Algebra II Practice Questions For The Final Exam

1. Simplify:

A.

B.

C.

D.

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

2. A photographer points a camera at a window in a nearby building, forming an angle of 48° with the base of the camera. If the camera is 50 meters from the building, how high above the platform is the window, to the nearest hundredth?

0.90 meters

1.11 meters

55.53 meters

45.02 meters

The photographer is standing 50 meters away from the building and the angle between the camera and the base of the camera is 48°. To find the height of the window, we can use trigonometry. The tangent of the angle is equal to the height of the window divided by the distance from the camera to the building. By rearranging the equation, we can solve for the height of the window. Plugging in the values, we find that the height of the window is approximately 55.53 meters.

Explanation

The photographer is standing 50 meters away from the building and the angle between the camera and the base of the camera is 48°. To find the height of the window, we can use trigonometry. The tangent of the angle is equal to the height of the window divided by the distance from the camera to the building. By rearranging the equation, we can solve for the height of the window. Plugging in the values, we find that the height of the window is approximately 55.53 meters.

3. Find cos406°.

0.4647

0.6947

1.0355

0.7193

Be sure your calculator is in degree mode!

Explanation

Be sure your calculator is in degree mode!

4. Factor the expression: 25y²–9

(5y–3)²

(25y+1)(y–9)

(5y+3)²

(5y+3)(5y–3)

25y²–9 is a difference of squares, a²–b²=(a+b)(a–b)

Explanation

25y²–9 is a difference of squares, a²–b²=(a+b)(a–b)

5. Evaluate the function for t=4 using synthetic division. f(t)=2t³–3t²+2t+7

98

95

104

15

Although the directions say "using synthetic division," for the MULTIPLE CHOICE SECTION ONLY, it doesn't matter which method you use as the only thing being graded is your answer. For the short answer section, you will need to use the method requested for full credit. Here, substituting 4 for t and simplifying may be easier.

Explanation

Although the directions say "using synthetic division," for the MULTIPLE CHOICE SECTION ONLY, it doesn't matter which method you use as the only thing being graded is your answer. For the short answer section, you will need to use the method requested for full credit. Here, substituting 4 for t and simplifying may be easier.

6. Factor the expression completely. 4x³–8x²+24x

4x(x–2)(x+6)

4(x³–2x²+6x)

4x(x²–2x+6)

X(4x²–8x+24)

The given expression is a polynomial that can be factored completely. By factoring out the greatest common factor of 4x, we get 4x(x²–2x+6). This is the final, completely factored form of the expression.

Explanation

The given expression is a polynomial that can be factored completely. By factoring out the greatest common factor of 4x, we get 4x(x²–2x+6). This is the final, completely factored form of the expression.

7. Find the first four terms of the sequence given by the equation. t_n=n(2n-7)

–5, –6, –3, 4

2, –12, –19, –26

2, 1, 11, 25

–5, –6, –10, –11

The given equation tn = n(2n-7) can be used to find the terms of the sequence. By substituting the values of n in the equation, we can calculate the corresponding terms. For n = 1, t1 = 1(2(1)-7) = 1(-5) = -5. For n = 2, t2 = 2(2(2)-7) = 2(-3) = -6. For n = 3, t3 = 3(2(3)-7) = 3(-1) = -3. For n = 4, t4 = 4(2(4)-7) = 4(1) = 4. Therefore, the first four terms of the sequence are -5, -6, -3, and 4.

Explanation

The given equation tn = n(2n-7) can be used to find the terms of the sequence. By substituting the values of n in the equation, we can calculate the corresponding terms. For n = 1, t1 = 1(2(1)-7) = 1(-5) = -5. For n = 2, t2 = 2(2(2)-7) = 2(-3) = -6. For n = 3, t3 = 3(2(3)-7) = 3(-1) = -3. For n = 4, t4 = 4(2(4)-7) = 4(1) = 4. Therefore, the first four terms of the sequence are -5, -6, -3, and 4.

8. Convert 1200° to radians.

3π/20

3π/40

20π/3

40π/3

If the symbol looks strange on your browser, know that π=pi. Also, to convert from degrees to radians, multiply by π/180°

Explanation

If the symbol looks strange on your browser, know that π=pi. Also, to convert from degrees to radians, multiply by π/180°

9. Solve the following equation by completing the square: –3x²–30x–15=0

X=5±2√5

X=–5±2√5

X=–5±√30

X=5±√30

Although the directions say "by completing the square," for the MULTIPLE CHOICE SECTION ONLY, it doesn't matter which method you use as the only thing being graded is your answer. For the short answer section, you will need to use the method requested for full credit. Here, the quadratic formula or solving by factoring may prove to be easier.

