Graphing Data - Algebra II
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What is the next number in this sequence: 8, 27, 64, 125 ?
What is the next number in this sequence: 8, 27, 64, 125 ?
Find the pattern of the sequence:
This pattern is 
so the next number in the sequence would be 
Find the pattern of the sequence:
This pattern is
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Find the next 2 numbers in this sequence: 33, 46, 72, 111
Find the next 2 numbers in this sequence: 33, 46, 72, 111
Find the pattern in this sequence of numbers:
In this case, the pattern is adding 13n to the previous number where n= how many numbers came before the current number.
so the first number we are looking for would be:
the second number we are looking for would be:
Find the pattern in this sequence of numbers:
In this case, the pattern is adding 13n to the previous number where n= how many numbers came before the current number.
so the first number we are looking for would be:
the second number we are looking for would be:
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The amount of water inside of a leaky boat is measured periodically after the boat has been in the water in different periods of time and are found to have a linear relationship. The results are given in the following chart:
Time in water (mins) Amount of water in the boat (gal) 0 0 6 4.8 19 15.2 28 22.4
Using the method of linear extrapolation based on the data from the table, how much water would you expect to be in the boat after 
minutes?
The amount of water inside of a leaky boat is measured periodically after the boat has been in the water in different periods of time and are found to have a linear relationship. The results are given in the following chart:
| Time in water (mins) | Amount of water in the boat (gal) |
|---|---|
| 0 | 0 |
| 6 | 4.8 |
| 19 | 15.2 |
| 28 | 22.4 |
Using the method of linear extrapolation based on the data from the table, how much water would you expect to be in the boat after 
To extrapolate the results of the study out to 53 minutes, first we have to determine an equation representing the relationship between time passed and amount of water; we can write our equation in slope-intercept form:
Where our y-axis represents amount of water and the x-axis represents time. We can pick 2 points and label them Point 1 and Point 2; looking back at the table:
Time in water (mins) Amount of water in the boat (gal) 0 0 6 4.8 19 15.2 28 22.4
We label Point 1 as 
and Point 2 as 
; we plug these points into the slope formula as follows:
So, the slope of our line that describes how much water is in the boat is 
; to find our 
term, the y-intercept, we need to pick a point on the graph and plug in our slope to solve for y-intercept. Let's once again choose the point 
:
Simplify the expression and we find that b=0, so our slope-intercept equation is:
Plugging in a value of 53 for 
, we find that:
So the answer is 42.4 gallons of water in the boat after 53 minutes.
To extrapolate the results of the study out to 53 minutes, first we have to determine an equation representing the relationship between time passed and amount of water; we can write our equation in slope-intercept form:
Where our y-axis represents amount of water and the x-axis represents time. We can pick 2 points and label them Point 1 and Point 2; looking back at the table:
| Time in water (mins) | Amount of water in the boat (gal) |
|---|---|
| 0 | 0 |
| 6 | 4.8 |
| 19 | 15.2 |
| 28 | 22.4 |
We label Point 1 as
So, the slope of our line that describes how much water is in the boat is
Simplify the expression and we find that b=0, so our slope-intercept equation is:
Plugging in a value of 53 for
So the answer is 42.4 gallons of water in the boat after 53 minutes.
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varies directly with the square root of 
. If 
, then 
. What is the value of 
if 
?
If 
varies directly with the square root of 
, then for some constant of variation 
,
If 
, then 
; therefore, the equation becomes
,
or
.
Divide by 5 to get 
, making the equation
.
If 
, then 
.
If
If
or
Divide by 5 to get
If
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If 
varies directly with 
and when 
, 
due to the effect of a constant, what is the value of 
when 
?
If
Since 
varies directly with 
, 
where 
is a constant.
1. Solve for 
when 
and 
.
2. Use your equation to solve for 
when 
.
Since
1. Solve for
2. Use your equation to solve for
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If 
varies indirectly with 
and when 
, 
due to the effect of a constant, what is the value of 
when 
?
If
Since 
varies indirectly with 
, 
1. Solve for 
when 
and 
.
2. Use the equation you found to solve for 
when 
.
Since
1. Solve for
2. Use the equation you found to solve for
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varies directly with 
. If 
, what is 
if 
?
1. Since 
varies directly with 
:
with K being some constant.
2. Solve for K using the x and y values given:
3. Use the equation you solved for to find the value of y:
1. Since
2. Solve for K using the x and y values given:
3. Use the equation you solved for to find the value of y:
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varies inversely with 
. If 
, then what is 
equal to when 
?
1. Since 
varies indirectly with 
:
2. Use the given x and y values to determine the value of K:
3. Using the equation along with the value of K, find the value of y when x=5:
1. Since
2. Use the given x and y values to determine the value of K:
3. Using the equation along with the value of K, find the value of y when x=5:
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varies directly with 
and when 
. What is 
when 
?
1. Since 
varies directly with 
:
2. Use the values given for x and y to solve for K:
3. Use your new equation with the K you solved for to solve for y when x=27:
1. Since
2. Use the values given for x and y to solve for K:
3. Use your new equation with the K you solved for to solve for y when x=27:
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varies inversely with 
. When 
. What is the value of 
when 
?
1. Since y varies indirectly with 
:
2. Solve for K using the x and y values given:
3. Using the equation you created by solving for K, find y when x=100:
1. Since y varies indirectly with
2. Solve for K using the x and y values given:
3. Using the equation you created by solving for K, find y when x=100:
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Given the two following points, use interpolation to determine the best estimate for the value 
, 
Given the two following points, use interpolation to determine the best estimate for the value
Using our two known points, we can use interpolation to determine the value at any point between them with the following formula:
Where 
is our first given point, 
is our second given point, and 
is the point we want to find. We know our two given points, as well as the x value of our unknown point, so now all we must do is plug in all of our known values and solve for y, our only unknown:
Using our two known points, we can use interpolation to determine the value at any point between them with the following formula:
Where
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The output of a factory in units per day versus the number of employees working is plotted on the graph below, with the following data points collected:
(Workers, Units of output per day):
Assuming a linear relationship, interpolate to find how many units will be made per day if 
workers are present.
The output of a factory in units per day versus the number of employees working is plotted on the graph below, with the following data points collected:
(Workers, Units of output per day):
Assuming a linear relationship, interpolate to find how many units will be made per day if
We want to do a linear interpolation since the relationship between workers and units can be assumed to be linear. This means there is a constant slope between the points, so the slope between two known points will be equal to the slope between the point we are trying to find and some known point. This is expressed in the relation:
,
where 
and 
are the points we want to find and 
and 
are known. We choose the known points to be those that are just to the left and right of the point we are trying to find,
and 
.
Plugging these into our interpolation formula and knowing 
, we can find 
, the units output per day.
.
Simplifying and rearranging to solve for 
:
.
So there are 
units produced when the number of workers is 
.
We want to do a linear interpolation since the relationship between workers and units can be assumed to be linear. This means there is a constant slope between the points, so the slope between two known points will be equal to the slope between the point we are trying to find and some known point. This is expressed in the relation:
where
Plugging these into our interpolation formula and knowing
Simplifying and rearranging to solve for
So there are
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Given the points
and 
, use linear interpolation to find the value of 
when 
.
Given the points
Use the formula for interpolation to determine the value of y:
We will use (30, 51) as our x2 and y2 and (20, 36) as our x1 and y1 and we will solve for y using 26.5 for x.
Use the formula for interpolation to determine the value of y:
We will use (30, 51) as our x2 and y2 and (20, 36) as our x1 and y1 and we will solve for y using 26.5 for x.
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Given 
and 
use linear interpolation to find 
when 
.
Given
Use the formula for interpolation:
We will use (30, 15) as x2 and y2, (15, 10) as x1 and y1, and solve for y when x=17.9:
Use the formula for interpolation:
We will use (30, 15) as x2 and y2, (15, 10) as x1 and y1, and solve for y when x=17.9:
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Mary measures her height every year on her birthday, starting at 11 until she turns 16. She wants to make a table with all the information gathered, but discovers she lost the piece of paper on which she wrote her height down on her 14th birthday. Her incomplete table looks like this:
Age (years) Height (inches) 11 47.5 12 50.25 13 53 14 ? 15 58.5 16 61.25
Using the method of linear interpolation, which of the following is the closest estimate of Mary's height on her 14th birthday?
Mary measures her height every year on her birthday, starting at 11 until she turns 16. She wants to make a table with all the information gathered, but discovers she lost the piece of paper on which she wrote her height down on her 14th birthday. Her incomplete table looks like this:
| Age (years) | Height (inches) |
|---|---|
| 11 | 47.5 |
| 12 | 50.25 |
| 13 | 53 |
| 14 | ? |
| 15 | 58.5 |
| 16 | 61.25 |
Using the method of linear interpolation, which of the following is the closest estimate of Mary's height on her 14th birthday?
Using linear interpolation means that we draw a line between the points on our data set and use that line to estimate a value that lies between two data points; in this case, we have the data from Mary's 13th and 15th birthdays, so we can describe a line between those two points and estimate her height at 14. Our line will be written in slope-intercept form:
Where the variable 
represents Mary's Age in years and the variable 
represents her height in inches. First, we need to find the slope. Using 2 points on our table 
and 
as point 1 and point 2, respectively, we plug these values into our slope formula:
Next, we find the y-intercept by plugging in our slope (which we just found) and a point from our table (we'll stick with 
) and solving for 
:
Simplify:
Subtract 35.75 from both sides to solve for 
:
The equation of our interpolation line is:
So, to get an estimate of Mary's height on her 14th birthday, we plug in 
and solve for 
:
Our estimate of Mary's height at 
is 
Using linear interpolation means that we draw a line between the points on our data set and use that line to estimate a value that lies between two data points; in this case, we have the data from Mary's 13th and 15th birthdays, so we can describe a line between those two points and estimate her height at 14. Our line will be written in slope-intercept form:
Where the variable
Next, we find the y-intercept by plugging in our slope (which we just found) and a point from our table (we'll stick with
Simplify:
Subtract 35.75 from both sides to solve for
The equation of our interpolation line is:
So, to get an estimate of Mary's height on her 14th birthday, we plug in
Our estimate of Mary's height at
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Find the value of 
when 
given the points 
and 
.
Find the value of
Write the interpolation formula.
Identify and substitute the values.
Simplify the fraction.
The answer is: 
Write the interpolation formula.
Identify and substitute the values.
Simplify the fraction.
The answer is:
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The ages of the students at GW High School are normally distributed with a mean of 
and a standard deviation of 
years.
What is the proportion of students that are younger than 
years old?
The ages of the students at GW High School are normally distributed with a mean of
What is the proportion of students that are younger than
This question relates to the 
rule of normal distribution. We know that 
of the data are within 
standard deviations from the mean.
In this case this means that 
of the students are between
and 
and 
and 
Therefore we have 
of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 
is the same as the proportion of students above 
.
Thus the right answer is 
or 
.
This question relates to the
In this case this means that
Therefore we have
Thus the right answer is
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The scores for your recent english test follow a normal distribution pattern. The mean was a 75 and the standard deviation was 4 points. What percentage of the scores were below a 67?
The scores for your recent english test follow a normal distribution pattern. The mean was a 75 and the standard deviation was 4 points. What percentage of the scores were below a 67?
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations of the mean.
In this case, 95% of the students' scores were between:
75-(2 x 4) and 75+(2 x 4)
or between a 67 and a 83, with equal amounts of the leftover 5% of scores above and below those scores. This would mean that 2.5% of the students scored below a 67% on the test.
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations of the mean.
In this case, 95% of the students' scores were between:
75-(2 x 4) and 75+(2 x 4)
or between a 67 and a 83, with equal amounts of the leftover 5% of scores above and below those scores. This would mean that 2.5% of the students scored below a 67% on the test.
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Your class just took a math test. The mean test score was a 78 with a standard deviation of 2 points. With this being the case, 99.7% of the class scored between what two scores?
Your class just took a math test. The mean test score was a 78 with a standard deviation of 2 points. With this being the case, 99.7% of the class scored between what two scores?
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations (in either direction) of the mean.
In this case, 99.7% of the students' scores were between 3 standard deviation above the mean and 3 standard deviations below the mean:
78-(2 x 3) and 78+(2 x 3)
or between a 72 and an 84.
Use the 68-95-99.7 rule which states that 68% of the data is within 1 standard deviation (in either direction) of the mean, 95% is within 2 standard deviations, and 99.7% is within 3 standard deviations (in either direction) of the mean.
In this case, 99.7% of the students' scores were between 3 standard deviation above the mean and 3 standard deviations below the mean:
78-(2 x 3) and 78+(2 x 3)
or between a 72 and an 84.
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All of the following statements regarding a Normal Distribution are true except:
All of the following statements regarding a Normal Distribution are true except:
The graph of a normally-distributed data set is symmetrical.
The graph of a normally-distributed data set has a single, central peak at the mean of the data set that it describes.
The graph of a normally-distributed data set will vary based only upon the mean and the standard deviation of the set that it describes.
The graph of a normally-distributed data set with a higher standard deviation will be wider than the graph of a normally-distributed data set with a lower standard deviation.
The question asks us to find the statement that is not true; however, all statements are true so the correct response is "All of these are true."
The graph of a normally-distributed data set is symmetrical.
The graph of a normally-distributed data set has a single, central peak at the mean of the data set that it describes.
The graph of a normally-distributed data set will vary based only upon the mean and the standard deviation of the set that it describes.
The graph of a normally-distributed data set with a higher standard deviation will be wider than the graph of a normally-distributed data set with a lower standard deviation.
The question asks us to find the statement that is not true; however, all statements are true so the correct response is "All of these are true."
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