College Algebra Midterm Test KEY - ALL Questions Name____________________________________

1.       Simplify (3x2y-2)2

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9x4y-4 or 9x4/y4
1.       Simplify the expression given below.

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(4x6y4)/9
1.       Simplify (3x1/2y3/4)2

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9xy3/2
1.       Simplify the expression given below.

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answer is 9/4
2.       Simplify the expression given below.

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2.       Simplify the expression given below.

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2.       Rationalize the denominator of the expression given below and simplify.

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2.       Multiply and simplify the expression given below.

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3.       Factor x2 - 2x - 80

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(x - 10)(x + 8)
3.       Factor x2 - 19x + 90

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(x - 10)(X - 9)
3.       Factor 6x2 + 5x - 4

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(3x + 4)(2x - 1)
3.       Factor 6x2 - 17x - 3

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(6x + 1)(x - 3)
4.       Factor 4x2 - 9y4

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(2x + 3y2)(2x - 3y2)
4.       Factor 9x2 - 4y6

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(3x + 2y3)(3x - 2y3)
4.       Factor the greatest common factor out of 6x2y2 + 9xy3

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3xy2(2x + 3y)
4.       Factor the greatest common factor out of 6x2y3 + 9x3y4

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3x2y3(2 + 3xy)
5.       Divide and simplify the expression given below. Hint: factor first!

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5.       Divide and simplify the expression given below. Hint: factor first!

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5.       Add the rational expressions given below.

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5.       Add the rational expressions given below.

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6.       Write the complex fraction given below in standard form by multiplying by the conjugate top and bottom.

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11/17 + (10/17) i
6.       Write the complex fraction given below in standard form by multiplying by the conjugate top and bottom.

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7/5 + (6/5) i
6.       Write the complex fraction given below in standard form by multiplying by the conjugate top and bottom.

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(-2/5) + (16/5) i
6.       Write the complex fraction given below in standard form by multiplying by the conjugate top and bottom.

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(1/5) + (-8/5) i
7.       Solve the linear equation given below.

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x = 26.25 or x = 105/4
7.       Solve the linear equation given below.

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x = -50/3
7.       In the first few years after purchase, the value of a car that is worth $20,000 new is given by V = 20000 - 2800t where t = number of years after the car is purchased. If such a car has a value of $12,500 , how long ago was it purchased? Round to the tenth of a year.

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t = 7500/2800 or 75/28 or 2.7 years (rounded)
7.       Solve the absolute value equation given below.

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x = 4 or x = -14/3
8.       Solve the equation 2x2 + x = 3 by factoring.

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x = -3/2 or x = 1
8.       Solve the equation 3x2 - x = 2 by factoring.

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x = -2/3 or x = 1
8.       Solve the equation 2x2 - x = 3 by factoring.

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x = 3/2 or x = -1
8.       Solve the equation x2 = 5x + 2 with the quadratic formula.

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9.       Solve the equation 2x2 + 6x - 1 = 0 by completing the square. You must complete the square!

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9.       Solve the equation 2x2 + 4x - 1 = 0 by completing the square. You must complete the square!

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9.       A box has a height of 2 feet and a length that is 4 ft more than its width, as shown below. If the volume of the box is 15 cubic feet, what is the width and length? Round answer to 2 decimal places.

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x = 1.39 ft and x+4 = 5.39 ft
9.       A box has a height of 2 feet and a length that is 3 ft more than its width, as shown below. If the volume of the box is 14 cubic feet, what is the width and length? Round answer to 2 decimal places.

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x = 1.54 ft and x+3 = 4.54 ft
10.       Solve the equation 3x3 - 3x2 - 18x = 0. Hint: Factor out GCF first.

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x=0, x=3, x=-2
10.       Solve 3x4/3 + 3 = 51

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x = 8 or x =-8
10.       Solve the equation given below.

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x=2 or x = 1/2
10.       Solve the equation given below.

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x = 2 or x = -5
11.       Solve the radical equation below. You must show the algebraic solution - not just check and guess!

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x = 6 ONLY!
11.       Solve the radical equation below. You must show the algebraic solution - not just check and guess!

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x = 6 ONLY!
11.       Solve the equation x4 - 2x2 - 24 = 0.

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11.       Solve the equation x4 - 3x2 - 18 = 0.

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12.       Solve the absolute value inequality given below.

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answer is -5/3 ≤ x ≤ 3
12.       Solve the absolute value inequality given below.