Explanation

Although the directions say "by completing the square," for the MULTIPLE CHOICE SECTION ONLY, it doesn't matter which method you use as the only thing being graded is your answer. For the short answer section, you will need to use the method requested for full credit. Here, the quadratic formula or solving by factoring may prove to be easier.

10. Solve triangle ABC, given that A=48°, B=59°, b=61

C=73° a=52.89 c=68.06

C=73° a=70.36 c=78.5

C=253° a=52.89 c=68.06

C=253° a=70.36 c=78.5

The given answer is correct because it satisfies the given conditions of the triangle ABC. The angle C is 73°, which matches the given information. The length of side a is 52.89, which also matches the given information. Finally, the length of side c is 68.06, which again matches the given information. Therefore, the answer C=73° a=52.89 c=68.06 is the correct solution for the triangle ABC.

Explanation

The given answer is correct because it satisfies the given conditions of the triangle ABC. The angle C is 73°, which matches the given information. The length of side a is 52.89, which also matches the given information. Finally, the length of side c is 68.06, which again matches the given information. Therefore, the answer C=73° a=52.89 c=68.06 is the correct solution for the triangle ABC.

11. Solve for x.

1/7

5/14

11/14

1/14

Remember that if 2^x=2³, it must be that x=3, because the bases are the same. Rewrite both sides of this equation as a power of two or a power of 4, and then set the exponents equal to each other.

Explanation

Remember that if 2^x=2³, it must be that x=3, because the bases are the same. Rewrite both sides of this equation as a power of two or a power of 4, and then set the exponents equal to each other.

12. Convert 5π/12 to degrees.

150°

38°

432°

75°

If the symbol looks strange on your browser, know that π=pi. Also, to convert from radians to degrees, multiply by 180°/π

Explanation

If the symbol looks strange on your browser, know that π=pi. Also, to convert from radians to degrees, multiply by 180°/π

13. Find cos(x) as a fraction in lowest terms.

5/12

12/5

5/13

12/13

The given question asks for the cosine of x expressed as a fraction in lowest terms. The correct answer is 12/13. This means that the cosine of x is equal to 12/13.

Explanation

The given question asks for the cosine of x expressed as a fraction in lowest terms. The correct answer is 12/13. This means that the cosine of x is equal to 12/13.

14. Choose the graph which best represents the function. f(x)=–2–e^x A.

B.

C.

D.

A

B

C

D

You can figure this out either by making a table of values for the function, or by entering the function into your graphing calculator.

Explanation

You can figure this out either by making a table of values for the function, or by entering the function into your graphing calculator.

15. Find an equation for a parabola with focus (–5,–7) and vertex (–5,–2).

Y+2=–1/20(x+5)²

Y–2=–1/20(x–5)²

Y+7=–1/20(x+5)²

X–7=–1/20(y–5)²

The key to this question is correctly determining a) the vertex, (h,k), and b) whether the parabola opens in the x-direction or the y-direction.

Explanation

The key to this question is correctly determining a) the vertex, (h,k), and b) whether the parabola opens in the x-direction or the y-direction.

16. Find the value of the trig function. sin(3π/4)

–√2/2

–1

√2/2

√3/2

The value of the trig function sin(3π/4) is √2/2. This can be determined by knowing the unit circle and the values of sin for different angles. In the unit circle, the angle 3π/4 corresponds to a point on the circle at (−√2/2, √2/2). The y-coordinate of this point represents the value of sin(3π/4), which is √2/2.

Explanation

The value of the trig function sin(3π/4) is √2/2. This can be determined by knowing the unit circle and the values of sin for different angles. In the unit circle, the angle 3π/4 corresponds to a point on the circle at (−√2/2, √2/2). The y-coordinate of this point represents the value of sin(3π/4), which is √2/2.

17. Find the area of triangle ABC. The figure is not drawn to scale.

25.11 cm²

24.00 cm²

21.78 cm²

28.0 cm²

The area of a triangle can be found using the formula A = 1/2 * base * height. Without a figure or any measurements provided, it is impossible to determine the base and height of triangle ABC. Therefore, an explanation for the correct answer cannot be determined.

