Algebra 1B Study Guide (with Answers and Solutions)

This study guide contains two sections--Section 1: Problems to Solve and Section 2: Solutions. First, solve the problems in Section 1 and then use Section 2 to check your work.

Section 1: Problems to Solve

Question 1Draw the graph that represents the solution set of y = x2 − 4 and x + y + 2 = 0 and give the solutions (remember there might be one, two, or none)

Question 2Ashton needs to know the how long a banner to order. He can find the length using the distance formula if he knows the points on the archway at which the ends of the banner are to be attached.

Question 3Solve the quadratic equation: 3x2 − 2x − 8 = 0

Question 4

Simplify the expression: (3xy)-2

Question 5Solve the equation

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Question 6

Simplify the expression:

Question 7

Factor completely: 16x2 − 25 = 0

Question 8: Find the quotient

Question 9Use the quadratic formula to solve the quadratic equation 2 x2 − 16x − 8 = 0.

Question 10 Does this graph shows a strong negative correlation, strong positive correlation or no correlation at all?

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Question 11This table contains data on the number of people visiting a historical landmark over a period of one week.

Day Number ofVisitors

1 120

2 124

3 130

4 131

5 135

6 132

7 135

Sketch a scatter plot. Which of these values comes closest to the correlation coefficient for this data? (a) 0.0 (b) 0.3 (c) -0.3 (d) 0.9

Question 12The data gives the responses of 15 students who were asked how many text messages They send in a month.{12, 100, 100, 102, 102, 104, 106, 108, 109, 110, 110, 110, 112, 112, 114}

What is the best measure of the central tendency of the data? (a) mean (b) mode (c) median (d) range (e) interquartile range

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Question 13Which data set has a greater spread? Why?

A: {38, 12, 23, 48, 55, 16, 18}

B: {44, 13, 24, 12, 56}

a) Set A has a greater spread because it has more data values than set B.

b) Set A has a greater spread because its mean is greater.

c) Set B has a greater spread because its median is greater.

d) Set B has a greater spread because its range is greater

Question 14Mrs. Miller has to make a histogram of the following test grades64, 89, 67, 65, 66, 82, 69, 93, 83, 81, 83, 69, 88, 94, 68, 82, 86, 83, 88, 62

Which frequency table should she use for the histogram? (a)

Interval 60–70

70–80

80–90

90–100

Frequency 8 0 10 2

(b)

Interval 56–63

63–70

70–77

77–84

84–91

91–98

Frequency 1 7 0 5 5 2

(c)

Interval 50–70

70–80

80–85

85–90

90–100

Frequency 8 0 6 4 2

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(d)

Interval 60–75

75–90

90–105

Frequency 8 10 2

(e)

Interval 55–75

75–90

90–110

Frequency 8 10 2

Question 15According to this box and whisker plot, what are the minimum, first quartile, median, third quartile and maximum of the set?

Vocabulary: Causation, correlation

Question 16Michelle collected data on the number of push-ups she could do in 2 minutes. The table below shows her data

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What is the independent variable?

Question 17The data set gives the number of hours it took each of the 10 students in a cooking class to master a particular technique. Which is the best measure of the central tendency of the data?{3, 3, 4, 4, 4, 5, 5, 5, 5, 30}

a) Meanb) Medianc) First Quartile d) Range

Question 18 An environmental agency studying the effects of drought on a lake found that the water level decreased at a constant rate during the first few months of the drought. When the agency began its research, the depth of the lake was 346 feet. The table represents the agency’s subsequent observations about the lake's depth.Which equation represents the depth of the lake, y, based on the number of weeks passed, x?

Time(week

s)

Lake Depth(feet)

0 346.02 344.85 343.0

6

7 341.810 340.012 338.8

a) y = -0.6x + 346 b) y = -1.2x + 346c) y = -0.6xd) y = -1.2x

Question 19A comic-strip writer churns out a different number of comic strips each day. For 16 days, the writer logged the number of comic strips written each day (sorted low to high): {1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7}. If the writer writes for one more day and comes up with 8 new comic strips, how will the skew be affected?

a) The distribution will be skewed to the negative side. b) The distribution will be skewed to the positive side.c) The distribution will have the same mean and median.d) The distribution will have a mean lower than the median.

Answer: b) The distribution will be skewed to the positive side.

Question 20

An object in a simulation accelerates from rest to a speed of 4 meters per second in 2 seconds. It then accelerates linearly for the next 3 seconds until it reaches 10 meters per second. The object remains at that speed (acceleration = 0) for 3 seconds and then de-accelerates linearly until it comes to rest 2 seconds later. Draw the graph that demonstrates the speed of the object for those 10 seconds.

Question 21What are the x-intercepts of the parabola x2 − 4x − 12?

Question 22

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What is the focus of the parabola given by the equation y = x2 − 2x − 3? Question 23

Which point is the vertex for the graph

f(x) = |x| + ?

Question 24

Graph y = |x2| − 2x − 8?

Question 25

What is the value of f(x) = 81x when x =1

2?

Question 26Find the inverse of y = 3x – 2

Question 27What is the value of log 26 rounded to the nearest hundredth?

Question 28Which one represents the equation log8 25 = 2 in exponential form?

Question 29Determine whether the parabola y = −x2 + 18x + 2 opens up, down, left, or right.

Question 30 What is the vertex of the parabola given by y = x2 − 20x + 5?

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Question 31If f(x) = x2 − 2x + 9 and g(x) = 8 − x, what is (f o g)(−7)?

END OF REVIEW

Section 2: Answers and Solutions

Question 1Draw the graph that represents the solution set of y = x2 − 4 and x + y + 2 = 0 and give the solutions (remember there might be one, two, or none)

Question 2Ashton needs to know the how long a banner to order. He can find the length using the distance formula if he knows the points on the archway at which the ends of the banner are to be attached.

a. Using the graphing method, find the points at which the equation of the angle of the banner will intersect the equation of the archway.

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The two equations intersect at (1, 5) and (4, 8), so the banner should be hung at these two points.

b. Using the algebraic method, confirm whether the two points obtained by the graphing method are correct.

Given these two equations:

y = -x2 + 6x (1)

y – x = 4 (2)

Modify equation (2) to get y = x + 4.

To find the points of intersection, equate the two equations:

-x2 + 6x = x + 4

x2 - 5x + 4 = 0

(x – 4)(x – 1) = 0

x = 4 or x = 1.

Substitute the values of x in equation (2) to get the values of y:

y = x + 4

10

= (4) + 4

= 8.

y = x + 4

= (1) + 4

= 5

The two points of intersection are (4, 8) and (1, 5). These are the same points obtained by graphing.

c. Using the distance formula find the distance between points A and B. How long a banner does Ashton need to order?

The distance formula is

x1 = 4

y1 = 8

x2 = 1

y2 = 5

Substituting these values in the formula:

d = 4.24 units.

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The distance between the two points of intersection of the archway and the banner is 4.24 units, so the banner must be 4.24 units long.

Question 3Solve the quadratic equation: 3x2 − 2x − 8 = 0

(a)

(b)

(c)

(d) [Answer

Question 4

Simplify the expression: (3xy)-2

1___9x2y2

Question 5Solve the equation

To solve we first square both sides of the equation. The result is x - 2 = 81. This equation is simple to solve. We have x = 83

Question 6

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Simplify the expression:

(a)

(b) [Answer]

(c)

(d)

Question 7

Factor completely: 16x2 − 25 = 0(a) (4x + 5) (4x − 5) [Answer]

(b) 16x2 − 25(c) (4x + 5) (4x + 5) (d) (4x2 + 5) (4x2 − 5)

Question 8: Find the quotient

Question 9Use the quadratic formula to solve the quadratic equation 2 x2 − 16x − 8 = 0.

(a) [Answer]

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(b)

(c)

(d)

Question 10 Does this graph shows a strong negative correlation, strong positive correlation or no correlation at all?

Question 11This table contains data on the number of people visiting a historical landmark over a period of one week.

Day Number ofVisitors

1 120

2 124

3 130

4 131

5 135

6 132

7 135

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Sketch a scatter plot. Which of these values comes closest to the correlation coefficient for this data? (a) 0.0 (b) 0.3

(c) 0.9 [Answer]

(d) -0.3 (e) -0.9

Question 12The data gives the responses of 15 students who were asked how many text messages They send in a month.{12, 100, 100, 102, 102, 104, 106, 108, 109, 110, 110, 110, 112, 112, 114}

What is the best measure of the central tendency of the data? (a) mean (b) mode

(c) median [Answer]

(d) range (e) interquartile range

Question 13Which data set has a greater spread? Why?

A: {38, 12, 23, 48, 55, 16, 18}

B: {44, 13, 24, 12, 56}

a) Set A has a greater spread because it has more data values than set B.

b) Set A has a greater spread because its mean is greater.

c) Set B has a greater spread because its median is greater.

d) Set B has a greater spread because its range is greater

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Answer d) Set B has a greater spread because its range is greater

Question 14Mrs. Miller has to make a histogram of the following test grades64, 89, 67, 65, 66, 82, 69, 93, 83, 81, 83, 69, 88, 94, 68, 82, 86, 83, 88, 62

Which frequency table should she use for the histogram? (a)

Interval 60–70

70–80

80–90

90–100

Frequency 8 0 10 2

(b)

Interval 56–63

63–70

70–77

77–84

84–91

91–98

Frequency 1 7 0 5 5 2

(c)

Interval 50–70

70–80

80–85

85–90

90–100

Frequency 8 0 6 4 2

(d)

Interval 60–75

75–90

90–105

Frequency 8 10 2

[Answer]

(e)

Interval 55–75

75–90

90–110

Frequenc 8 10 2

16

y

Question 15According to this box and whisker plot, what are the minimum, first quartile, median, third quartile and maximum of the set?

Answers :

Minimum = 35

First quartile = 76Median = 82Third quartile = 90Maximum = 112

Vocabulary: Causation, correlation

Question 16Michelle collected data on the number of push-ups she could do in 2 minutes. The table below shows her data

a. What is the independent variable?

Answer: Time (in seconds)

Question 17

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The data set gives the number of hours it took each of the 10 students in a cooking class to master a particular technique. Which is the best measure of the central tendency of the data?{3, 3, 4, 4, 4, 5, 5, 5, 5, 30}

e) Meanf) Mediang) First Quartile h) Range

Question 18 An environmental agency studying the effects of drought on a lake found that the water level decreased at a constant rate during the first few months of the drought. When the agency began its research, the depth of the lake was 346 feet. The table represents the agency’s susequent observations about the lake's depth.Which equation represents the depth of the lake, y, based on the number of weeks passed, x?

Time(week

s)

Lake Depth(feet)

0 346.02 344.85 343.07 341.810 340.012 338.8

e) y = -0.6x + 346 f) y = -1.2x + 346g) y = -0.6xh) y = -1.2x

Answer: a) y = -0.6x + 346

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Question 19A comic-strip writer churns out a different number of comic strips each day. For 16 days, the writer logged the number of comic strips written each day (sorted low to high): {1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7}. If the writer writes for one more day and comes up with 8 new comic strips, how will the skew be affected?

e) The distribution will be skewed to the negative side. f) The distribution will be skewed to the positive side.g) The distribution will have the same mean and median.h) The distribution will have a mean lower than the median.

Answer: b) The distribution will be skewed to the positive side.

Question 20

An object in a simulation accelerates from rest to a speed of 4 meters per second in 2 seconds. It then accelerates linearly for the next 3 seconds until it reaches 10 meters per second. The object remains at that speed (acceleration = 0) for 3 seconds and then de-accelerates linearly until it comes to rest 2 seconds later. Draw the graph that demonstrates the speed of the object for those 10 seconds.

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(d) [Answer]

Question 21What are the x-intercepts of the parabola x2 − 4x − 12? Factor and set to zero (x-6) (x+2) = 0 Either x-6 =0 or x+ 2 =0 x=6 x= -1Write the x- intercepts as coordinates (6, 0) (-1, 0)

Question 22

What is the focus of the parabola given by the equation y = x2 − 2x − 3?

(e) [Answer]

Question 23

Which point is the vertex for the graph

f(x) = |x| + ?20

Answer: (0, )

Question 24

Graph y = |x2| − 2x − 8?

Question 25

What is the value of f(x) = 81x when x =1

2?

Answer f(x) = 9

Question 26Find the inverse of y = 3x – 2

First get “x” alone

Once you have "x =", switch x and y; the "y =" is the inverse.

Then the inverse is y = (x + 2) / 3

Question 27What is the value of log 26 rounded to the nearest hundredth?

Answer:= log 10 6 ≈ .77815 ≈ 2.585 log 102 .30103

Question 28Which one represents the equation log8 25 = 2 in exponential form?

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Answer : 8 2 =25

Question 29Determine whether the parabola y = −x2 + 18x + 2 opens up, down, left, or right.

Answer: Look at the coefficient of the x squared. If it is positive then opens up. If it is negative, then opens down. This has a -1 in front of x squared so it opens down.

Question 30 What is the vertex of the parabola given by y = x2 − 20x + 5? First find -b 2(a) this is the x part of the vertex

= - -20 2(5) = 2 now you have the x so plug that into equation for y.

y = 22 – 20(2) + 5 = 4- 40 +5 = -31 so the vertex is (2, -31)

Question 31If f(x) = x2 − 2x + 9 and g(x) = 8 − x, what is (f o g)(−7)?

Answer : First find g(-7) g(x) = 8 – xg(-7)= 8- - 7g(-7)= 15 Now plug this into f(x)

f(x) = x2 − 2x + 9f(x)= 152 - 2(15) +9 f(g(x)) = 225 -30 + 9 ( f o g )(−7) = 204

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