What are the first four terms of the sequence represented by the expression n(n – 2) – 3?
To find the first four terms of the sequence represented by the expression n(n – 2) – 3, we need to substitute the first four positive integers (1, 2, 3, and 4) for n and evaluate the expression.
For n = 1:
n(n – 2) – 3 = 1(1 – 2) – 3 = -2
For n = 2:
n(n – 2) – 3 = 2(2 – 2) – 3 = -3
For n = 3:
n(n – 2) – 3 = 3(3 – 2) – 3 = 0
For n = 4:
n(n – 2) – 3 = 4(4 – 2) – 3 = 5
Therefore, the first four terms of the sequence are -2, -3, 0, and 5.
A car traveling at 46 mi/h slows to a speed of 23 mi/h in 5 seconds. It maintains that speed for 5 seconds and then slows to a stop after 5 more seconds. Which graph shows the car’s speed over time?
We can break down the car's motion into three parts:
1. The car slows down from 46 mi/h to 23 mi/h in 5 seconds.
2. The car maintains a constant speed of 23 mi/h for 5 seconds.
3. The car slows down from 23 mi/h to a stop in 5 seconds.
To create a graph of this motion, we can plot the car's speed (in mi/h) on the y-axis and time (in seconds) on the x-axis.
For the first part of the motion, the car's speed is decreasing, so the graph will have a negative slope. It starts at 46 mi/h and ends at 23 mi/h after 5 seconds.
For the second part of the motion, the car's speed is constant, so the graph will be a horizontal line at y = 23.
For the third part of the motion, the car's speed is decreasing again, so the graph will have a negative slope. It starts at 23 mi/h and ends at 0 mi/h after 5 seconds.
Using this information, we can eliminate options A and C right away, since they do not show a constant speed in the second part of the motion.
Option B shows a steep line at the beginning, indicating a rapid decrease in speed, which is accurate for the first part of the motion. It then shows a horizontal line for the constant speed, which is correct. Finally, it shows a less steep line towards the end, indicating a slower decrease in speed, which is also accurate for the third part of the motion. Therefore, Option B is the correct graph for the car's speed over time.
Option D is a bit tricky, as it shows the correct slopes for the first and third parts of the motion. However, it shows a non-zero speed during the second part of the motion, which is incorrect. Therefore, Option D is not the correct graph.
Therefore, the correct graph is Option B.
Which statement describes the speed of the remote-control car over time?
A. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 5 mi/h in the next 2 seconds, and then remains at 5 mi/h for the last 5 seconds.
B. The speed of the car increases from 4 mi/h to 2 mi/h in the first 3 seconds, decreases to 5 mi/h in the next 2 seconds, and then remains at 5 mi/h for the last 5 seconds.
C. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 6 mi/h in the next second, and then remains at 6 mi/h for the last 6 seconds.
D. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 5 mi/h in the next 5 seconds, and then remains at 5 mi/h for the last 10 seconds.
The correct statement that describes the speed of the remote-control car over time is:
A. The speed of the car decreases from 4 mi/h to 2 mi/h in the first 3 seconds, increases to 5 mi/h in the next 2 seconds, and then remains at 5 mi/h for the last 5 seconds.
This matches the information given in the graph, where the car initially slows down, then speeds up, and then maintains a constant speed for the last part of the motion.
Given the function rule f(x) = x² – 5x + 1, what is the output of f(–3)?
A. 7
B. –5
C. 25
D. 5
Given the function rule f(x) = x² – 5x + 1, we need to find the output of f(-3), which means we need to substitute -3 for x in the expression for f(x) and evaluate.
f(-3) = (-3)² - 5(-3) + 1
= 9 + 15 + 1
= 25
Therefore, the output of f(-3) is 25.
Hence, the option is (C).
Suppose you earn $15 for each car you wash. Which function describes the relationship between your total earnings E and the number of cars you wash, c?
A. E(c) = c + 15
B. E(c) = 15c
C. c = 15E
D. E(c) = c – 15
Since you earn $15 for each car you wash, your total earnings E will be equal to 15 times the number of cars you wash, c.
Therefore, the function that describes the relationship between your total earnings E and the number of cars you wash, c, is:
B. E(c) = 15c
Option B states that E(c) (total earnings) is equal to 15c (15 times the number of cars you wash). This is the correct relationship between E and c, so option B is the correct answer.
Hence, the answer is (B).
The data in the table illustrate a linear function.
x –3 0 3 6
y –6 –2 2 6
What is the slope of the linear function? Which graph represents the data?
To determine the slope of the linear function represented by the data in the table, we can use the formula for the slope of a line:
slope = (change in y) / (change in x)
We can use any two points from the table to calculate the slope, but we'll use the first and last points for convenience.
slope = (6 - (-6)) / (6 - (-3))
slope = 12 / 9
slope = 4/3
Therefore, the slope of the linear function is 4/3.
To determine which graph represents the data, we can look at the slope and the y-intercept. The slope of the function is 4/3, which means that as x increases by 3, y increases by 4. The y-intercept of the function is -2, which means that the line passes through the point (0, -2).
Option A shows a line with a negative slope that does not pass through the point (0, -2), so it cannot be the correct graph.
Option B shows a line with a positive slope that passes through the point (0, -2) and has a slope that looks like it could be around 4/3. However, the line does not pass through the point (-3, -6), so it is not the correct graph.
Option C shows a line with a positive slope that passes through the points (-3, -6) and (6, 6), which are the two points given in the table. The line also passes through the point (0, -2) and has a slope of approximately 4/3. Therefore, Option C is the correct graph that represents the data.
Option D shows a line with a negative slope and a y-intercept of 5, which is not consistent with the data in the table. Therefore, it is not the correct graph.
Therefore, the slope of the linear function is 4/3 and the correct graph that represents the data is Option C.
Which hill described in the table is the steepest and why?
Street
Horizontal Distance (ft)
Vertical Rise of Street (ft)
Dixie Hill
60
20
Bell Hill
60
40
Liberty Hill
60
30
A. Bell Hill; it rises two-thirds foot for every 1 foot of horizontal travel.
B. Dixie Hill; it rises 1 foot for every 3 feet of horizontal travel.
C. Bell Hill; it rises 3 feet for every 2 feet of horizontal travel.
D. Liberty Hill; it rises 2 feet for every 1 foot of horizontal travel.
To determine which hill is the steepest, we need to calculate the slope of each hill. The slope is equal to the vertical rise divided by the horizontal distance.
For Dixie Hill:
Slope = Vertical Rise / Horizontal Distance = 20/60 = 1/3
For Bell Hill:
Slope = Vertical Rise / Horizontal Distance = 40/60 = 2/3
For Liberty Hill:
Slope = Vertical Rise / Horizontal Distance = 30/60 = 1/2
Therefore, the hill with the steepest slope is Bell Hill, with a slope of 2/3. This means that for every horizontal distance of 1 foot, the street rises by 2/3 of a foot.
Option A is incorrect, as it states that Bell Hill rises by two-thirds of a foot for every 1 foot of horizontal travel, which is the reciprocal of the slope.
Option B is incorrect, as it describes the slope of Dixie Hill, not Bell Hill.
Option C is also incorrect, as it describes the slope of Bell Hill but with the numbers reversed.
Option D is incorrect, as Liberty Hill has a slope of 1/2, which is less than the slope of Bell Hill.
Therefore, the answer is: Bell Hill; it rises three feet for every two feet of horizontal travel (with a slope of 2/3).
Which graph represents the linear function y = start fraction 1 over 5 end fractionx – 2?
The given linear function can be written in the slope-intercept form of a linear equation as:
y = (1/5)x - 2
This means that the y-intercept of the line is -2 and the slope of the line is 1/5.
To graph the line, we can use the y-intercept of -2 to plot the point (0,-2), and then use the slope of 1/5 to find additional points and connect them to graph the line.
Starting from the y-intercept of (0,-2), we can use the slope to move up one unit and right five units to find another point on the line. This gives us the point (5, -1). We can repeat this process and use the slope to move up one unit and over five units to the right to find another point on the line. This gives us the point (10, 0).
Now we can plot these points (0,-2), (5,-1), and (10,0) and connect them to graph the line.
The graph of the linear function y = (1/5)x - 2 is a line that passes through the points (0,-2), (5,-1), and (10,0), and extends in both directions.
Looking at the options given, the correct graph that represents the function y = (1/5)x - 2 is option C, as it shows a line with a slope of 1/5 and a y-intercept of -2, passing through the points (0,-2), (5,-1), and (10,0).
Therefore, the correct graph that represents the linear function y = (1/5)x - 2 is Option C.
Which graph represents the linear function y = –5x + 2?
The given linear function can be rewritten in slope-intercept form as:
y = -5x + 2
This means that the y-intercept of the line is 2 and the slope of the line is -5.
To graph the line, we can use the y-intercept of 2 to plot the point (0,2), and then use the slope of -5 to find additional points and connect them to graph the line.