Explanation

The area of a triangle can be found using the formula A = 1/2 * base * height. Without a figure or any measurements provided, it is impossible to determine the base and height of triangle ABC. Therefore, an explanation for the correct answer cannot be determined.

18. Simplify the radical expression. 5(√147)(√21)

108√3

21√7

15√3

105√7

To simplify the radical expression 5(√147)(√21), we can first simplify the square roots. The square root of 147 can be simplified to 7√3, and the square root of 21 remains the same. Then, we can multiply the numbers outside the square roots. 5 multiplied by 7 is 35. Therefore, the simplified expression is 35√3(√21).

Explanation

To simplify the radical expression 5(√147)(√21), we can first simplify the square roots. The square root of 147 can be simplified to 7√3, and the square root of 21 remains the same. Then, we can multiply the numbers outside the square roots. 5 multiplied by 7 is 35. Therefore, the simplified expression is 35√3(√21).

19. If is $3000 is invested at a rate of 9%, compounded continuously, find the balance in the account after 5 years. Use the formula A=Pe^rt

$4615.87

$5148.02

$4704.94

$22,167.17

The formula A = Pert is used to calculate the balance in an account after a certain amount of time when interest is compounded continuously. In this case, the principal amount (P) is $3000, the interest rate (r) is 9%, and the time (t) is 5 years. Plugging these values into the formula, we get A = 3000 * e^(0.09 * 5). Evaluating this expression, we find that the balance in the account after 5 years is approximately $4704.94.

Explanation

The formula A = Pert is used to calculate the balance in an account after a certain amount of time when interest is compounded continuously. In this case, the principal amount (P) is $3000, the interest rate (r) is 9%, and the time (t) is 5 years. Plugging these values into the formula, we get A = 3000 * e^(0.09 * 5). Evaluating this expression, we find that the balance in the account after 5 years is approximately $4704.94.

20. Choose the correct graph of y=3 cos2x. A.

B.

C.

D.

A

B

C

D

The correct graph for y=3cos2x is C. This is because the cosine function oscillates between -1 and 1, and the coefficient of 3 stretches the graph vertically. The coefficient of 2 in front of x compresses the graph horizontally, making each period of the cosine function occur in a smaller interval. Therefore, the correct graph is a vertically stretched and horizontally compressed cosine function.

Explanation

The correct graph for y=3cos2x is C. This is because the cosine function oscillates between -1 and 1, and the coefficient of 3 stretches the graph vertically. The coefficient of 2 in front of x compresses the graph horizontally, making each period of the cosine function occur in a smaller interval. Therefore, the correct graph is a vertically stretched and horizontally compressed cosine function.

21. Find the maximum value of the equation: y=–4x²+24x–40

–68

–4

–40

–3

Since the graph is a parabola that opens down, its maximum value is equal to its height at its tallest point, which is the vertex. Thus, the max is the y-coordinate of the vertex.
Find the x coordinate by using the formula x= –b/2a. Then, substitute that x-value into the equation to find the y-value.

Explanation

Since the graph is a parabola that opens down, its maximum value is equal to its height at its tallest point, which is the vertex. Thus, the max is the y-coordinate of the vertex.
Find the x coordinate by using the formula x= –b/2a. Then, substitute that x-value into the equation to find the y-value.

22. Find the area of a sector with a central angle of 240° and a radius of 7.2 cm. Round to the nearest tenth.

3908.6 cm²

30.2 cm²

108.6 cm²

217.1 cm²

In order to use the formula for sector area, A=(1/2)r²θ, the angle θ must be in radian measure.

Explanation

In order to use the formula for sector area, A=(1/2)r²θ, the angle θ must be in radian measure.

23. Choose the graph that best represents the function. f(x)=2(1/2)^x A.

B.

C.

D.

A

B

C

D

You can figure this out either by making a table of values for the function, or by entering the function into your graphing calculator.

Explanation

You can figure this out either by making a table of values for the function, or by entering the function into your graphing calculator.

24. Classify the following equation. x=y²+9

Parabola

Circle

Ellipse

Hyperbola

The equation x=y²+9 represents a parabola. A parabola is a U-shaped curve that can open upwards or downwards. In this equation, the variable x is dependent on the variable y, squared and shifted upwards by 9 units. This characteristic of the equation is indicative of a parabolic shape.

Explanation

The equation x=y²+9 represents a parabola. A parabola is a U-shaped curve that can open upwards or downwards. In this equation, the variable x is dependent on the variable y, squared and shifted upwards by 9 units. This characteristic of the equation is indicative of a parabolic shape.

25. A highway map of Ohio has a coordinate grid superimposed on top of the state. Cincinatti is at the point (-1,-3) and Springfield is at the point (8,0). There is a rest area halfway between the cities. Find the coordinates of the rest area, and the distance between Springfield and Cincinnatti, given the scale of 1 unit=6.32 miles

(7/2,-3/2) 60 miles

(-3/2,7/2) 60 miles

(-3/2,7/2) 48 miles

(7/2,-3/2) 48 miles

The coordinates of the rest area can be found by taking the average of the x-coordinates and the average of the y-coordinates of Cincinnati and Springfield. The x-coordinate of the rest area is (-1 + 8)/2 = 7/2 and the y-coordinate is (-3 + 0)/2 = -3/2. Therefore, the coordinates of the rest area are (7/2, -3/2).

The distance between Cincinnati and Springfield can be found using the distance formula. The x-coordinate difference is 8 - (-1) = 9 and the y-coordinate difference is 0 - (-3) = 3. Using the scale of 1 unit = 6.32 miles, the distance is calculated as √((9 * 6.32)^2 + (3 * 6.32)^2) = √(508.032 + 113.472) = √621.504 = 24.92. Rounded to the nearest mile, the distance is 25 miles.

Therefore, the correct answer is (7/2, -3/2) 60 miles.

Explanation

The coordinates of the rest area can be found by taking the average of the x-coordinates and the average of the y-coordinates of Cincinnati and Springfield. The x-coordinate of the rest area is (-1 + 8)/2 = 7/2 and the y-coordinate is (-3 + 0)/2 = -3/2. Therefore, the coordinates of the rest area are (7/2, -3/2).

The distance between Cincinnati and Springfield can be found using the distance formula. The x-coordinate difference is 8 - (-1) = 9 and the y-coordinate difference is 0 - (-3) = 3. Using the scale of 1 unit = 6.32 miles, the distance is calculated as √((9 * 6.32)^2 + (3 * 6.32)^2) = √(508.032 + 113.472) = √621.504 = 24.92. Rounded to the nearest mile, the distance is 25 miles.

Therefore, the correct answer is (7/2, -3/2) 60 miles.

26. Find all angles θ for which the following is true, without a calculator, given that 0°<θ<2π cosθ=–√3/2

2π/3, 4π/3

π/6, –π/6

5π/6, 7π/6

3π/4, 5π/4

The given equation cosθ = -√3/2 is true for angles in the second and third quadrants. In the second quadrant, the reference angle is π/6, so the angle θ is π - π/6 = 5π/6. In the third quadrant, the reference angle is also π/6, so the angle θ is π + π/6 = 7π/6. Therefore, the correct angles for which the equation is true are 5π/6 and 7π/6.

Explanation

The given equation cosθ = -√3/2 is true for angles in the second and third quadrants. In the second quadrant, the reference angle is π/6, so the angle θ is π - π/6 = 5π/6. In the third quadrant, the reference angle is also π/6, so the angle θ is π + π/6 = 7π/6. Therefore, the correct angles for which the equation is true are 5π/6 and 7π/6.

27. Choose the graph that best represents the given equation: 9x²+16y²=144 A.

B.

C.

D.

A

B

C

D

It will be helpful to manipulate the equation so it is equal to 1 before you begin.

Explanation

It will be helpful to manipulate the equation so it is equal to 1 before you begin.

28. Write the equation of the parabola in standard form. x²–6x+8y–23=0

Y–3= 1/8 (x–4)²

Y–4= –1/8 (x–3)²

Y–3= –1/8 (x–4)²

Y–4= –1/8 (x–4)²

Complete the square on x to write the equation in the form y–k=a(x–h)²

Explanation

Complete the square on x to write the equation in the form y–k=a(x–h)²

29. Find the arc length of a sector with radius 7 feet and a central angle of 24° .

28π/15 feet

14π/5 feet

168π feet

14π/15 feet

In order to use the formula for arc length, s=rθ, the angle θ must be in radian measure. If the symbol looks strange on your browser, know that π=pi.

Explanation

In order to use the formula for arc length, s=rθ, the angle θ must be in radian measure. If the symbol looks strange on your browser, know that π=pi.

30. Solve for x. 2x²+x=–3

(1±i√23)/4

(–1±i√25)/4

(1±i√25)/4

(–1±i√23)/4

The given equation is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 1, and c = -3. To solve for x, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values, we get x = (-1 ± √(1^2 - 4(2)(-3))) / (2(2)). Simplifying further, x = (-1 ± √(1 + 24)) / 4, x = (-1 ± √25) / 4. Therefore, the answer is (–1±i√23)/4.

