Question 1

You can get the change in a linear function from the formula
Change in function = Slope × Change in variable.

Accepted Answer

The change in a linear function is determined by multiplying the slope of the function by the change in the variable, which aligns with the formula Δy = mΔx, where Δy is the change in the function, m is the slope, and Δx is the change in the variable.

Question 2

If a population doubles in size each day, then the population is a linear function of time with slope 2.

Accepted Answer

The population is an example of exponential growth, not linear growth, because it multiplies by a constant factor (in this case, 2) over equal intervals of time.

Question 3

The distance  , in miles, I can drive without stopping for gas depends on the amount  , in gallons, of fuel in the tank.The relationship is ​  .
​
If I add 5 gallons of fuel to my tank, how much farther can I drive?

Accepted Answer

A

Question 4

When you order merchandise from a web site you pay shipping costs  , in dollars.The shipping costs depend on the weight  , in pounds, of the merchandise.The relationship is ​  ​
If my order weighs 5 pounds more than yours, how do shipping costs compare?

Accepted Answer

The answer of When you order merchandise from a web...

Question 5

The rate of change of a linear function depends on the variable of the function.

Accepted Answer

The rate of change of a linear function is constant and is determined by the slope of the line, not by the variable of the function.

Question 6

My business pays tax on income, and the tax paid is a linear function of income.If my income is $10420 then the tax owed is $929.60.If my income is $11309 then the tax owed is $1000.72.How much tax do I owe if my income is $11874? ​

Accepted Answer

The tax function can be represented as T(x) = mx + b, where T(x) is the tax owed, x is the income, m is the slope of the line, and b is the y-intercept. To find m, use the two given points: (10420, 929.60) and (11309, 1000.72). The slope m = (1000.72 - 929.60) / (11309 - 10420) = 71.12 / 889 = 0.08. To find b, substitute one of the points into the equation: 929.60 = 0.08(10420) + b, solving for b gives b = 929.60 - 833.60 = 96. Using the equation T(x) = 0.08x + 96, for x = 11874, T(11874) = 0.08(11874) + 96 = 950.92 + 96 = 1045.92.

Question 7

You pay $10 to register at a web site.After registering you pay 59 cents for each song you download.Then the total amount you pay for music at this site is a linear function of the number of songs you download.

Accepted Answer

The total amount paid is a linear function because it follows the form y = mx + b, where m is the constant rate of 59 cents per song, x is the number of songs, and b is the initial $10 registration fee.

Question 8

You can get the rate of change of a linear function by dividing the change in the variable by the change in the function.

Accepted Answer

The rate of change of a linear function is obtained by dividing the change in the function by the change in the variable, not the other way around.

Question 9

If I add $5 to a cookie jar each week, then the amount of money in the cookie jar is a linear function of time with slope 5 dollars per week.

Accepted Answer

The amount of money increases by a constant rate ($5) for each unit increase in time (week), which is the definition of a linear function, where the slope represents the rate of change.

Question 10

The money I pay for music is a linear function of the number of songs I download.If I download 7 songs, then I pay $11.98.If I download 13 songs, then I pay $18.82.How much do I pay if I download 23 songs? ​
​

Accepted Answer

The answer of The money I pay for music is...

Question 11

A linear function is a function with a constant rate of change.

Accepted Answer

A linear function is defined by a linear equation, which represents a straight line on a graph. This line has a constant slope, which means it has a constant rate of change.

Question 12

The sales tax  , in dollars, I have to pay when I buy  dollars worth of merchandise is given by  . ​
If I decide to buy $44 more in merchandise, how much additional sales tax will I have to pay?

Accepted Answer

The answer of The sales tax  , in dollars,...

Question 13

It is a fact that if you drop a rock, its velocity increases by 32 feet per second for each additional second it falls.We can conclude that the velocity of the rock is a linear function of time since the rock was dropped.

Accepted Answer

The velocity of the rock increases by a constant amount (32 feet per second) for each second of time, which is the definition of a linear function.