Starting from the y-intercept of (0,2), we can use the slope to move down five units and to the right one unit to find another point on the line. This gives us the point (1,-3). We can repeat this process and use the slope to move down five units and over one unit to the right to find another point on the line. This gives us the point (2,-8).
Now we can plot these points (0,2), (1,-3), and (2,-8) and connect them to graph the line.
The graph of the linear function y = -5x + 2 is a line that passes through the points (0,2), (1,-3), and (2,-8), and extends in both directions.
Looking at the options given, the correct graph that represents the function y = -5x + 2 is option D, as it shows a line passing through the points (0,2), (1,-3), and (2,-8) and extending in both directions, with a slope of -5.
Therefore, the correct graph that represents the linear function y = -5x + 2 is option D.
Which function rule represents the data in the table below?
Input (x)
1
2
3
4
5
Output (y)
9
12
15
18
21
A. y = 4 + 5x
B. y = 3 + 6x
C. y = 5 + 4x
D. y = 6 + 3x
To determine the function rule that represents the data in the table, we need to find the relationship between the input (x) and the output (y). Notice that when the input increases by 1, the output increases by a constant amount.
Specifically, when x increases by 1, y increases by 3. Therefore, the slope of the linear function that represents this relationship is 3.
Additionally, we can see that when x=1, y=9. This means that the y-intercept of the linear function is 9.
Putting this information together, the function rule that represents the data in the table is:
D. y = 6 + 3x
This is the only option that matches both the slope and the y-intercept of the linear function that represents the data. If we substitute the input values x=1, x=2, x=3, x=4, and x=5 into this function rule, we obtain the output values given in the table.
Therefore, y = 6 + 3x is the correct function rule that represents the data in the table.
Hence, the option is (D).
Max charges $3.50 per hour when he mows lawns, plus $6.00 for transportation expenses. Which function rule represents the amount y Max charges to mow lawns for x hours?
A. y = 9.50x
B. y = 6.00x + 3.50
C. y = 3.50x + 6.00
D. y = 2.5x
Max charges $3.50 per hour when he mows lawns, plus $6.00 for transportation expenses. This means that the total amount y that Max charges to mow lawns for x hours can be found by adding the amount he charges for the hours worked and the amount he charges for transportation expenses.
The amount he charges for the hours worked is found by multiplying the hourly rate of $3.50 by the number of hours x:
3.50x
The amount he charges for transportation expenses is a fixed amount of $6.00.
Therefore, the function rule that represents the amount y Max charges to mow lawns for x hours is:
y = 3.50x + 6.00
Option C is the only option that matches this function rule, so it is the correct answer.
Hence, the option is (C).
The sale price of ground beef at a local grocery store is $1.49 for the first pound and $1.09 for each additional pound. Which function rule shows how the cost of ground beef, y, depends on the number of pounds, x?
A. y = 1.49x + 1.09
B. y = 1.09(x – 1) + 1.49
C. y = (1.09 + 1.49)x
D. y = 1.09x + 1.49
The cost of ground beef depends on the number of pounds purchased. We know that the first pound costs $1.49 and each additional pound costs $1.09. This means that for any number of pounds purchased, we need to add the cost of the first pound ($1.49) to the cost of the remaining pounds ($1.09 times the number of additional pounds).
We can write this relationship as the following function rule:
y = 1.49 + 1.09(x - 1)
Where x is the number of pounds of ground beef and y is the total cost. We subtract 1 from x to account for the first pound, which has a different cost than each additional pound.
This function rule simplifies to:
y = 1.09x + 0.4
Therefore, the function rule that shows how the cost of ground beef, y, depends on the number of pounds, x, is:
D. y = 1.09x + 1.49
So option D is the correct answer.
Hence, the option is (D).
Which function rule represents the data in the table?
x
–3
–2
–1
0
1
y
–1
2
5
8
11
A. y = –3x – 8
B. y = one-thirdx – 8
C. y = one-thirdx + 8
D. y = 3x + 8
Which function rule represents the data in the table?
x –3 –2 –1 0 1
y –1 2 5 8 11
A. y = –3x – 8
B. y = one-thirdx – 8
C. y = one-thirdx + 8
D. y = 3x + 8
Which function rule is represents
x –3 –2 –1 0 1
y –1 2 5 8 11
A. y = –3x – 8
B. y = one-thirdx – 8
C. y = one-thirdx + 8
D. y = 3x + 8
Which quadratic rule represents the data in the table?
x
–1
0
1
2
3
y
–4
–5
–4
–1
4
A. y = –2x² + 5
B. y = –x² + 5
C. y = x² – 5
D. y = x² + 5