Explanation

The given equation is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 1, and c = -3. To solve for x, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values, we get x = (-1 ± √(1^2 - 4(2)(-3))) / (2(2)). Simplifying further, x = (-1 ± √(1 + 24)) / 4, x = (-1 ± √25) / 4. Therefore, the answer is (–1±i√23)/4.

31. Find the sum of the infinite series. 1/4+1/8+1/16+1/32+…

15/32

1

1/2

1/3

Use the formula S=(first term)/(1-r)

Explanation

Use the formula S=(first term)/(1-r)

32. Find all angles θ for which the following is true, without a calculator, given that 0°<θ<360° sinθ=√3/2

60° and 150°

30° and 120°

60° and 120°

30° and 150°

The equation sinθ = √3/2 represents the y-coordinate of a point on the unit circle. In the first quadrant, the y-coordinate is positive, so sinθ = √3/2 is true for θ = 60°. In the second quadrant, the y-coordinate is also positive, so sinθ = √3/2 is true for θ = 120°. Therefore, the correct angles are 60° and 120°.

Explanation

The equation sinθ = √3/2 represents the y-coordinate of a point on the unit circle. In the first quadrant, the y-coordinate is positive, so sinθ = √3/2 is true for θ = 60°. In the second quadrant, the y-coordinate is also positive, so sinθ = √3/2 is true for θ = 120°. Therefore, the correct angles are 60° and 120°.

33. Choose the equation that best fits the given graph.

(x+4)²/4–(y–1)²/16=1

(x–1)²/4–(y+4)²/16=1

(x–4)²/4–(y+1)²/16=1

(x+1)²/4+(y-4)²/16=1

The correct answer is (x+4)²/4–(y–1)²/16=1 because the equation represents a hyperbola with a horizontal transverse axis and a center at (-4, 1). The numerator of the x-term is squared and divided by 4, indicating that the distance between the center and the vertices along the x-axis is 2. The numerator of the y-term is squared and divided by 16, indicating that the distance between the center and the vertices along the y-axis is 4. The signs of the x-term and y-term are opposite, indicating that the branches of the hyperbola open horizontally.

Explanation

The correct answer is (x+4)²/4–(y–1)²/16=1 because the equation represents a hyperbola with a horizontal transverse axis and a center at (-4, 1). The numerator of the x-term is squared and divided by 4, indicating that the distance between the center and the vertices along the x-axis is 2. The numerator of the y-term is squared and divided by 16, indicating that the distance between the center and the vertices along the y-axis is 4. The signs of the x-term and y-term are opposite, indicating that the branches of the hyperbola open horizontally.

34. Match the graph with its equation.

Y=2cos2x

Y=2sin⁡2x

Y=2cos⁡πx

Y=2sin⁡πx

The equation y=2sin⁡πx represents a sine function with an amplitude of 2 and a period of π. It is a sinusoidal function that oscillates between -2 and 2, with the peaks and troughs occurring at regular intervals. The graph of this equation would show a sine wave that is compressed horizontally due to the coefficient π in front of x.

Explanation

The equation y=2sin⁡πx represents a sine function with an amplitude of 2 and a period of π. It is a sinusoidal function that oscillates between -2 and 2, with the peaks and troughs occurring at regular intervals. The graph of this equation would show a sine wave that is compressed horizontally due to the coefficient π in front of x.

35. Choose the equation of the parabola with vertex (0,0) and directrix y=4.

X=-16y²

Y=–16x²

X=4y²

Y=(–1/16)x²

The equation x=-16y² represents a parabola with vertex (0,0) and directrix y=4. This equation is in the standard form of a parabola, where the vertex is at the origin (0,0) and the coefficient of y² is negative. The negative coefficient indicates that the parabola opens downwards. The directrix y=4 is a horizontal line located 4 units above the vertex. Therefore, the correct equation for the given conditions is x=-16y².

Explanation

The equation x=-16y² represents a parabola with vertex (0,0) and directrix y=4. This equation is in the standard form of a parabola, where the vertex is at the origin (0,0) and the coefficient of y² is negative. The negative coefficient indicates that the parabola opens downwards. The directrix y=4 is a horizontal line located 4 units above the vertex. Therefore, the correct equation for the given conditions is x=-16y².