Question 14

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity  , in feet per second, after  seconds is given by ​  ​
By how much does velocity increase over a 4-second period?

Accepted Answer

The answer of If a rock is thrown downward with...

Question 15

If  is increased by 5 units from  , then  is increased by 9 units.If  is increased by an additional 5 units, then  is increased by 11 units.We can conclude that  a linear function of  .

Accepted Answer

The answer of If  is increased by 5 units...

Question 16

It is a fact that if you drop a rock, it will fall 16 feet the first second of the fall.The next second it will fall 48 feet.In view of this fact, I can conclude that the distance a rock falls is a linear function of the time since the rock was dropped.

Accepted Answer

The distance a rock falls due to gravity is not linear but quadratic, as the acceleration due to gravity causes the velocity (and thus the distance fallen) to increase with the square of the time elapsed.

Question 17

If snow is falling at a rate of 0.5 inch each hour, then the depth of snow on the ground is a linear function of time as long as the snow continues to fall.

Accepted Answer

The depth of snow increases by a constant amount (0.5 inch) for each hour that passes, which is the definition of a linear function.

Question 18

The slope of the graph of a linear function is the same as the rate of change of the linear function.

Accepted Answer

The slope of the graph of a linear function represents how much the y-value changes for a unit change in the x-value, which is exactly the definition of the rate of change for the function.

Question 19

The height of a staircase is a linear function of the number of steps.

Accepted Answer

The height of a staircase increases in a constant manner with each step added, making it a linear function of the number of steps.

Question 20

The graph of a linear function is a straight line.

Accepted Answer

A linear function has the form f(x) = mx + b, where m and b are constants, and its graph is always a straight line.

Quiz 12: Linear Functions

You can get the change in a linear function from the formula Change in function = Slope × Change in variable.

If a population doubles in size each day, then the population is a linear function of time with slope 2.

The distance , in miles, I can drive without stopping for gas depends on the amount , in gallons, of fuel in the tank.The relationship is . If I add 5 gallons of fuel to my tank, how much farther can I drive?

When you order merchandise from a web site you pay shipping costs , in dollars.The shipping costs depend on the weight , in pounds, of the merchandise.The relationship is If my order weighs 5 pounds more than yours, how do shipping costs compare?

The rate of change of a linear function depends on the variable of the function.

My business pays tax on income, and the tax paid is a linear function of income.If my income is $10420 then the tax owed is $929.60.If my income is $11309 then the tax owed is $1000.72.How much tax do I owe if my income is $11874?

You pay $10 to register at a web site.After registering you pay 59 cents for each song you download.Then the total amount you pay for music at this site is a linear function of the number of songs you download.

You can get the rate of change of a linear function by dividing the change in the variable by the change in the function.

If I add $5 to a cookie jar each week, then the amount of money in the cookie jar is a linear function of time with slope 5 dollars per week.

The money I pay for music is a linear function of the number of songs I download.If I download 7 songs, then I pay $11.98.If I download 13 songs, then I pay $18.82.How much do I pay if I download 23 songs?

A linear function is a function with a constant rate of change.

The sales tax , in dollars, I have to pay when I buy dollars worth of merchandise is given by . If I decide to buy $44 more in merchandise, how much additional sales tax will I have to pay?

It is a fact that if you drop a rock, its velocity increases by 32 feet per second for each additional second it falls.We can conclude that the velocity of the rock is a linear function of time since the rock was dropped.

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity , in feet per second, after seconds is given by By how much does velocity increase over a 4-second period?

If is increased by 5 units from , then is increased by 9 units.If is increased by an additional 5 units, then is increased by 11 units.We can conclude that a linear function of .

It is a fact that if you drop a rock, it will fall 16 feet the first second of the fall.The next second it will fall 48 feet.In view of this fact, I can conclude that the distance a rock falls is a linear function of the time since the rock was dropped.

