# Algebra II Final

Hans pays \$205 in advance on his account at the athletic club. Each time he uses the club, \$10 is deducted from the account. Write an equation that represents the value remaining in his account after x visits to the club. Find the value remaining in the account after 11 visits.
V = 205 – 10x; \$95

Solve the proportion (x−8)/8 = 4/5
x=14 or 72/5

Solve for x.

x-7=1

x=8

Solve the equation or formula for the variable specified. B = 7a²b, for b.
b = B/(7a²)

Which equation best describes the graph below? (Evernote)
y = -2x + 5

solve for x. 3x – 3 = x – 9
x = -3

Which inequality has the solution shown in the graph? (Evernote)
m + 7 ≥ 9

What inequality describes the graph? (Evernote)
x ≤ -5

For the data given, find the equation of the line of best fit. (Evernote)
Incorrect: None of the listed answers are correct

Correct: y = 0.63x + 1.90

Write a proportion that models the statement 10 is to 5 as 16 is to 8.
10/5 = 16/8

Write an equation in slope-intercept form for a line that passes through the given pair of points.
(4,-7), (-2,-10)
y = 1/2x – 9

Solution
Slope Formula:
slope(m) = (y2 – y1) / (x2 – x1)
Substitute and Solve
m = (-10 +7) / (-2 – 4)
m = (-3) / (-6)
m = 1/2 or 0.5
Slope intercept form:
y = mx + b
Pick one of the two points (4,-7) or (-2,-10), substitute and solve
-7 = 1/2(4) + b
-7 = 2 + b
-7 – 2 = b
-9 = b
y = 1/2x – 9

Write an equation in slope-intercept form for a line that passes through the given pair of points. (-6,5), (-9,0)
y = 5x/3 + 15

Write an equation in slope-intercept form of the line that passes through (- 5, 4) and is parallel to the graph of y = -4x + 1.
y = -4x -16

When Spheres-R-Us ships bags of golf balls, each bag must be within 5 balls of 370. Identify the inequality which results in an acceptable number of golf balls in each bag.
365 ≥ 375

Abbiville is a small town that has been steadily growing since 1960. Use the table below to create a linear equation that estimates Abbiville’s population over time. What will the population be in 2014 if the growth remains constant? (Evernote)
364 people

Which is the equation of the line with slope -4 and y-intercept 5?
4x + y -5 = 0
y = -4x + 5

A server at a restaurant kept track of the number of requests for the daily special, and the time of day the request was made. The data is displayed below. (Evernote)
Incorrect: It is a linear model with a positive correlation.

Write the equation of the line, in slope-intercept form, that contains the point (-1, – 5) and is perpendicular to the line -8x – 4y = 3.
y = -2x -7

Write the slope-intercept form of an equation of the line that passes through the point (3, – 6) and has the slope m = -4.
y = -4x + 6

If x = 78 when y = 130 and x varies directly as y, then find x when y = 190.
x = 114 when y = 190

Formula: y = ky(solving for y) or x = ky(solving for x)
Substitute
78 = k(130)
Divide by 130 on both sides
k = 0.6
x = 0.6(190)
x = 114

Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
move it 3 to the right

Evaluate the expression. (2/5)⁵
32/3125

Which of the following is an example of the Commutative Property of Addition?
5 + 2 = 2 + 5

Evaluate the expression. 64 × 4² – 2 × 2²
1016

Ivan Bogdanovich plans to decorate stuffed animals to sell at a crafts fair. The decorations cost \$44.00 and the stuffed animals cost \$5.25 each.
a. Write a function expressing the cost, C(x), of the project in terms of the number of stuffed animals decorated, x.
b. Determine the cost of decorating 25 stuffed animals.
c. How many stuffed animals can be decorated with a budget of \$227.75?
a. C(x) = 5.25x + 44.00
b. \$175.25
c. 35

For the pair of functions, f and g, find (g o f)(x) and (f o g)(x).

f(x) = 6 – x, g(x) = x² – 2

(g o f)(x) = x² – 12x + 34
Solution
(6-x)² – 2
36 – 6x – 6x + x² – 2
x² – 6x – 6x + 36 – 2
x² – 12x + 34

(f o g)(x) = -x² + 8
Solution
-x(x² – 2) + 6
-(x² – 2) + 6
-x² + 2 + 6
-x² + 8

Evaluate the expression. (3/8)⁻²
64/9

Describe how the graph of the function y = 15(x + 2) + 4 can be obtained from the graph of y = 3(x + 2).
shift up 4 units, and vertically stretch by a factor of 5

Simplify the expression.
((-u)³(-u⁹)⁸)/(u⁵)³
-u⁶⁰

Solution
(-u³ × u⁷²)/u¹⁵
-u⁷⁵/u¹⁵ = -u⁶⁰

Evaluate the expression. (-3/8)²
9/64

The graph of f(x) = x² is stretched vertically by a factor of 3, translated 4 units to the right, and translated 8 units downward. Determine the equation of the transformed function, g(x).
g(x) = 3(x – 4)² – 8

A T-shirt company estimates that the average cost per shirt can be approximated by the function
A(x) = (3.75x + 200)/x, where x is the number of T-shirts made. Find the average cost per T-shirt when the company makes 10 shirts.
\$23.75

Find the inverse of the function and determine whether the inverse is a function. y = 7x²
y = ±√(x/7), y is not a function.

Find the inverse of the function and determine whether the inverse is a function. f(x) = 19x²
f⁻¹(x) = ±√(x/19), f⁻¹(x) is not a function.

Since 1993, Daphne Hamilton has owned a franchise of take-out restaurants called The Burger Barn. The number of customers, C, in thousands, that The Burger Barn has served each year can be modeled by the function C(t) = t² + 38t + 600, where t is the number of years from 1993. Using this model, estimate the number of customers served in 1999.
864,000

If f(x) = 9 – x² and g(x) = 3 – x, which is the rule of function (f × g)(x)?
x³ – 3x² – 9x + 27

Evaluate the expression. 3 – 14 × 6 ÷ 7 + 3
-6

If f(x) = 3x + 4 and g(x) = 2 + x, which is the rule of function 3g(x) – 3f(x)?
-6x – 6

Which property is shown by the following statement?
(13 + 3)+ 8 = 13 + (3 + 8)

Given the graph of y = 3(x + 1), what function would be obtained by moving the graph up 4 and moving it 6 to the right?
f(x) = 3(x – 5) + 4

Use elimination to solve the system of equations.
{7x – 3y = -43
{5x + 6y = -47
(-7, -2)

Write the pair of parametric equations as a single equation in x and y.
{x = 8t – 4
{y = 6t² – 2
y = 6((x+4)/8)² – 2

Solution
x = 8t – 4
x + 4 = 8t
Divide by 8 on both sides
t = (x + 4)/8
Substitute
y = 6((x+4)/8)² – 2

Mr. Reaich, the ticket agent for a small commuter airline, has developed a model to forecast the total monthly revenue from the airline’s passenger service. Under this model, the total revenue, R, is equal to \$368 times the number of first-class, F, plus \$324 times the number of coach passengers, C. Their flight schedule and overall aircraft capacity limit the total number of passengers carried during a single month to a maximum of 30,775. The number of coach class passengers is at least seven times the number of first-class passengers, and logically, the number of airline passengers in either class cannot be negative. Express the model in mathematical form.
\$368F + \$324C = R
F + C ≤ 30,775
C ≥ 7F
F ≥ 0; C ≥ 0

Solve the system of equations by substitution.

