The vertex of the graph of

g(x)

is

(2, 2)

The horizontal shift results in an

x

-coordinate of

2

, and the vertical shift results in a

y

-coordinate of

2

.

Key Concept

Quadratic Transformations and Vertex Shifts

Question 1957Question →

In a certain school district, the number of students enrolled in high school chemistry was $20\%$ of the total number of high school students. After a new curriculum was introduced, the number of chemistry students increased by $35\%$ , while the total number of high school students in the district remained the same. What percent of the total high school student population was enrolled in chemistry after the curriculum change?

7\%

27\%

48\%

55\%

Show answer & explanation

Answer: $27\%$

Answer

27\%

To find the new percentage of students enrolled in chemistry, we can express the initial enrollment in terms of the total student population,

T

. Initially, the number of chemistry students is

0.20T

. An increase of

35\%

on this enrollment is calculated by multiplying the initial enrollment by

1.35

. This gives the new enrollment as

0.20T \times 1.35 = 0.27T

. Since the total number of students

T

remains unchanged, the new enrollment represents

27\%

of the total population.

Step-by-Step Solution

1

Define variables for the initial values.

Let the total number of high school students in the district be

T

. The initial number of chemistry students is

0.20T

.

Establishing algebraic representations helps organize the relation between the variables.

2

Calculate the new number of chemistry students after the

35\%

increase.

The new number of chemistry students is

0.20T \times (1 + 0.35) = 0.20T \times 1.35 = 0.27T

.

An increase of

35\%

means multiplying the initial value by

1.35

.

3

Calculate the new percentage of the total student population.

Since the total student population

T

remained the same, the new fraction of chemistry students is

\frac{0.27T}{T} = 0.27

, which is

27\%

.

Dividing the new number of chemistry students by the total student population gives the final percentage.

Key Concept

Calculating percentage changes on a subgroup relative to a constant total population.

Question 1958Question →

If $-3(2x - 5) + 4x \geq -7$ , what is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 11

Answer

The maximum possible value of

x

is

11

.

By applying the distributive property, combining like terms, and dividing by

-2

(while reversing the inequality sign), we find that the solution is

x \leq 11

. Thus, the maximum possible value of

x

is

11

.

Step-by-Step Solution

1

Apply the distributive property to simplify the left side of the inequality.

-6x + 15 + 4x \geq -7

Multiplying

-3

by each term inside the parentheses

(2x - 5)

yields

-6x

and

+15

.

2

Combine the variable terms on the left side.

-2x + 15 \geq -7

Combining

-6x

and

4x

gives

-2x

.

3

Subtract

15

from both sides of the inequality.

-2x \geq -22

To isolate the variable term

-2x

on the left side.

4

Divide both sides by

-2

and flip the inequality sign.

x \leq 11

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

5

Determine the maximum value from the solution set.

11

The solution set consists of all values less than or equal to

11

, so the greatest value is

11

.

Key Concept

Solving multi-step linear inequalities, including applying the distributive property and reversing the inequality sign when dividing by a negative number.

Question 1959Question →

The function $f$ is defined by $f(x) = 3^x - 4$ . In the $xy$ -plane, the graph of the function $g$ is obtained by first reflecting the graph of $f$ across the $x$ -axis, then translating the graph vertically up by 10 units, and finally translating the graph horizontally to the right by 2 units. If $g(c) = 5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 4

Answer

The value of c is 4.

Reflecting the function

f(x) = 3^x - 4

across the

x

-axis changes its sign to

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

. Translating this graph vertically up by 10 units adds 10 to the function, yielding

-3^x + 14

. Finally, translating horizontally to the right by 2 units replaces

x

with

x - 2

, producing the function

g(x) = -3^{x-2} + 14

. Setting

g(c) = 5

gives the equation

-3^{c-2} + 14 = 5

. Subtracting 14 from both sides results in

-3^{c-2} = -9

, which simplifies to

3^{c-2} = 9

. Since

9 = 3^2

, the exponent

c-2

must equal 2, which gives

c = 4

.

Step-by-Step Solution

1

Reflect the function

f(x) = 3^x - 4

across the

x

-axis

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

A reflection across the

x

-axis replaces

y

with

-y

, meaning the entire function is multiplied by

-1

.

2

Translate the reflected function vertically up by 10 units

-3^x + 4 + 10 = -3^x + 14

A vertical translation upward by

k

units adds

k

directly to the function expression.

3

Translate the function horizontally to the right by 2 units

g(x) = -3^{x-2} + 14

A horizontal translation to the right by

h

units replaces

x

with

x-h

in the function expression.

4

Set

g(c) = 5

and solve the exponential equation for

c

-3^{c-2} + 14 = 5 \implies 3^{c-2} = 9 \implies c - 2 = 2 \implies c = 4

Substitute

c

into

g(x)

, set the output to 5, isolate the exponential term, and equate exponents to find

c

.

Key Concept

Applying sequential function transformations (reflections, vertical translations, horizontal translations) algebraically and solving exponential equations.

Question 1960Question →

The table below summarizes the results of a survey conducted by a market research firm. The survey asked $160$ randomly selected adults about their age group and their primary source of news.

Age group	Digital/Social Media	Print/Television	Total
Under 40	$68$	$12$	$80$
40 and older	$32$	$48$	$80$
Total	$100$	$60$	$160$

Given that a randomly selected person from the survey has digital or social media as their primary news source, what is the probability that the person is under 40 years old?

\frac{17}{40}

\frac{17}{25}

\frac{17}{20}

\frac{5}{8}

Show answer & explanation

Answer: $\frac{17}{25}$

Answer

The correct answer is

\frac{17}{25}

(representing the ratio of individuals under 40 years old among those who use digital/social media as their primary news source).

The correct answer is

\frac{17}{25}

. To find the conditional probability that a randomly selected person is under 40 years old given that their primary news source is digital or social media, the denominator must represent the total number of people who satisfy the given condition (

100

). The numerator must represent the number of people who satisfy both conditions (are under 40 years old and have digital/social media as their primary news source, which is

68

). Dividing these values yields

\frac{68}{100}

, which simplifies to

\frac{17}{25}

.

Step-by-Step Solution

1

Identify the size of the restricted sample space based on the conditional statement in the question.

The total number of surveyed individuals whose primary news source is digital or social media is

100

.

The phrase 'Given that a randomly selected person... has digital or social media as their primary news source' restricts the denominator to the column total for digital/social media.

2

Identify the number of favorable outcomes within the restricted sample space.

Within the group of

100

individuals who use digital/social media, the number of individuals who are under 40 years old is

68

.

This value serves as the numerator for the conditional probability.

3

Divide the number of favorable outcomes by the size of the restricted sample space and simplify the resulting fraction.

The probability is

\frac{68}{100} = \frac{17}{25}

.

Both numbers are divided by their greatest common divisor,

4

, to simplify the fraction to its lowest terms.

Key Concept

Conditional Probability in Two-Way Tables

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution