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
D. y = x² + 5
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
D. y = 3x + 8
A 154-lb person burns 420 calories per hour riding an exercise bicycle at a rate of 15 mi/hr. Write a function rule to represent the total calories burned over time by that person. Explain how the information in the problem relates to the function.
Let's start by identifying the variables involved in this problem. We have:
- x = time spent riding the exercise bicycle (measured in hours)
- y = total calories burned by the person
According to the problem, the person burns 420 calories per hour riding the exercise bicycle at a rate of 15 mi/hr. So we can say that the rate of calories burned (in calories per hour) is constant and equal to 420. We also know that the person is riding at a rate of 15 miles per hour, but we don't need that value to calculate the total calories burned.
To find the function rule, we need to express the total calories burned as a function of the time spent riding. We can do this by multiplying the rate of calories burned by the time spent riding, like this:
y = 420x
This function rule represents the total calories burned over time by the person. It tells us that for each hour spent riding the exercise bicycle, the person burns 420 calories. To find the total calories burned for any amount of time, we simply plug in the value of x (in hours) into the function rule.
In summary, the information in the problem gives us the rate of calories burned (in calories per hour) and the time spent riding (in hours), which we use to calculate the total calories burned. This relationship is expressed by the function rule y = 420x.
Explain how to write a function rule from the table below. Then write a function rule.
x 0 2 4 6
y 2 1 0 –1
To write a function rule from a table, we need to look for a pattern in the x and y values. In other words, we need to figure out how the y values are related to the x values.
Looking at the table, we can see that as x increases by 2, y decreases by 1. Specifically, y starts at 2, then goes to 1, then to 0, then to -1 as x increases by 2 each time. This tells us that the function is decreasing by a constant amount as x increases. We can conclude that the function is linear, and we can use the slope-intercept form of the equation to write the function rule.
The slope-intercept form of a linear equation is: y = mx + b, where m is the slope and b is the y-intercept. We can use the information from the table to find the slope and y-intercept.
First, we can find the slope by calculating the change in y over the change in x. From x = 0 to x = 2, the change in y is -1 - 2 = -3, and the change in x is 2 - 0 = 2. So the slope is -3/2.
Next, we can find the y-intercept by looking at the y value when x = 0. From the table, we see that y = 2 when x = 0. So the y-intercept is 2.
Putting these pieces together, the function rule for this table is:
y = -3/2 x + 2
This function represents a linear relationship between the x and y values in the table. It tells us that for every increase of 2 in x, y decreases by 3/2. The y-intercept of 2 tells us that when x = 0, y is equal to 2.
What are the next three terms in the sequence?
–1, 9, 19, 29, …
A. 38, 37, 32
B. 40, 51, 62
C. 39, 49, 59
D. 38, 47, 56
To find the pattern in this sequence, we should look at the differences between the terms:
1. The difference between the first and second terms is 9 - (-1) = 10.
2. The difference between the second and third terms is 19 - 9 = 10.
3. The difference between the third and fourth terms is 29 - 19 = 10.
Since the differences between the terms are all the same, we can conclude that the sequence is an arithmetic sequence with a common difference of 10.
Using this information, we can find the next three terms in the sequence:
1. To get to the fifth term, we add 10 to the fourth term, which is 29. So the fifth term is 39.
2. To get to the sixth term, we add 10 to the fifth term, which is 39. So the sixth term is 49.
3. To get to the seventh term, we add 10 to the sixth term, which is 49. So the seventh term is 59.
Therefore, the next three terms in the sequence are 39, 49, and 59, which is option C.
Geoff planted dahlias in his garden. Dahlias have bulbs that divide and reproduce underground. In the first year, Geoff’s garden produced 8 bulbs. In the second year, it produced 16 bulbs, and in the third year it produced 32 bulbs. If this pattern continues, how many bulbs should Geoff expect in the sixth year?
A. 64 bulbs
B. 512 bulbs
C. 128 bulbs
D. 256 bulbs
The pattern suggests that the number of bulbs is doubling each year. We can see this as follows:
Year 1: 8 bulbs
Year 2: 2 * 8 bulbs = 16 bulbs
Year 3: 2 * 16 bulbs = 32 bulbs
Year 4: 2 * 32 bulbs = 64 bulbs
Year 5: 2 * 64 bulbs = 128 bulbs
Year 6: 2 * 128 bulbs = 256 bulbs
Therefore, Geoff should expect 256 bulbs in the sixth year, which is option D.
What are the first four terms of the sequence represented by the expression n(n – 2) – 3?
A. –5, –2, 1, 4
B. –4, –3, 0, 5
C. –3, 0, 3, 6
D. –2, 0, 2, 4
To find the first four terms of this sequence, we can evaluate the expression for n = 1, 2, 3, and 4:
n = 1: 1(1 - 2) - 3 = -4
n = 2: 2(2 - 2) - 3 = -3
n = 3: 3(3 - 2) - 3 = 0
n = 4: 4(4 - 2) - 3 = 5
Therefore, the first four terms of the sequence are -4, -3, 0, and 5, which is option B.
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?
A. A number of points are shown on a graph. The points are connected by line segments.The y-axis is labeled Speed left-parenthesis miles per hour right-parenthesis and is numbered from 0 to 60 with a step size of 10. The x-axis is labeled Time left-parenthesis s right-parenthesis and is numbered from 0 to 15 with a step size of 5. The line segments connect these approximate points: left-parenthesis 0 comma 22 right-parenthesis and go through approximately left-parenthesis 5 comma 22 right-parenthesis, left-parenthesis 10 comma 46 right-parenthesis, and left-parenthesis 15 comma 46 right-parenthesis.
B. A number of points are shown on a graph. The points are connected by line segments.The y-axis is labeled Speed left-parenthesis miles per hour right-parenthesis and is numbered from 0 to 60 with a step size of 10. The x-axis is labeled Time left-parenthesis s right-parenthesis and is numbered from 0 to 15 with a step size of 5. The line segments connect these approximate points: left-parenthesis 0 comma 46 right-parenthesis and go through approximately left-parenthesis 5 comma 22 right-parenthesis, left-parenthesis 10 comma 22 right-parenthesis, and left-parenthesis 15 comma 0 right-parenthesis.
C. A number of points are shown on a graph. The points are connected by line segments.The y-axis is labeled Speed left-parenthesis miles per hour right-parenthesis and is numbered from 0 to 60 with a step size of 10. The x-axis is labeled Time left-parenthesis s right-parenthesis and is numbered from 0 to 15 with a step size of 5. The line segments connect these approximate points: left-parenthesis 0 comma 24 right-parenthesis and go through approximately left-parenthesis 5 comma 46 right-parenthesis, left-parenthesis 10 comma 46 right-parenthesis, and left-parenthesis 15 comma 0 right-parenthesis.
D. A number of points are shown on a graph. The points are connected by line segments.The y-axis is labeled Speed left-parenthesis miles per hour right-parenthesis and is numbered from 0 to 60 with a step size of 10. The x-axis is labeled Time left-parenthesis s right-parenthesis and is numbered from 0 to 15 with a step size of 5. The line segments connect these approximate points: left-parenthesis 0 comma 0 right-parenthesis and go through approximately left-parenthesis 5 comma 23 right-parenthesis, left-parenthesis 10 comma 23 right-parenthesis, and continues on through just above left-parenthesis 15 comma 45 right-parenthesis.
The car traveling at 46 mi/h slows to a speed of 23 mi/h in 5 seconds, maintains that speed for 5 seconds, and then slows to a stop after 5 more seconds. We can use this information to create the graph of the car’s speed over time.
At the beginning of the 5-second interval, the car is going 46 mi/h, and at the end of that interval, it is going 23 mi/h. This suggests that the speed of the car is decreasing at a constant rate of 4.6 mi/h per second over that time period (since it is covering a difference of 23-46 = -23 miles in 5 seconds). This rate of decrease continues for another 5 seconds, during which the speed remains constant at 23 mi/h. Finally, the car slows to a stop over another 5 seconds, decreasing at the same rate as before (4.6 mi/h per second).
Putting all of this information together, we can see that the graph that shows the car’s speed over time should look like this:
A. A number of points are shown on a graph. The points are connected by line segments. The y-axis is labeled Speed (miles per hour) and is numbered from 0 to 60 with a step size of 10. The x-axis is labeled Time (s) and is numbered from 0 to 15 with a step size of 5. The line segments connect these approximate points: (0, 22) and go through approximately (5, 22), (10, 46), and (15, 46).
Therefore, option A is the correct choice.
Use the graph below to answer the question that follows.
A graph titled 'Remote-Control Car' shows 'Time' in seconds from 0 to 10 on the x-axis and 'Speed' in miles per hour from 0 to 10 on the y-axis.
A line segment connects left-parenthesis 0 comma 4 right-parenthesis and left-parenthesis 3 comma 2 right-parenthesis.
A second line segment connects left-parenthesis 3 comma 2 right-parenthesis to left-parenthesis 5 comma 5 right-parenthesis.
A horizontal line segment connects left-parenthesis 5 comma 5 right-parenthesis to the point left-parenthesis 10 comma 5 right-parenthesis.
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 graph shows that the speed of the remote-control car changes at different times. Let's break down each section of the graph:
1. From 0 to 3 seconds, the speed of the car decreases from 4 mi/h to 2 mi/h. This is shown by the line segment that connects the point (0, 4) to the point (3, 2).
2. From 3 to 5 seconds, the speed of the car increases from 2 mi/h to 5 mi/h. This is shown by the line segment that connects the point (3, 2) to the point (5, 5).
3. From 5 to 10 seconds, the speed of the car remains constant at 5 mi/h. This is shown by the horizontal line segment that connects the point (5, 5) to the point (10, 5).
Based on this information, we can see that 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.
Therefore, option A is the correct choice.
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?
A. Negative Start Fraction 4 over 3 End Fraction semicolon. A line passes through a graph.The line passes through the points left-parenthesis negative 3 comma negative 6 right-parenthesis left-parenthesis zero comma negative 2 right-parenthesis left-parenthesis 3 comma 2 right-parenthesis left-parenthesis 6 comma 6 right-parenthesis.
B. Negative Start Fraction 3 over 4 End Fraction semicolon. A line passes through a graph.The line passes through the points left-parenthesis negative 6 comma negative 3 right-parenthesis left-parenthesis negative 2 comma zero right-parenthesis left-parenthesis 2 comma 3 right-parenthesis left-parenthesis 6 comma 6 right-parenthesis.
C. Start Fraction 3 over 4 End Fraction semicolon. A line passes through a graph.The line passes through the points left-parenthesis negative 6 comma negative 3 right-parenthesis left-parenthesis negative 2 comma zero right-parenthesis left-parenthesis 2 comma 3 right-parenthesis left-parenthesis 6 comma 6 right-parenthesis.
D. Start Fraction 4 over 3 End Fraction semicolon. A line passes through a graph.
To find the slope of the linear function represented by the data in the table, we can use the formula for finding the slope of a line passing through two points. Let's use the first two points:
(x1, y1) = (-3, -6)
(x2, y2) = (0, -2)
The slope of the line passing through these points is:
slope = (y2 - y1) / (x2 - x1)
slope = (-2 - (-6)) / (0 - (-3))
slope = 4 / 3
So the slope of the linear function is 4/3.
The graph that represents the data should be a line passing through the points in the table. We can use the slope-intercept form of the equation for a line to write the equation for this linear function:
y = mx + b
where m is the slope and b is the y-intercept. Since we know the slope is 4/3, we just need to find the y-intercept. We can use any point in the table to do this. Let's use (0, -2):
-2 = (4/3)(0) + b
Solving for b, we get:
b = -2
So the equation for the linear function is:
y = (4/3)x - 2
Based on this equation, we can see that the correct graph that represents the data is:
A. Negative (4/3); A line passes through a graph. The line passes through the points (-3, -6), (0, -
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.