{3x + 4y = 32
{3x + y = 17

(4, 5)

Use a graph to solve the system of equations.

{x + y = -1
{y = 3x + 19

(-5, 4)

Graph and classify the system of equations as independent, inconsistent, or dependent. If the system is independent, find the solution from the graph.

{3x + 3y = 2
{3x – y = 8

Independent (2, -2)

Use elimination to solve the system of equations.

{2x – 5y = 1
{5x -2y = -4

(-22/21, -13/21)

Use elimination to solve the system of equations.

{-16x – 6y = 170
{8x + 3y = -85

Incorrect: no solutions

Correct: Infinitely Many Solutions

Use a graph to solve the system of equations.

{x + y = -4
{y = 3x – 8

(1, -5)

Which of the given ordered pairs are NOT solutions to the system of inequalities?

{2x + 3y < 1 {x -3y > 2

(3, 1), (-3, -2), (5, -5), (-2, 4), (-4, 5)

(-4, 5), (3, 1), (-2, 4)

Write the system of inequalities whose solution is graphed as the shaded region.
Incorrect:
y < 3x - 2 y > -2x + 5

Use mathway; Graph the intersection (and)

Use the graph of the feasible region for the set of constraints to find the maximum and minimum values of the objective function C = 3x + 5y.

Constraints:
{x + y ≥ 1
{4x – 2y ≤ 4
{3y ≤ 3x + 3

The maximum value of C is 29 at (3, 4) and the minimum value is 3 at (1, 0) .

Determine which ordered pair (x, y) is a solution of 3x – y ≤ 15.
(5, 4)

Solution
Substitute
3(5) – 4 ≤ 15
11 ≤ 15
True

Which inequality has (1, 7) as a solution?
Incorrect: -2x – 5y ≥ -32

Solution
Substitute
-2(1) – 5(7) ≥ -32
-2 – 35 ≥ -32
-37 ≥ -32
False, not a solution of the inequality

Use elimination to solve the system of equations.

{4x + 2y = -12
{3x – 2y = 5

(-1, -4)

An airplane, flying at an altitude of 3.21 miles above the ground, has a horizontal speed of 350 miles per hour and is descending at a rate of 14 miles per hour. Use parametric equations to find the airplane’s altitude above ground after it has traveled 11 miles, as measured along the ground.
2.77 mi

Solution
x(t)=350t
y(t)=3.21-14t
Solve either equation for “t”
x=350t
t=x/350
Substitute
y=3.21-14(x/350)
Substitute for “x”, x=11
y=3.21-14(11/350)
y=2.77

Use the set of constraints to find the maximum and minimum values of the objective function C = 5x + 11y

Constraints:
{x + y ≥ 2
{6x – 6y ≤ 12
{8y ≤ 4x + 16

Incorrect: The maximum value of C is 139 at (8, 9) and the minimum value is 10 at (0, 2).

Correct: The maximum value of C is 106 at (8,6) and the minimum value is 10 at (2,0).

Ms. Kim, an analyst at Multi-Fastener Corp, has developed a model to forecast the company’s total production of metal fasteners. Under this model, production, P, is equal to 4500 times the man-hours spent producing machine screws (S) plus 2900 times the man-hours spent producing nuts and bolts (P). Manpower constraints mean that total number of man-hours per year cannot exceed 44,000; logically, the number of man-hours assigned to either task cannot be negative. Express the model in mathematical form.
P = 4500S + 2900B
{S + B ≤ 44,000
{S ≥ 0
{B ≥ 0

Solve the system of equations by substitution.

{4y = 4
{x + 2y + z = 10
{y – 2z = -5

Incorrect: None of the listed answers are correct

Correct:
x=5
y=1
z=3

Write the pair of parametric equations as a single equation in x and y.

{x = 8t² -3
[y = 6t – 7

x = 8((y + 7)/6)² – 3

Find
[1 17 14][11 3]
[19 7]
[9 5]
[460 192]

Which system of equations can be classified as independent?
none of these

Write the system of equations as a matrix equation. Then solve the system, if possible, by using a
matrix equation. If not possible, classify the system.

{4x + y = -24
{2x + 2y = -18

(-5, -4)

You can just solve by using Elimination or Substitution

Find the determinant, and tell whether the matrix has an inverse.

det [10 -7]
[-5 -6 ]

-95; Yes

Write the augmented matrix for the system of equations.

{x – y = -9
{5x = 0

[1 -1| -9]
[5 0| 0]

Solution
x=1, y=-1
5x=5

The band and the cheerleading squad at a local school are ordering supplies. The supplies they need are listed in the table. (Evernote)
[10 13 5] [ 5 ]
[10 16 8] [12]
[ 2 ] =
[216]
[258]

Perform the indicated operations on the given matrices.

[3 -6 -7] [-2 -1 7]
[-3 -2 4] + [ 3 5 -7]
[-4 -1 7] [4 -9 -3]

[ 1 -7 0 ]
[ 0 3 -3 ]
[0 -10 4]

Write the augmented matrix for the system of equations.

{8x + 2y + 6z = -3
{3x – y + 10z = 3
{-9x -2y -4z = -7

[8 2 6| -3 ]
[3 -1 10| 3 ]
[-9 -2 -4 |-7 ]

The Student Government secretary, and two teacher’s aides are ordering supplies. The supplies they need are listed in the table. (Evernote)
(Evernote)

Kozinski played in 16 basketball games this season, Hussein played in 32 games, and Johnson played in 55 games. Kozinski averaged 6 points and 2 rebounds per game, Hussein averaged 8 points and 11 rebounds, and Johnson averaged 18 points and 16 rebounds. Multiply the following matrices to get the total number of points scored and the total number of rebounds made, by all three players combined.

[ 6 2 ]
[16 32 55] [ 8 11]
[18 16]

1342 points, 1264 rebounds

Solve: (Evernote)

det[x+3 2]
[3 x-2]
=0

Incorrect: x =-4

Correct: x = 0

Mathway: Find the Determinant

Classify the following system of equations as inconsistent, dependent, or independent:

{-3x -7y -2z = -25
{-6x +21y -6z = 57
{-3x +14y -4z = 40

independent, because there is an intersection

Find the reduced row-echelon form of the augmented matrix that represents the following system of equations: (Evernote)

{-6x + 9y = -5
{-8x – 4y = 0

Incorrect:
[1 0| 5/24 ]
[0 1|-5/12]

Correct:
[1 0| -5/24]
[0 1| 5/12

Perform the indicated operations on the given matrices. (Evernote)

[-8 -8] + [-2 3]
[-6 4] + [3 -1]

(2×2 matrice + 2×2 matrice)

[-10 -5]
[-3 3]

Find AC – CB (Evernote)
[-13 19]
[-12 -44]

Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system.

{3x + 2y + z = -2
{3x -2y -z = -22
{3x + 2y -z = -10

[3 2 1] [x] [ -2 ]
[3 -2 -1] [y] = [-22]
[3 2 -1] [z] [-10]
;(-4, 3, 4)

Find the inverse of the following matrix (if it exists):

[2 -13 -6]
[0 5 0]
[0 -24 0]

The matrix has no inverse.

Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system.

{x + 2y + z = -17
{3x + 7y + 2z = -56
{x – y + 2z = 0

x = -4, y = -6, z = -1

In May, Victor bought 24 styrofoam balls and decorated them as toy figurines. In June, he sold 17 figurines. In May, Wanda bought 28 styrofoam balls to decorate and in June, she sold 12 figurines. Which matrix represents all of their May purchases and their June sales?
May June
Victor [24 -17]
Wanda [28 -12]

Which matrix represents the graph of polygon PQRS? (Evernote)
P Q R S
[2 -3 -3 2]
[6 2 -7 3]

Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 4x² = 8
±√2; ±1.41

Use factoring and the Zero-Product Property to solve the quadratic equation

2x² + 5x -3 = 0

1/2, -3

(-b ± √((b² – 4ac))/2a

Find a quadratic function that fits the set of data points exactly, in the form y = ax² + bx + c with values of a, b, and c to two decimal places.

(1, 4.1), (5, 2.3), (11, 4.9)

y = 0.09x² – 0.98x + 4.99

Mathway:
Find the Equation of the Parabola

Which function is the correct match with the the given graph and vertex (Evernote)
f(x) = -3(x + 1)²

Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry.

y = -6x² – 24x – 31

Incorrect:
y = 6(x + 2)² – 7
x = -2; (2,-7)

Correct:
y = -6(x + 2)² – 7
x = -2; (-2, -7)

Mathway:
Find the Vertex Form
Find the Axis of Symmetry
Find the Vertex (Coordinates)

Use the quadratic formula to solve the equation.

6x² – 24x – 30 = 0

5, -1

Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 2(x + 4)² – 48 = 0
Incorrect: 4 ± 2√6; 5.8 or -13.9

Correct: -4 ± 2√6; 0.9 or -8.9

Simplify: (2 + 5i)/(5 + 4i)
(30/41) + (17/41)i

Solve the quadratic inequality, and graph the solution on a number line.

x² – 2x ≥ 24

x ≤ -4 or x ≥ 6

Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. y = x² + 14x + 4
Incorrect: None of the listed answers are correct

Correct:
y = (x+7)² – 45
x = -7; (-7, -45)

Which inequality is best represented by the graph shown: (Evernote)
y ≤ -x² + 5x

Use factoring and the Zero-Product Property to find the zeros of the quadratic function. f(x) = x² -7x – 8
8, -1

Which of the following is a quadratic function?
f(x) = -1 + 9x² + 7x

Find the value of x to the nearest hundredth.
17.2

Solve the quadratic inequality, and graph the solution on a number line. (Evernote)

x² – 4x + 3 < 0

1 < x < 3

Factor the quadratic expression. x² – 11x + 30
(x – 5) (x – 6)

Gupta threw a baseball off a cliff into an open field 50 feet below. The chart gives the horizontal distance, x (in feet), the baseball traveled from Gupta and the height, y (in feet), of the baseball above the field. (Evernote)

distance, x | 8 | 18 | 33 | 43 |
height, y | 65 | 76 | 79 | 71 |

Choose the quadratic equation that best fits the baseball’s trajectory from Gupta to the open field below.

y = -0.0383x² + 2.143x + 50

Show that the function is a quadratic function by writing it in the form f(x) = ax² + bx + c and indentifying a, b and c.

f(x) = (5x – 4) (5x + 3)

f(x) = 25x² – 5x – 12
a = 25, b =5, c = -12

The turnstiles at the entrance to the State Fair kept track of the number of people entering the fairgrounds, for the first seven hours following the openning of the fair. Find a quadratic equation that models the data shown. (Evernote)
0.083x² – 0.197x + 7.62

Graphing Calculator
STAT→Edit
Put x values into L1 and y values into L2.
2ND→MODE→STAT→CALC

Solve the equation by completing the square. Give exact solutions. x² – 2x – 24 = 0
-4, 6

Solve the equation for x. Write the exact solution and the approximate solution to the nearest hundredth, when appropriate.

In(8x – 6) = 3

x = (3/4) + (e³/8)
x ≈ 3.26

Exponential form
e³= 8x – 6

Solve the equation for x: 10^x = 18
1.26

Evaluate 8^log₈ 9
9

Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: In 10
2.303

Solve log₇ 8 – log₇ (x + 3) = log₇ 3, for x
Incorrect: None of the listed answers are correct

Correct: x=-1/3

Solution
log(x) – log(y) = log(x/y)
log₇ 8 – log₇ (x+3) = log₇ (8/(x+3))
log₇ (8/(x+3)) = log₇ 3
(Mathway can solve once the left terms are combined)

If a principal of \$1250 is invested at an annual interest rate of 4% compounded annually, what is the account balance at the end of 6 years?
\$1581.648

Solve the equation 3^4x = 27^x+3
x = 9

The magnitude of an earthquake is found by the equation M = 2/3 log E/10^11.8 , where M is the magnitude and E is the energy released. Find the magnitude of an earthquake that released 10^24.4 ergs of energy.
Incorrect: None of the listed answers are correct

Correct: M = 8.4

Solution
M = 2/3 log₁₀ (10^24.4/10^11.8)

The formula for estimating the number, N, of a certain product sold is N = 7400 In(7t + 3), where t is the number of years after the product is introduced. What is the expected number of sales 7 years after the product is introduced? Round to the nearest whole number.
29,239

Solve the equation. Round your answers to the nearest hundredth. 5^x-1 = 25
3

Find the final amount of the investment: \$3000 at 8% interest compounded quarterly for 4 years.
\$4118.36

Describe the value of b for the function y(x) = 1/3(1/b)^x to represent exponential growth.
Incorrect: None of the listed answers are correct

Correct: “b” has to be greater than 0 and less than 1. (0 < b < 1)

Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: e^¼
1.284

Solve log₃(x+4) – log₃(x-4) = log₃5 for x
6

The inflation rate of the U.S. dollar is 3.1 percent. What this means is that every year, prices increase by 3.1 percent. If a pound of meat cost \$2.37 nine years ago, what does it cost now?
\$3.12

Solution
2.37(1+0.031)⁹

Erosion gradually reduces the size of a small Pacific island that has a current area of just 460 acres. If the island’s area decreases at an annual rate of 0.05%. Find the multiplier for the rate of exponential decay.
Incorrect: 0.95

Correct: 0.9995

Solution
100 – 0.05 = 99.95
99.95/100 = 0.9995

Write the equation in logarithmic form.

6^-4 = 1/1296

log₆ 1/1296 = -4

Which function represents exponential decay?
y(x) = 5(0.44)^x

Evaluate the logarithmic expression to the nearest thousandth.

log₈ 1/2

-0.333

If \$1500 is invested at an interest rate of 10%, compounded continuously, determine the balance in the account after 2 years. Use the formula A = Pe^rt.
\$1832.10

Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary.

f(x) = x⁴ – 9x³ + 10x² + 90x – 200

4, 5, ±3.16

Divide x³ + 5x² + 9x + 9 ÷ x + 3
x² + 2x + 3

Use a graph, synthetic division, substitution, and factoring to solve the equation.

x³ – 7x² + 14x – 8 = 0

2, 1, 4

Find all the zeros of the polynomial function.

f(x) = x² – 10x + 34

Incorrect: 5 ± 6i

Correct: 5 ± 3i

The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1987. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 1998. (Evernote)
T(x) = 0.2x³ + 0.3x² + 0.2x + 0.1; 304.8 thousand

Use a graph, synthetic division, substitution, and factoring to solve the equation.

x³ – x² – 14x + 24 = 0

-4, 2, 3

Find all the zeros of the polynomial function.

f(x) = x³ + 4x² + 6x + 9

-3, (-1 ±i√11)/2

Write the product as a polynomial in standard form.

(5x² – 2x + 5)(x² – 8x – 2)

5x⁴ – 42x³ + 11x² – 36x – 10

Write a polynomial equation in standard form by using the given information.

P is of degree 3; P(0) = 24; zeros: 3, -4, 2

P(x) = x³ – x² – 14x + 24

Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary.

f(x) = x³ – 4x² + 7x – 12

None of the listed answers are correct

Simplify 3x² – 4x – 5 + 4x² + 8x – 7
7x² + 4x – 12

Classify the polynomial by degree and number of terms. Describe the shape of its graph. 3x³
cubic monomial; ‘S’ shaped with 2 turns

Evaluate 2x² + 3x + 4 when x=2
18

For the function, use synthetic division and substitution to determine whether the given value is a zero of the function.

P(x) = 3x⁴ – 10x³ – 41x² + 68x + 60

Zeroes are 5, 2, -2/3, -3

Evaluate x⁴ – 10x² + 24 when x=3
15

Write the product as a polynomial in standard form.

(x – 6)(x + 5)(x – 4)

x³ – 5x² – 26x + 120

Use substitution to determine which of the given linear expressions is a not a factor of 3x⁴ – 16x³ – 15x² + 88x + 60
x + 5

Mathway (Algebra)
Factor

Determine the end behavior of the graph of the function

f(x) = 3x⁴ – 2x³ + 2x + 2

rises to the left; rises to the right

Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary.

f(x) = x⁴ – x³ – 16x² + 10x + 60

3, -2, ±3.16

Classify the polynomial by degree and number of terms. Describe the shape of its graph. 7x² + 7x + 7

Determine whether the function is a rational function. If so, find the domain and identify the horizontal and vertical asymptotes, and any holes in the graph. If the function is not rational, state why not.

f(x) = (x – 1)/((x – 7)(x + 3))

Incorrect:
rational; x≠1, 7 or -3; asymptotes at y=0, x=1, x=7 and x=-3

Correct:
rational; x≠-3,7
asymptotes at y=0, x=-3, and x=7

Simplify the rational expression. (Evernote)

( 3/(x² + 7x + 12) + 1/(x² + 11x + 28) ) ÷ ( 2/(x² + 10x + 21) + 1/x² + 13x + 36) )

( 4(x + 9)(x + 6) ) / ( 3(x² + 12x + 31) )

Simplify.

( (x – 2)/(2x²) ) – ( (2x + 1)/(9x) ) + ( (7x)/(12) )

( 21x³ – 8x² + 14x – 36 ) / 36x²

³√(3x-6) + 6 ≥ 7

x ≥ 7/3

Simplify the rational expression.

( (x² + 14x + 49)/8x ) ÷ ( (x+7)/4x )

(x+7)/2

-256/7

Rationalize the denominator.

2 ÷ √5

(2√5)/5

Rationalize the denominator.

(√11) ÷ (9 + √5)

( 9√11 – √55) / 76

Simplify.

(-3x + 8)/(x² – 49) – (-2x + 1)/(x² – 49)

-1/(x+7)

Use the information to write the appropriate variation equation, and find y for the given values. y varies jointly as x and the inverse of z.

y=-99.4 when x=-11 and z=4. Find y when x=2 and z=-8.

y = 9x/z; -9/4

Simplify the sum, difference, product, or quotient. Assume that the value of any variable is positive

³√(-27x) – 4 ³√(x⁴) + 9 ³√(x) + 9x ³√(x)

6 ³√(x) + 5x ³√(x)

72

Solve the equation or inequality.

4/(x-3) – 2/(x+4) = 0

x=-11

Find the domain of the radical function. f(x) – √(9-x)
x ≤ 9

Solution
f(x) – √(9-x) = 0
f(x) = √(9-x)

Mathway: Find the Domain

Use the information to write the appropriate variation equation, and find y for the given values.

y varies jointly as x and z. y=98/3 when x=2 and z=7. Find y when x=8 and z=3

y = 7/3(x)(z); 56

Designer Dolls, Inc. found that the number of dolls sold, N, varies directly as their advertising budget, A, and inversely as the price of each doll, P. Designer Dolls, Inc. sold 8400 dolls when \$84,000 was spent on advertising and the price of a doll was set at \$70. Determine the number of dolls sold when the amount spent on advertising is increased to \$98,000.
9800

Write an equation that can be used to solve the problem. Then answer the question asked.

A group of college students are volunteering for Habitat for Humanity during their spring break. They are putting the finishing touches on a house they built. Working alone, Dale Horton can paint a certain room in 3 hours. Kathy Garcia can paint the same room in 9 hours. How long will it take them working together to paint the room?

x/3 + x/9 = 1; 2.25 hr

The wattage rating of an appliance is given as “W”, in watts, and varies jointly as the resistance, “R”, in ohms, and as the square of the current, “I”, in amperes. If the wattage is 2 watts when the resistance is 200 ohms and the current is 0.1 amperes, find the wattage when the resistance is 150 ohms and the current is 0.4 amperes.
Incorrect: None of the listed answers are correct

Correct: W (wattage) = 24

Solve the equation or inequality.

(2x-2)/(x²-36) ≥ 1/(x+6)

-6 ≤ x ≤ -4 or x ≥ 6

√(x + 14) ≤ x – 16

x ≥ 22

Find the standard equation for the hyperbola with the given characteristics.

vertices: (2, 0) and (-2, 0)
asymptote: y=1/2x

x²/4 – y² = 1

Mathway: Find the Vertex Form, till the verices and asymptote matches up

Which equation describes a parabola?
6y² + 11x – 13y = -1

Find the center and the radius of the circle that has a diameter with the given endpoints.

Diameter CD, C(-1, -4), D(5, 4)

center: (2, 0)

Find the standard equation for the ellipse, using the given characteristics taken from the graph. (Evernote)
(x+4)²/16 + (y+3)²/9 = 1

Write the equation in standard form and classify the conic section it defines.

2x² + 2y² – 28x + 24y + 162 = 0

(x-7)² + (y+6)² = 4; circle

Find the center and the radius of the circle that has a diameter with the given endpoints.

Diameter CD, with endpoints C(- 1, 5), D(5, 9)

center: (2, 7)

Mathway: Find the Midpoint

Find the standard equation of a circle with the given radius and center.

center: (1, 1)

(x – 1)² + (y – 1)² = 49

Solve the nonlinear system of equations.

{x² + y² = 144
{x² – 4y² = 64

four solutions: (±8√2, ±4)

Which set of characteristics matches the given standard equation and parabola graph? (Evernote)

x = 1/24 y²

vertex: (0, 0)
directrix: x=-6

Solve the nonlinear system of equations.

{x² + y² = 256
{x + y = 16

two solutions: (0, 16) and (16, 0)

Which set of characteristics matches the given parabola graph and standard equation? (Evernote)

x = -1/4 y²

axis of symmetry: y=0
focus: (-1, 0)
vertex: (0, 0)

Mathway: Graph

Find the standard equation for the hyperbola with the given characteristics.

center: (5, 4)
one focus: (-25, 4)
one vertex: (23, 4)

(x-5)²/324 – (y-4)²/576 = 1

The two hyperbolas in the graph are conjugates, meaning they share the same asymptotes. Given the following equation of the hyperbola represented by the thin curve, find the equation of its conjugate, which is represented by the thick curve. (Evernote)

x²/9 – y²/25 = 1

y²/25 – x²/9 = 1

Identify the conic section produced by the following diagram. (Evernote)
hyperbola

Find the standard equation for the hyperbola with the given characteristics.

vertices: (2, 0) and (−2, 0)
asymptote: y = 3x

x²/4 – y²/36 = 1

Which set of characteristics matches the given standard equation and parabola graph? (Evernote)
focus: (-1, -4)
vertex: (-1, 2)

Identify the conic section produced by the intersection of the plane and the cones in the following diagram.
circle

Find the standard equation of a circle with the given radius and center.

center: (3, – 5)

(x – 3)² + (y + 5)² = 16

Mathway: Find the Equation (put coordinate points comma r=4)

Find the distance between points B(5, 7) and F(1, 2), and the coordinates of the midpoint of BF.
distance = √41; 6.4
midpoint = (3, 9/2)

Mathway

Find the standard equation for the ellipse, using the given characteristics.

vertices: (0, ±8)
foci: (0, ±√55)

x²/9 + y²/64 = 1

Find: ₇P₄
840

A hat contains 22 names, 10 of which are female. If five names are randomly drawn from the hat, what is the probability that at least four female names are drawn?
0.105

((₁₀C₄ × ₁₂C₁)+(₁₀C₅ × ₁₂C₀))÷(₂₂C₅)

((210 × 12)+(252 × 1))÷(26334)

2772÷26334 = 0.105

Teesha is in the bowling club. There are 33 students in the club. Five of them will be picked at random to attend an awards banquet. What is the probability that Teesha will not be randomly chosen to attend the banquet?
28/33

Solution
Probability of getting chosen is 5/33, so the the probability of not getting chosen is 33 – 5 = 28

How many different arrangements can be made using all of the letters in the word GAME?
24

Solution
₄P₄

A spinner is evenly divided into 8 equal areas and numbered from 1 through 8. What is the probability of spinning a number less than 4 in a single spin?
3/8

A lunch menu consists of 4 different kinds of sandwiches, 2 different kinds of soup, and 5 different drinks. How many choices are there for ordering a sandwich, a bowl of soup, and a drink?
40

4 × 2 × 5 = 40

Two cards are randomly drawn in succession from a deck of 52 playing cards. Find the probability that the king of spades and any ace are drawn, in that order, without replacement.
1/663 or 0.0015082

Solution
1/52 × 4/51 = 0.0015082

Evaluate: (₁₂C₇ × ₁₄C₁₀)/₁₃C₅
616

Solution
₁₂C₇ = 792
₁₄C₁₀ = 1001
₁₃C₅ = 1287

(792 × 1001)/1287 = 616

If 2 blocks are randomly taken from a bag containing 10 blue blocks, 10 red blocks, and 7 yellow blocks, what is the probability of drawing a blue block and a red block?
50/351 or 0.1424501

Solution
10/27 × 10/26 = 0.1424501

Determine the probability that you will roll a number greater than 1 on a number cube.
5/6

Evaluate: ₇C₄
35

A circular, rotating, serving tray has 5 different desserts placed around its circumference. How many different ways can all of the desserts be arranged on the circular tray?
24

Solution
(n-1)!
(5-1)!
4! = 4 × 3 × 2 × 1 = 24

How many different ways can 6 people be seated around a circular table?
120

Solution
(n-1)!
(6-1)!
5! = 5 × 4 × 3 × 2 × 1 = 120

Three cards are drawn in succession and without replacement from a standard deck of 52 cards. How many sets of three cards are possible?
22,100

Solution
52 choose 3
₅₂C₃ = 22,100

How many distinct committees of 14 people can be formed if the people are drawn from a pool of 23 people? Use factorials to express the answer.
₂₃C₁₄ = 23!/(9! 14!)

Solution
nCr = n!/(r!(n-r)!)
23!/(14!(23-14)!)
23!/(14!(9)!)
23!/ (9! 14!)

Jamila spins a spinner with 3 sections of equal area, like the one below, 25 times. It lands on the 1eight times. What is the experimental probability of spinning a 1?
8/25

In a bag there are 3 green jelly beans, 2 black jelly beans, and 8 yellow jelly beans. Once a jelly bean is drawn, it is not replaced. Find the probability of randomly drawing a green jelly bean and then a black jelly bean in two consecutive draws.
1/26

Suppose you mix-up the cards below and choose one without looking. What is the probability of selecting neither “E” nor “G”?

H E H E G W I

4/7

Solution
Probability of selecting “E” or “G” is 3/7.
7/7 – 3/7 = 4/7

Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 diamonds or 2 face cards?
Incorrect: 0.108

Correct: 0.106 or 47/442

Solution
(13/52 × 12/51) + (12/52 × 11/51) – (3/52 × 2/51)

A box contains 2 green, 6 yellow, and 3 purple balls. Find the probability of obtaining a purple ball in a single random draw.
3/11

Find the four arithmetic means between -3 and 102.
18, 39, 60, 81

Solution
t(n) = t(1) + (n – 1)d
102 = -3 + (6 -1)d
102 = -3 + 5d
105 = 5d
d = 21
Substitute
t(2) = -3 + (2 – 1)21
-3 + (1)21
-3 + 21
18
t(3) = -3 + (3 – 1)21
-3 + (2)21
-3 + 42
39
And so on

Use Pascal’s Triangle to determine the probability that you will get three green lights in a row of five lights. Assume red and green are equally likely occurrences.
Incorrect: 7/32

Correct: 5/16 (10/32)

Solution
(n C k)(p∧k)(1 – p)∧(n-k)
(₅C₃)(½)³(1 – ½)⁵⁻³
(₅C₃)(½)³(1 – ½)²
10 × 1/8 × 1/4 = 5/16

Write the first five terms of the sequence defined by the given recursive or explicit formula. (Evernote)
4, -3, 31, -167, 959

Expand the binomial raised to a power. (3a – b)⁵
243a⁵ – 405a⁴b + 270a³b² – 90a²b³ + 15ab⁴ – b⁵

Mathway: Expand Using the Binomial Theorem

Express 0.7312 as a geometric series, and write its sum as the ratio of two integers.
(Evernote)

Evaluate the sum. (Evernote)
-3003

Find the sum of the geometric series 0.2 + 0.02 + 0.002 + . . . given the formula S=a/(1 – r), where “a” is the first term, “r” is the common ratio, and “S” is the sum.
2/9

Solution
S = 0.2/(1 – 0.1)
S = 0.222… or 2/9

Find the 6th and 8th entries in the row 12 of Pascal’s triangle.
792;792

Solution
Use nCk-₁ to love for nth entry

₁₂C₆-₁ = ₁₂C₅ = 792
₁₂C₈-₁ = ₁₂C₇ = 792

Use Pascal’s triangle to find the number of ways to choose 3 boxes from 7 boxes.
Incorrect: 21

Correct: 35

Solution
₇C₃ = 35

Find the three positive geometric means between 6 and 1875/8.
15, 75/2, 375/4

Solution
t₁=6 and t₅=1875/8
t(n) = t₁r^(n-1)
t₅ = 6r⁴
1875/8 = 6r⁴
r = 2.5

t₂ = 6(2.5)¹
15
t₃ = 6(2.5)²
37.5 or 75/2
t₄ = 6(2.5)³
9375 or 375/4

A 50-row theater has 30 seats in the front row. The second row has 31 seats. If each row has one more than the row in front of it, how many seats are there in the theater?
2725

Solution
t₁ = 30 and d = 1
t(n) = t₁ + (n-1)d
t₅₀ = 30 + (49)1
t₅₀ = 79

S = n/2[2t₁ + (n – 1)d]
S = 50/2[2(30) + 49]
S = 25(60 + 49)
S = 25(109)
S = 2725

Find the sum of the infinite geometric series, if it exists.

3 – 9/5 + 27/25 – 81/125 + 243/625 – …

15/8

Solution
(9/5) ÷ 3 = 0.6
r = 0.6 (common ration)
t₁ = 3
S = t₁ ÷ (1 – r)
S = 3 ÷ (1 – 0.6)
S = 3 ÷ 0.4
S = 7.5 or 15/8

Use the given formula to find the first four terms of the arithmetic sequence.

t(n) = 12 – 3n

9, 6, 3, 0

Solution
t(1) = 12 – 3(1)
9
t(2) = 12 – 3(2)
6

There are 8 cars in a parking lot on a very cold day. Suppose the probability of any one of them not starting is 0.14. What is the probability that exactly 3 of the cars will not start?
Incorrect: ≈ 0.003

Correct: ≈ 0.072

Solution
(nCk) × (p)^n × (1 – p)^(n – k)
(₈C₃) × (0.14)³ × (1 – 0.14)⁵
= 0.072

Write an explicit formula for the nth term of the geometric sequence.

7/4, 49/24, 343/144, 2401/864, …

(Evernote)

Solution
explicit formula for nth term in a geometric sequence is
t(n) = t₁r^(n-1)
t₁ = 7/4 and r = 7/6
t(n) = 7/4(7/6)^(n-1)

Find the sum of the infinite geometric series, if it exists.
(Evernote)

∑3(-½)^(k-1)

2

Solution
∑3(-½)^(k-1)

t₁ = 3(-½)⁰
3
t₂ = 3(-½)¹
-1.5

r = t₂/t₁
r = -1.5 ÷ 3
r = -½

S = t₁/(1 – r)
S = 3/(1 – (-½))
S = 3/1.5
S = 2

Find the sum of the first 7 terms of the geometric series 8 + 8/3 + 8/9 + 8/27 + …
11.99

Solution
r ≠ 1
r = 8/3 ÷ 8
r = 1/3 & t₁ = 8

S(n) = t₁[(1 – r^n) ÷ (1 – r)]
S₇ = 8[(1 – (1/3)⁷) ÷ (1 – 1/3)]
S₇ = 11.99

Use Pascal’s triangle to solve for the value of n: 2(n C 3) = n+4 C ₇ (Evernote)
Incorrect: 2

Correct: 4

Make a conjecture about the pattern of the given data. Find the sum of the 7th row.

3
3 + 6 + 3
3 + 6 + 9 + 6 + 3
3 + 6 + . . . + ( 3n − 3) + ( 3n ) + ( 3n − 3) + . . . + 3

Incorrect: The sums of the rows are 3, 12, 27, . . . . The sum of the nth row is n(2n +3). The sum of the 7th row is 119.

Correct: The sum of the rows are 3, 12, 27, . . . . The sum of the nth row is 3(n²). The sum of the 7th row is 147.

Find the two geometric means between 7 and 3584.
Incorrect: 63, 567

Correct: 56, 448

Solution
t₁=7 and t₄=3584
Formula
t(n) = t₁r^(n – 1)
Substitute
t(4) = 7r⁴⁻¹
3584 = 7r³
divide both sides by 7
r³ = 512
Take cube root of both sides
r = 8
t(2)=7 × 8
56
t(3)=7 × 8²
448
7, 56, 448, 3584

Find the mean, the median, and mode of the data set: 14, 6, 21, 16, 16, 7, 20, 19, 7
Mean: 14
Median: 16
Mode: 7, 16

Using the given information, which data set is the correct choice? (Evernote)
20, 12, 27, 14, 16, 13, 12, 13, 26, 27, 21, 28, 20, 22, 20

Find the minimum and maximum values, quartiles, range, and interquartile range for the data set.

29, 14, 40, 12, 16, 46, 22, 23, 19, 31, 7, 9

Minimum = 7
Maximum = 46
Q₁=13; Q₂=20.5; Q₃=30
Range = 39
IQR = 17

Q₂ = mean

Last year, the personal best high jumps of track athletes in a nearby state were normally distributed with a mean of 205 cm and a standard deviation of 13 cm. What is the probability that a randomly selected high jumper has a personal best between 218 and 244 cm? (Evernote)
0.1574

Solution
(Graphing calculator)

Between, mean, then standard deviation

The bowling scores of all of the bowlers in 12 bowling leagues are normally distributed. Their mean score is 201 points, with a standard deviation of 33 points. If one league has 120 bowlers, how many of them are likely to score more than 168 points?
101

Solution
(Graphing calculator)
2nd→VARS→DRAW→1:ShadeNorm(168, 10⁹⁹, 201, 33)→Enter = 0.84

.84 or 84%
84% of 120 is 100.8
rounded to the nearest whole number equals 101

Find the variance and the standard deviation for the following data.

7, 20, 7, 15, 21, 4, 22, 17, 13

variance ≈ 39.78; standard deviation ≈ 6.31

https://www.easycalculation.com/statistics/standard-deviation.php

Take the Variance (Population Standard deviation) and Population Standard deviation

Find the minimum and maximum values, quartiles, range, and interquartile range for the data set.

27, 8, 9, 48, 19, 10, 29, 42, 2, 50, 44

Minimum = 2
Maximum = 50
Q₁=9; Q₂=27; Q₃=44
Range = 48
IQR = 35

Mathway
Find the Five Number Summary, range, and Find the Interquartile Range (H-Spread)

Using the given stem and leaf plot and and it’s median and mode, select the data set on which it was created. (Evernote)
7, 6.9, 8.2, 7.6, 6, 7.2, 8.4, 7.6, 7.3, 7.8, 6.2, 9.5, 8.9, 6.5, 9

An English class was taken by 28 students. Their final scores were: three As, five Bs, nine Cs, six Ds, and five Fs. Using the given frequency table, find the mean grade point score, using A = 4, for the entire group of 28 students. Give the answer to the nearest hundredth, if necessary. (Evernote)
Incorrect: [none]

Correct: 1.82 (Close to a “C”)

NCUse the frequency table below, with the a total of 18 catalog items sold at four different prices, to find the average price per item. (Evernote)
\$16.50

The relative frequency histogram below represents the age in years of the first 100 children to have their portraits taken at the “See What Develops” photography studio. What is the probability that the next child to have portraits taken will be between 1 and 2 years old?
Incorrect: None of the listed answers are correct

Correct: 31/100 or 31% (maybe 0.315 or 32%)

Solution
.11 + .20 + .31 + 30 + .15 + 24 = 1 (or 100%)

Age 1= .20
Age 2= .11
.20 + .11 = .31 (or 31%)

31/100 or 31% Probability

A number cube is tossed 10 times with the following results.
5, 6, 1, 5, 5, 2, 6, 1, 2, 2
Find the range and the mean deviation of the data.
None of the listed answers are correct

To find mean deviation minus the mean from each data value, add everything up, then divide by the number of data values.

Example:
54, 49, 47, 48, 52
mean of data = 50

54 – 50 = | 4 | = 4
49 – 50 = |-1| = 1
47 – 50 = |-3| = 3
48 – 50 = |-2| = 2
52 – 50 = | 2 | = 2

4 + 1 + 3 + 2 + 2 = 12
12 ÷ 5 = 2.4
Mean deviation = 2.4

A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of the purchases are returned. What is the probability that no more than 1 of the next 6 purchases will be returned?
0.954

Solution
((₆C₁)(0.06)¹(1 – 0.06)⁵) + ((₆C₀)(0.06)⁰(1 – 0.06)⁶)
= 0.954

In Sean’s school there are 96 families which have 4 children. The circle graph shows the probability of each combination of girls and boys in a family with four children. Use the circle graph to predict the probability that one of these 96 families, chosen at random, will only have children of the same sex. (Evernote)
12.5%

A biologist is collecting insects in a field. Beetles represent 67 percent of all the insects that have been collected so far. What is the probability that exactly half of the next 6 insects collected will be beetles?
0.21617

Solution
(₆C₃)(0.67)³(1-0.67)³
= 0.21617

The depth of snow at seven different mountain lodges is 13 in., 15 in., 21 in., 17 in., 90 in., 13 in., and 19 in. Find the mean, median, and mode. Tell which measure is the most useful for predicting how deep the snow will be at an eighth lodge.
Mean = 26.9 in
Median = 17 in
Mode = 13
The median is the most useful.

The personal savings of the Young Saver Club were normally distributed with a mean of \$525 and a standard deviation of \$64. What is the probability that a randomly selected saver has an account total between \$525 and \$589?
0.3413

Solution
(Graphing calculator)
2nd→VARS→DRAW→1:ShadeNorm(525, 589, 525, 64)→Enter = 0.3413

A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of all purchases are returned. What is the probability that no more than 1 of the next 7 purchases will be returned?
≈ 0.938

((₇C₁)(0.06)¹(1 – 0.06)⁶) + ((₇C₀)(0.06)⁰(1 – 0.06)⁷)
≈ 0.938

The class average on a math test was 77.5 and the standard deviation was 5.8. Find the z-score for a test score of 83, and the percentage of the class who scored below 83.
≈ 0.95; 82.89%

Solution
z-score
z = (x – x̅) ÷ σ
z = (83 – 77.5) ÷ 5.8
= 0.948 → 0.95

(Graphing calculator)
2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 83, 77.5, 5.8)→Enter = 0.83 or 83%

In a certain normal distribution of scores, the mean is 30 and the standard deviation is 5.5. Find the z-score corresponding to a score of 37 and find the percentage of the scores that are below 37.
1.27; 89.80%

Solution
z-score
z = (x – x̅) ÷ σ
z = (37 – 30) ÷ 5.5
= 1.27

(Graphing calculator)
2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 37, 30, 5.5)→Enter = 0.8984 or 89.80%

Refer to ∆ ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. Find tan x (Evernote)
3/4 or 0.75

Solution
24/32 = 0.75

Evaluate the trigonometric expression. Sin⁻¹(-½)
-30°

Solution
π = 180°
-π/6
-180°/6 = 30°

Mathway
Evaluate (Do not use Find the Exact Value)

Refer to ∆ ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. (Evernote)
8/17 or 0.4706

sinθ = opp./hyp.

Use the following information to find the unknown sides and angles. m∠B = 28°; c=18
b = 8.5; a = 15.9; m∠A = 62°; m∠C = 90°

Solution
We have two angles
m∠A + m∠B + m∠C = 180
m∠A + 28° + 90° = 180°
m∠A = 62°

Use Law of Sines to solve for side “a” or “b’

a/sin A = c/sin C
a/sin(62°) = 18/sin(90°)
a = 18sin(62°)/sin(90°)
a = 15.89 → 15.9

b/sin B = a/sin A
b/sin(28°) = 15.9/sin(62°)
b = 15.9sin(28°)/sin(62°)
b = 8.45 → 8.5

What is the reference angle for 516° ?
24°

Mathway (Trigonometry)
Find the Reference Angle

Find tan 421°.
1.804

Mathway (Trigonometry)
Find the Exact Value

For ∆ ABC, find the measure of ∠A to the nearest degree. (Evernote)
19°

Solution
Use Law of Cosines for “c”
c = √(a² + b² – 2ab cos C)
c = 37

a = 12, b = 35, and c = 37

Use Law of Sines
sin A/a = sin C/c
sin A/12 = sin(90°)/37
sin A = 12sin(90°)/37
sin A ≈ 12/37

Convert from radians to degrees ∠A = 18.58° → 19°

Evaluate the trigonometric expression. cos(Cot⁻¹1)
√2/2

Mathway (Trigonometry)
Find the Exact Value

Given the quadrant of θ in standard position and a trigonometric function value of θ, find exact values for the remaining functions.

cosθ = -3√5/7
tanθ = -2√5/15
cotθ = -3√5/2
secθ = -7√5/15
cscθ = 7/2

Mathway (Trigonometry)
Find the Other Trig Values in Quadrant II

Find tan 382°.
0.404

Mathway (Trigonometry)
Find the Exact Value

(31/20)π

Mathway (Trigonometry)

Find the reference angle for 525°.
15°

Mathway (Trigonometry)
Find the Reference Angle

Point P is located at the intersection of a circle with a radius of r and the terminal side of an angle θ. Find the coordinates of P to the nearest hundredth. θ = 120° , r = 13
P(- 6.5, 11.26)

Solution
Coordinates:
(cos θ = x/r, sin θ = y/r)

θ = 120° and r = 13

cos(120°) = x/13
-½ = x/13
x = -13/2 or -6.5

sin(120°) = y/13
√3/2 = y/13
y = 13√3/2 or 11.26

P(-6.5, 11.26)

The function d = – 6 cos 3t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the frequency?
Incorrect: 3π/2 cycles/seconds

Correct: 3/2π

Solution
d = A cos wt
d = -6 cos 3t
w = 3

f (frequency) = w/2π
f = 3/2π

For a circle of radius 4 feet, find the arc length “s” subtended by a central angle of π/30 radians. Round to the nearest hundredth.
None of the listed answers are correct

Solution
r = 4 and θ = π/30
s = rθ
s = 4(π/30)
s = 2π/15

The function d = 9 cos 2t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the maximum displacement of the object from its resting position?
9 m

Solution
A = displacement
d = A cos wt
d = 9 cos 2t
A = 9 or 9m

Find cot (-290°).
0.364

Mathway (Trigonometry)
Find the Exact Value

For a circle of radius 8 feet, find the arc length of a central angle of 30°.
(4/3)π feet

Solution
r = 8 and θ = π/6 (change 30° to radians)

s = rθ
s = 8(π/6)
s = (4/3)π or 4π/3

Find the amplitude, the period, and the frequency of the graph. Then write an equation for the sine function for the graph.
3/4, 180°, 2, y = 3/4 sin 2x

Graph all multiple choices and pick the one that fits

Find the exact value of the sine, cosine, and tangent of -210°.
sin = 1/2
cos = -√3/2
tan = -√3/3

Mathway (Trigonometry)
Find the Exact Value

Suppose the depth of the tide in a certain harbor can be modeled by y = 25 – 4 cos(π/6t), where “y” is the water depth in feet and “t” is the time in hours. Consider a day in which t = 0 represents 12:00 midnight. For that day, when are high tide and low tide and what is the depth of each?
Incorrect: high tide at 12:00 noon and 12:00 midnight, depth 29 ft
low tide at 6:00 a.m. and 6:00 p.m., depth 21 ft

Correct: high tide at 12:00 noon and 12:00 midnight, depth 21 ft
low tide at 6:00 a.m. and 6:00 p.m., depth 29 ft

Use either the Law of Sines or the Law of Cosines to solve for “a” to the nearest tenth. (Evernote)
28.8

Solution
∠A + 61° + 76° = 180°
∠A = 43°

Use Law of Sines
a/sin(43°) = 41/sin(76°)
a = 41sin(43°)/sin(76°)
a = 28.8

Find cos((1/2)A) and sin((1/2)A) if sin(A)=4/5 and Ais a first-quadrant angle. (Evernote)
Incorrect:
cos(A/2) = √3/5
sin(A/2) = 2/5