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x ≥ 3 or x ≤ -4
12.       Solve 3x + 1 ≤ 10

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x ≤ 3
12.       Solve 5x + 3 ≥ 10

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x ≥ 7/5
13.       Solve x3 - x2 - 20x ≥ 0 using the critical value method.

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answer: -4 ≤ x ≤ 0 or x ≥ 5
13.       Solve (x - 2)(x+1)2 ≤ 0 using the critical value method.

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x ≤ 2
13.       Solve the rational inequality shown below using the critical value method.

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x < -3 or x = 2
13.       Solve the rational inequality shown below using the critical value method.

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x < -3 or x ≥ 2
14.       The number of inches per hour I that drain from a pool varies directly as the square of the diameter D of the hose used. If a 3/4" diameter hose is used, I = 1.2 inches per hour. How many inchese per hour I will drain with a hose of diameter D = 2". Round values to 2 significant decimal places.

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I = kD2, k=2.13, when D=0.75 and I =1.2. , When D=2, I = 2.13x22 = 8.52 inches per hour
14.       The number of inches per hour I that drain from a pool varies directly as the square of the diameter D of the hose used. If a 3/4" diameter hose is used, I = 0.9 inches per hour. How many inchese per hour I will drain with a hose of diameter D = 2.5". Round values to 2 significant decimal places.

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I = kD2, k=1.6, when D=0.75 and I =1.2. , When D=2.5, I = 1.6x2.52 = 10 inches per hour
14.       The sound intensity I varies inversely as the square of the distance D from the source of the sound. If the intensity of a sound is I = 0.008 Watts per square meter at a distance of 2.1 meters, what will the intensity be when the distance is 4 meters? Round values to 2 significant decimal places.

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I = k/D2, k=0.0353, when D=2.1 and I =0.008. , When D=4, I = 0.0353/42 = 0.00221 watts per square meter
14.       The sound intensity I varies inversely as the square of the distance D from the source of the sound. If the intensity of a sound is I = 0.006 Watts per square meter at a distance of 1.8 meters, what will the intensity be when the distance is 3 meters? Round values to 2 significant decimal places.

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I = k/D2, k=0.0194, when D=1.8 and I =0.006. , When D=3, I = 0.0194/32 = 0.00216 watts per square meter
15.       Find the standard form of the equation of a circle that contains fhe point (1,1) and has center (3,2).

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(x - 3)2 + (y - 2)2 = 5
15.       Find the standard form of the equation of a circle that contains fhe point (1,2) and has center (3,3).

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(x - 3)2 + (y - 3)2 = 5
15.       A circle has endpoints of a diameter as shown below. What is its equation?

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(x - 2)2 + (y - 1)2 = 4
15.       A circle has endpoints of a diameter as shown below. What is its equation?

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(x - 2)2 + (y - 1)2 = 9
16.       Graph the equation given below and show ALL intercepts.

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16.       Graph the equation given below and show ALL intercepts.

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16.       Graph the equation given below and show ALL intercepts.

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16.       Graph the equation given below and show ALL intercepts.

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17.       If f(x) = x2 + x, evaluate and simplify f(a + 2).

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a2 + 5a + 6
17.       If f(x) = x2 + 3, evaluate and simplify f(a - 2).

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a2 - 4a + 7
17.       Find the domain of the function given below.

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All real numbers except x = -2
17.       Find the domain of the function given below.

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All real numbers except x = 4 and x = -3
18.       A backyard has dimensions as shown below. Give the function for its area. Then use this function to find the width x for a backyard with area of 500 square feet. Round to 2 places.

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Area = 2x2, x = Width = 15.81 ft
18.       A backyard has dimensions as shown below. Give the function for its area. Then use this function to find the width x for a backyard with area of 250 square feet. Round to 2 places.

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Area = x(x + 10), x = Width = 11.58 ft
18.       A special triangle has dimensions as shown below with base always 3/4 the height. Give the function for the area of this special triangle (note that area = 1/2 times base times height). Then use this function to find the height x when this particular triangle has an area of 20 square inches. Round to 2 places.

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Area = (3/8)x2, x = height = 7.30 ft
18.       A special triangle has dimensions as shown below with base always 2/3 the height. Give the function for the area of this special triangle (note that area = 1/2 times base times height). Then use this function to find the height x when this particular triangle has an area of 24 square inches. Round to 2 places.

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Area = (1/3)x2, x = height = 8.485 ft
19.       A person sells 200 units per day and makes a daily net profit of $10. If 300 unit are sold per day the daily net profit is $30. Assuming a linear relationship between sales (x) and profit (y), find the equation that relates units sold per day (x) and net daily profit (y).

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y = (1/5)x - 30
19.       A person sells 220 units per day and makes a daily net profit of $10. If 300 unit are sold per day the daily net profit is $30. Assuming a linear relationship between sales (x) and profit (y), find the equation that relates units sold per day (x) and net daily profit (y).

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y = (1/4)x - 45
19.       A person sells 180 units per day and makes a daily net profit of $10. If 300 unit are sold per day the daily net profit is $30. Assuming a linear relationship between sales (x) and profit (y), find the equation that relates units sold per day (x) and net daily profit (y).

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y = (1/6)x - 20
19.       A person sells 200 units per day and makes a daily net profit of $20. If 300 unit are sold per day the daily net profit is $30. Assuming a linear relationship between sales (x) and profit (y), find the equation that relates units sold per day (x) and net daily profit (y).

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y = (1/10)x
20.       The price P of an item is a function of x, the number of items produced daily, and is given by the formula P = 0.02x2 - x + 20. What number of items x results in the lowest price?

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x=25
20.       The price P of an item is a function of x, the number of items produced daily, and is given by the formula P = 0.04x2 - 2x + 20. What number of items x results in the lowest price?

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x=25
20.       The profit P made by producing x items is a given by P = -0.02x2 + 2x. What number of items x results in the maximum profit?

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x=50
20.       The profit P made by producing x items is a given by P = -0.01x2 + x. What number of items x results in the maximum profit?

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x=50
21.       Sketch the graph of f(x) = 2(x + 3)2 - 2. Show the vertex and at least two other points.

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21.       Sketch the graph of f(x) = ½ (x - 1)2 + 2. Show the vertex and at least two other points.

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21.       Sketch the graph of f(x) = 3(x + 1)2 - 2. Show the vertex and at least two other points.

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21.       Sketch the graph of f(x) = 2(x + 3)2 + 2. Show the vertex and at least two other points.

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22.       Sketch the graph of g(x) = 2f(x), given the graph shown below.

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22.       Sketch the graph of g(x) = ½ f(x), given the graph shown below.

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22.       Sketch the graph of g(x) = ½ f(x), given the graph shown below.

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22.       Sketch the graph of g(x) = ½ f(x), given the graph shown below.

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23.       The graph of y = x3 and a shift of this graph are shown below. Identify the formula of the shifted graph.

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y = (x + 2)3 - 1
23.       The graph of y = x3 and a shift of this graph are shown below. Identify the formula of the shifted graph.

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y = (x - 2)3 + 1
23.       The graph of y = x3 and a shift of this graph are shown below. Identify the formula of the shifted graph.

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y = (x - 4)3 + 1
23.       The graph of y = x3 and a shift of this graph are shown below. Identify the formula of the shifted graph.

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y = (x + 4)3 - 1
24.       Given f(x) = x2 + 2, evaluate and simplify the difference quotient expression given below.

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2x + h
24.       Given f(x) = x2 + x, evaluate and simplify the difference quotient expression given below.

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2x + h + 1
24.       Given f(x) = x2 + 2x, evaluate and simplify the difference quotient expression given below.

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2x + h + 2
24.       Given f(x) = x2 + 1, evaluate and simplify the difference quotient expression given below.

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2x + h
25.       Find the inverse of the function f(x) = ¼ x + 3. Also, show that (f o f -1)(x) = x and (f -1 o f)(x) = x.

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f-1(x) = 4(x - 3), (f o f-1)(x) = ¼ [4(x - 3)] + 3 = x - 3 + 3 = x and (f-1 o f)(x) = 4[( ¼ x + 3) - 3] = 4( ¼ )x = x
25.       Find the inverse of the function f(x) = ½ x - 4. Also, show that (f o f -1)(x) = x and (f -1 o f)(x) = x.

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f-1(x) = 2(x + 4), (f o f-1)(x) = ½ [2(x + 4)] - 4 = x + 4 - 4 = x and (f-1 o f)(x) = 2[( ½ x - 4) + 4] = 2 ( ½ )x = x
25.       Find the inverse of the function f(x) given below. Also, show that (f o f -1)(x) = x and (f -1 o f)(x) = x.

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25.       Find the inverse of the function f(x) given below. Also, show that (f o f -1)(x) = x and (f -1 o f)(x) = x.

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