12. Translate the given recursive formula into an explicit formula.
an+1=an−5, a1=−2
A. an=−5−2(n−1)
B. an=−2−5n
C. an=−5−2n
D. an=−2−5(n−1)
13. Use the paragraph and table to choose a response. When a bungee cord jumper jumps, the cord stretches and then rebounds upward. The distances of the first four upward rebounds for a particular bungee jumper are given in the table below.
Rebound 1 2 3 4
Distance (ft.)50403225.6
Write both a recursive formula and an explicit formula for the distance on the nth rebound. Then find the approximate distance of the 7th rebound to the nearest tenth of a foot.
A. Recursive formula: an+1=an⋅1.25, a1=25.6; Explicit formula: an=25.6⋅1.25n−1; Distance of the 7th rebound: a7=97.7 ft.
B. Recursive formula: an+1=an⋅0.8, a1=50; Explicit formula: an=50⋅0.8n−1; Distance of the 7th rebound: a7=10.5 ft.
C. Recursive formula: an+1=an⋅0.8, a1=50; Explicit formula: an=50⋅0.8n−1; Distance of the 7th rebound: a7=13.1 ft.
D. Recursive formula:an+1=an−8, a1=50; Explicit formula: an=50−8(n−1); Distance of the 7th rebound: a7=2.0 ft.
14. Use the table to answer the question.
Hours Number of Bacteria
0 20 million
4 22 million
8 25 million
12 28 million
16 32 million
20 36 million
24 40 million
28 45 million
The scatterplot of these data, when graphed, reveals what relationship between hours and number of bacteria? Explain
A. The scatterplot reveals that hours and number of bacteria have a strong linear relationship since the points closely follow a line. Since the number of bacteria decreases as time increases, the relationship is negative.
B. The scatterplot reveals that hours and number of bacteria have a strong exponential relationship since the points closely follow an exponential curve. Since the number of bacteria increases as time increases, the relationship is positive.
C. The scatterplot reveals that hours and number of bacteria have a strong exponential relationship since the points closely follow an exponential curve. Since the number of bacteria increases as time increases, the relationship is negative.
D. The scatterplot reveals that hours and number of bacteria have a strong linear relationship since the points closely follow a line. Since the number of bacteria increases as time increases, the relationship is negative.
15. A residual plot for a linear model y=ax+b is given above. What information does the residual plot provide?
A. The strong pattern in the residual plot indicates that the linear model is a good fit.
B. The absence of a pattern in the residual plot indicates that the linear model is not a good fit.
C. The strong pattern in the residual plot indicates that the linear model is not a good fit.
D. The absence of a pattern in the residual plot indicates that the linear model is a good fit.
16. 6 students taking a college entrance exam scored 53, 64, 87, 95, 96, and 99. What is the standard deviation of entrance exam scores? Round your answer to the nearest whole number
S=
17. A marketing analyst has been tracking the attendance of baseball games for two minor league baseball teams. The distribution of each is given in the box plots below.
Given these distributions, which of the following are true? Select the two correct answers.
A. The spread of Team 1's distribution is higher than Team 2's.
B. Team 1's distribution is more symmetrical than Team 2's.
C. Team 2's distribution is more symmetrical than Team 1's.
D. The center for Team 2's distribution is higher than Team 1's.
E. The center for Team 1's distribution is higher than Team 2's.
18. How many lines of symmetry does a regular octagon have? Explain.
A. A regular octagon has 16 lines of symmetry, one for each vertex and one for each side.
B. A regular octagon has 4 lines of symmetry, one for each pair of opposite vertices.
C. A regular octagon has 8 lines of symmetry, one for each pair of opposite vertices and one for each pair of opposite sides.
D. A regular octagon has 4 lines of symmetry, one for each pair of opposite sides.
19. The given image depicts a transformation performed on △CDE to get its image. Which of the following transformations were performed on △CDE?
A. a translation followed by a reflection across AB←→
B. a rotation about the origin followed by a reflection across AB←→
C. a reflection across AB←→ followed by a translation
D. a reflection across AB←→ followed by a rotation about the origin
20. △ABC is given with A(−2, 5), B(−6, 1), and C(3, −3). A mapping is applied to the coordinates of △ABC to get A′(2, −5), B′(6, −1), and C′(−3, 3). What transformation was performed? Explain.
A. The mapping (x, y)→(−y, −x) was used, resulting in a rotation of 90° counter clockwise about the origin.
B. The mapping (x, y)→(−x, −y) was used, resulting in a rotation of 180° about the origin.
C. The mapping (x, y)→(−y, −x) was used, resulting in a reflection across y=−x.
D. The mapping (x, y)→(x, −y) was used, resulting in a reflection across y=x.
.
12. C. The recursive formula an+1=an−5 can be rewritten as an=an−1−5. Substituting repeatedly, we get an=a1−5(n−1), which simplifies to an=−5−2n.
13. A. To find the recursive formula, we see that each distance is 0.8 times the previous distance, so an+1=0.8an. The first distance is 25.6, so a1=25.6. To find the explicit formula, we can use the formula for geometric sequences, which is an=a1r^(n-1), where r is the common ratio. Plugging in the values, we get an=25.6*1.25^(n-1). To find the 7th distance, we calculate a7=25.6*1.25^(7-1)=97.7, rounded to the nearest tenth.
14. B. The scatterplot reveals a strong exponential relationship, as the points follow a curve that looks like an exponential function. Since the number of bacteria increases as time increases, the relationship is positive.
15. D. The absence of a pattern in the residual plot indicates that the linear model is a good fit. The residual plot measures the errors or residuals between the observed values and the predicted values of a linear model, and we want to see a random scattering of points with no visible pattern.
16. The standard deviation can be calculated using the formula s=sqrt[(Σ(x-μ)^2)/n], where Σ(x-μ)^2 is the sum of the squared deviations from the mean, and n is the sample size. We first find the mean of the scores: (53+64+87+95+96+99)/6=82.33. Then we calculate the sum of the squared deviations: (53-82.33)^2 + (64-82.33)^2 + (87-82.33)^2 + (95-82.33)^2 + (96-82.33)^2 + (99-82.33)^2 = 7134.67. Dividing by the sample size and taking the square root, we get s=22, rounded to the nearest whole number.
17. C and D. The box plot for Team 2 is more symmetrical than Team 1's, and the center for Team 2's distribution is higher than Team 1's.
18. C. A regular octagon has 8 lines of symmetry, one for each pair of opposite vertices and one for each pair of opposite sides. To see this, consider drawing the perpendicular bisectors of each side and the diagonals connecting opposite vertices, which divide the octagon into 8 congruent triangles that can be reflected onto each other.
19. B. A rotation of 180° about the origin followed by a reflection across AB←→ maps △CDE onto its image.
20. C. The mapping (x, y)→(−y, −x) was used, resulting in a reflection across y=−x. To see this, we can plot the original and transformed points on a coordinate plane and find that they are reflections across the line y=−x.