Correct:
cos(A/2) = (2√5)/5
sin(A/2) = √5/5

Solution
Pathagorean Identities
sin A = 4/5
cos²A = 1 – sin²A
cos A = 1 – (4/5)²
cos A = 3/5

Use Half-Angle Identities
sin(A/2) = √((1 – cos A)/2)
sin(A/2) = √((1 – 3/5)/2)
sin(A/2) = √5/5

cos(A/2) = √((1 + cos A)/2)
cos(A/2) = √((1 + 3/5)/2)
cos(A/2) = (2√5)/5

A = θ and the cos((1/2)A) is the same as cos(A/2)

Find all solutions of cos(x) – √(1 – 3cos²(x)) = 0
Incorrect: None of the listed answers are correct

Correct: No Solution

Use the Law of Sines to solve for m∠A to the nearest tenth. (Evernote)
29.8

Solution
sin B/b = sin C/c
sin B/43 = sin(80°)/45
sin B = 43sin(80°)/45
sin B = 0.94103

sin⁻¹(sin B) = ∠B
sin⁻¹((43sin¹(80°)/45))
sin⁻¹(0.94103) = 70.2° (rounded to the nearest tenth)
∠B = 70.2°

∠A + 70.2° + 80° = 180°
∠A = 29.8°

Use graphing calculator on degree mode or use Mathway and change answer to degrees.

Write the expression csc(2x) + cot(2x) in terms of tan x or cot x.
Incorrect: None of the listed answers are correct

Correct: cot(x)

Use the Law of Cosines to solve for “A” to the nearest tenth of a degree. (Evernote)
35.3°

Law of Cosines
cos A = (b² + c² – a²)/(2bc)
cos A = (43² + 50² – 29²)/(2 × 43 × 50)
cos A = 0.8158

cos⁻¹(0.8158) = 35.3°

Just do this ↓
∠A = cos⁻¹((b² + c² – a²)/(2 × 43 × 50))
∠A = 35.3°

Find the exact value of sin(- 450° + 300°).
-1/2

Solution
Use Sum and Diffrence Identities:
sin(A + B) = sinAcosB + cosAsinB
sin(450°)cos(300°) + cos(450°)sin(300°) = -1/2

Mathway (Trigonometry)
Find the Exact Value

Solve sin(x) – √(1 – 3sin²(x)) = 0 given that 0° ≤ x ≤ 360°.
Incorrect: 90°, 270°

Incorrect: None of the listed answers are correct

Correct: 30°, 150°, 210°, 330°

Solution
sin(x) = √(1 – 3sin²(x))
sin²(x) = 1 – 3sin²(x)
sin²(x) + 3sin²(x) – 1
4sin²(x) – 1
(2sin(x) + 1)(2sin(x) – 1)
sin(x) = -1/2 and 1/2

x = -30°, 30°, 150°, 210°
-30° + 360° = 330°
x = 30°, 150°, 210°, 330°

Or graph
y = sin²(x)
y = 1 – 3sin²(x)
Intersections:
30°, 150°, 210°, 330°

Find the image of (5, -6) after a counterclockwise rotation of 270° about the origin. (Evernote)
(-6, -5)

Wilfred is checking his bicycle’s wheel. The wheel has a radius of 25 cm, and its center is 54 cm above the ground. Wilfred is rotating the wheel at a constant speed of 230° /s. The height of a point on the tire as a function of time is given by h = 54 + 25 sin(230t), where h is the height in centimeters and t is the time in seconds. Find h when t = 5.5 s. Round your answer to the nearest tenth of a centimeter.
Incorrect: None of the listed answers are correct

Correct: 51.8

Solution
h = 54 + 25 sin(230t)
h = 54 + 25 sin(230 × 5.5)
h = 51.8

Solve sin(x) – √(1 – 3sin²(x)) = 0 given that 0° ≤ x ≤ 360°
Incorrect: 90°, 270°

Incorrect: None of the listed answers are correct

Correct: 30°, 150°, 210°, 330°

Solution
sin(x) = √(1 – 3sin²(x))
sin²(x) = 1 – 3sin²(x)
sin²(x) + 3sin²(x) – 1
4sin²(x) – 1
(2sin(x) + 1)(2sin(x) – 1)
sin(x) = -1/2 and 1/2

x = -30°, 30°, 150°, 210°
-30° + 360° = 330°
x = 30°, 150°, 210°, 330°

Or graph
y = sin²(x)
y = 1 – 3sin²(x)
Intersections:
30°, 150°, 210°, 330°

Given that F = 32°, G = 55°, and f = 10, solve the triangle. If no such triangles exist, write none. Round to the nearest tenth. (Evernote)
E = 93° , g = 15.5, e = 18.8

Solution
F = 32°, G = 55° , E = ?°
f = 10, g = ?, e = ?

32° + 55° + ∠E = 180°
∠E = 93°

Use Law of Sines
10/sin(32) = g/sin(55)
g = 10sin(55)/sin(32)
g = 15.5

10/sin(32) = e/sin(93)
e = 10sin(93)/sin(32)
e = 18.8

Given sinθ = 5/9, where π/2 < θ < π, find the exact values of sin 2θ and cos 2θ.
Incorrect: -31/81, (20√14)/81

Correct: 10/9, -(4√14)/9

Which of the following is equal to sin²(x)/(1 – cos(x))
Incorrect: None of the listed answers are correct

Correct: 1 + cos(x) or 1 + cosθ

Mathway
Rationalize the Denominator

Mathway
Verify the Identity+

Find the exact value of cos(345°).
(√6 + √2)/4

Mathway (Trigonometry)
Find the Exact Value

Whitney starts the engine on her small private airplane. The engine drives a propeller with a radius of 7.5 feet and its centerline 12.5 feet above the ground. At idle, the propeller rotates at a constant speed of approximately 850 revolutions per minute. The height of one propeller tip as a function of time is given by h = 12.5 + 7.5 sin(850t), where h is the height in feet and t is the time in minutes. Find h when t = 2.5
8.2 ft

Solution
h = 12.5 + 7.5 sin(850t)
h = 12.5 + 7.5 sin(850 × 2.5)
h = 8.2

Solve triangle ABC given that a = 11, b = 13, and c = 15.
A = 45.6° , B = 57.6° , C = 76.9°

Solution
Use Law of Cosines

cos A = (b² + c² – a²)/2bc
cos A = (13² + 15² – 11²)/(2 × 13 ×15)
cos A = 7/10
cos⁻¹(7/10) = 45.6°

cos B = (c² + a² – b²)/2ca
cos B = (15² + 11² – 13²)/(2 × 15 × 11)
cos B = 59/110
cos⁻¹(59/110) = 57.6°

cos C = (a² + b² – c²)/2ab
cos C = (11² + 13² – 15²)/(2 × 11 × 13)
cos C = 5/22
cos⁻¹(5/22) = 76.9°

csc(2x) + cot(2x) forms an identity with which of the following?
cot x

Use Mathway to “Verify the Identity”

csc(2x) + cot(2x) = cot(x)

Which of the following is an identity?
(cos(x))/(1 + sin(x)) = (1 – sin(x))/(cos(x))

Mathway
Verify the Identity

√((6x + 2)² – 4(9)(x²+1))/(2(9))

I got
√(12x – 16)/9 → √(4(3x – 4))/9
But the book has
√(6x – 8)/9 → √(2(3x -4))/9