If snow is falling at a rate of 0.5 inch each hour, then the depth of snow on the ground is a linear function of time as long as the snow continues to fall.

The slope of the graph of a linear function is the same as the rate of change of the linear function.

The height of a staircase is a linear function of the number of steps.

The graph of a linear function is a straight line.

Quiz 12: Linear Functions

You can get the change in a linear function from the formula Change in function = Slope × Change in variable.

If a population doubles in size each day, then the population is a linear function of time with slope 2.

The distance , in miles, I can drive without stopping for gas depends on the amount , in gallons, of fuel in the tank.The relationship is ​ . ​ If I add 5 gallons of fuel to my tank, how much farther can I drive?

When you order merchandise from a web site you pay shipping costs , in dollars.The shipping costs depend on the weight , in pounds, of the merchandise.The relationship is ​ ​ If my order weighs 5 pounds more than yours, how do shipping costs compare?

The rate of change of a linear function depends on the variable of the function.

My business pays tax on income, and the tax paid is a linear function of income.If my income is $10420 then the tax owed is $929.60.If my income is $11309 then the tax owed is $1000.72.How much tax do I owe if my income is $11874? ​

You pay $10 to register at a web site.After registering you pay 59 cents for each song you download.Then the total amount you pay for music at this site is a linear function of the number of songs you download.

You can get the rate of change of a linear function by dividing the change in the variable by the change in the function.

If I add $5 to a cookie jar each week, then the amount of money in the cookie jar is a linear function of time with slope 5 dollars per week.

The money I pay for music is a linear function of the number of songs I download.If I download 7 songs, then I pay $11.98.If I download 13 songs, then I pay $18.82.How much do I pay if I download 23 songs? ​ ​

A linear function is a function with a constant rate of change.

The sales tax , in dollars, I have to pay when I buy dollars worth of merchandise is given by . ​ If I decide to buy $44 more in merchandise, how much additional sales tax will I have to pay?

It is a fact that if you drop a rock, its velocity increases by 32 feet per second for each additional second it falls.We can conclude that the velocity of the rock is a linear function of time since the rock was dropped.

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity , in feet per second, after seconds is given by ​ ​ By how much does velocity increase over a 4-second period?

If is increased by 5 units from , then is increased by 9 units.If is increased by an additional 5 units, then is increased by 11 units.We can conclude that a linear function of .

It is a fact that if you drop a rock, it will fall 16 feet the first second of the fall.The next second it will fall 48 feet.In view of this fact, I can conclude that the distance a rock falls is a linear function of the time since the rock was dropped.

If snow is falling at a rate of 0.5 inch each hour, then the depth of snow on the ground is a linear function of time as long as the snow continues to fall.

The slope of the graph of a linear function is the same as the rate of change of the linear function.

The height of a staircase is a linear function of the number of steps.

The graph of a linear function is a straight line.

The distance , in miles, I can drive without stopping for gas depends on the amount , in gallons, of fuel in the tank.The relationship is . If I add 5 gallons of fuel to my tank, how much farther can I drive?

When you order merchandise from a web site you pay shipping costs , in dollars.The shipping costs depend on the weight , in pounds, of the merchandise.The relationship is If my order weighs 5 pounds more than yours, how do shipping costs compare?

My business pays tax on income, and the tax paid is a linear function of income.If my income is $10420 then the tax owed is $929.60.If my income is $11309 then the tax owed is $1000.72.How much tax do I owe if my income is $11874?

The money I pay for music is a linear function of the number of songs I download.If I download 7 songs, then I pay $11.98.If I download 13 songs, then I pay $18.82.How much do I pay if I download 23 songs?

The sales tax , in dollars, I have to pay when I buy dollars worth of merchandise is given by . If I decide to buy $44 more in merchandise, how much additional sales tax will I have to pay?

If a rock is thrown downward with an initial velocity of 8 feet per second, then its velocity , in feet per second, after seconds is given by By how much does velocity increase over a 4-second period?