Which equation is a linear function?

1) The function described in the table is a linear function.

This can be determined by seeing if there is a line that fits the entire table. To find this we first have to find the line of two points and then we can see if it fits all of the other points.

For the first points we'll use (0, 8) and (1, 10). First we have to determine the slope of these two points using the slope formula.
m = (y1 - y2)/(x1 - x2)
m = (10 - 8)/(1 - 0)
m = 2/1
m = 2.

Given a slope of 2 and a y-intercept of 8 (which we can see by the fact that 0,8 is a point on the line) we know the line between these two points is y = 2x + 8.

Now we can test all of the other points using that linear model. Each of which give us a true answer.

2) The second table gives you a linear function.

Similarly to the first problem, you need to determine the equation of a line given two points in the table and then try the rest of the values. In the second one, we'll choose the answers (0, 3) and (1, 5). First we have to determine the slope of these two points using the slope formula.

m = (y1 - y2)/(x1 - x2)
m = (5 - 3)/(1 - 0)
m = 2/1
m = 2.

Knowing this and the y-intercept as 3 (given the point 0,3 in the table) we can write the equation as y = 2x + 3. Each of the other values work in this equation.

3) This is non-linear since the exponential value of the lead x is greater than 1. All equations with exponential higher than 1 are non-linear. 

4) This is also non-linear since the exponential value of the lead x is greater than 1.

5) This is non-linear. We can tell this by finding the equation to the line similarly to how we found the line in #1 and #2.

For the first points we'll use (14, 18) and (10, 14). First we have to determine the slope of these two points using the slope formula. 
m = (y1 - y2)/(x1 - x2) 
m = (18 - 14)/(14 - 10) 
m = 4/4 
m = 1. 

Given a slope of 1,we can solve for the y-intercept using a point and slope intercept form. 

y = mx + b
14 = 1(10) + b
14 = 10 + b 
4 = b

Using this we can model the line between these two points as y = x + 4

Now we can test all of the other points using that linear model. If we choose the second set, you will see this does not work. 

y = x + 4
22 = 20 + 4
22 = 24

Which is not true. Therefore it is non-linear. 

1)

Answer

a) linear

Explanation

There is a simple way to tell if a function is linear from a table: look at the x and y-values; if the y-values are increasing or decreasing by the same amount when their corresponding x-values increases or decreases by the same amount, you have a linear function; otherwise, you don't.

Look at the table

From 0 to 1 x is increasing by 1; from 8 to 10 (the corresponding values), y is increasing by 2

From 1 to 2 x is increasing by 1; from 10 to 12, y is increasing by 2

So, every time that x increases by 1, y increases by 2; therefore, we have a linear function.

Notice that form -2 to 0 x is increasing by 2; from 4 to 8 increasing by 4, which is the same rate as before (when x increases 1, y increases 1)

Answer

xy

03

15

27

39

Explanation

When x increases by 1, y increases by 2; therefore we have a linear function.

If you look at the first and third tables,  y increases at different amounts every time x increases by 1; therefore, they are not linear functions.

In the first table when x increases by 1, y increases by 2, 4, or 10. Therefore, the table is not a linear function.

Similarly, in the third table when x increases by 1, y increases by 1, 3, or 5. Therefore, the table is not a linear function.

Answer

a) nonlinear

Explanation

A linear function is function of the form: [tex]y=mx+b[/tex] or [tex]Ay+Bx=C[/tex] where [tex]x[/tex] is the independent variable and [tex]y[/tex] is the dependent variable.

In a linear function the coefficient of the variables is always 1.

Notice that the coefficient of the independent variable [tex]x[/tex], in the function [tex]y=2x^2-4[/tex], is 2; therefore the function is nonlinear.

Answer

2=x3+y     Nonlinear

y+1=5(x−9)   Linear

7y + 2x = 12 Linear

4y = 24        Linear  

Explanation

2=x3+y The coefficient of the independent variable, x, is 3; therefore, the function is not linear.

y+1=5(x−9) We can simplify the expression to get:

y+1=5x-45

y=5x-46

Since y=5x-46 is in the form y=mx+b, we have a linear function.

7y + 2x = 12

Since 7y + 2x = 12 is a function of the form Ay + Bx = C, it is a linear function

4y = 24 We can siplify to get:

[tex]y=\frac{24}{4}[/tex]

[tex]y=6[/tex]

Since y=6 is a function of the form y = mx+b (with m=0), it is a linear function.

Answer

b) is not

Explanation

From 22 to 20, x decreases by 2; from 26 to 22, y decreases by 4. So, when x decreases by 2, y decreases by 4.

From 16 to 14, x decreases by 2; from 20 to 18, y decreases by 2. So, when x decreases by 2, y decreases by 2.

When x decreases by 2, y decreases by 4 or 2; therefore the function represented in the table is not a linear function.

(Warning) Not sure this is completley correct but this is just what I did.

Part A

Does the data for Amit’s puppy show a function? Why or why not?

It does show a function because it passes the vertical line test (no two points have the same x value).

Part B

Is the relationship for Amit’s puppy’s weight in terms of time linear or nonlinear? Explain your response.

Nonlinear because the line isn’t straight

Part C

Is the relationship between Amit’s puppy’s weight in terms of time increasing or decreasing? Explain your response.

Increasing because it is gaining weight

Part D

Does the data for Camille’s puppy show a function? Why or why not?

Yes, it does, because each input value has a unique output value

Part E

Is the relationship for Camille’s puppy’s weight in terms of time linear or nonlinear? Explain your response.

It is a linear function because the line has no curve, and the line is constant.

Part F

Is the relationship between Camille’s puppy’s weight in terms of time increasing or decreasing? Explain your response.

Increasing because as the puppy gets older it gains weight.

Part G

Does the data for Olivia’s puppy show a function? Why or why not?

Yes, it does, because each input value has a unique output value. The graph attached ( which shows the data for Camille’s puppy), that each x-value (Weeks) has a unique y-value (Weight in pounds).

Therefore, based on this and keeping in mind the explanation before, you can conclude that the data for Camille’s puppy shows a function.

Part H

Is the relationship for Olivia’s puppy’s weight in terms of time linear or nonlinear? Explain your response

Yes, it linear because it’s a straight line.

Part I

Is the relationship for Olivia’s puppy’s weight in terms of time increasing or decreasing? Explain your response. Increasing. For every week that goes by, Olivia's puppy is gaining one pound. 6-5= 1  14-13= 1. Gaining a pound every week makes the puppy’s weight increase.

Part J

Which two relationships have a y-intercept and a constant rate of change?

They all have y-intercepts and only Olivia and camilles have a constant rate of change.

Part A

To compare the linear functions, you first need to find their equations. For each of the linear functions, write an equation to represent the puppy’s weight in terms of the number of weeks since the person got the puppy.

Linear equation, y=mx+b

Exponential equation, y=a(b)×

Part B

Now you can compare the functions. In each equation, what do the slope and y-intercept represent in terms of the situation?

The y-intercept in the situation is 2/6.

Part C

Whose puppy weighed the most when the person got it? How much did it weigh?

Olivia’s puppy, it weighed 5 pounds

Part D

Whose puppy gained weight the slowest? How much did it gain per week?

Olivia’s puppy gained weight the slowest because it started off with more weight but only gained around 1 pound every week.

Part E

You can also graph the functions to compare them. Using the Edmentum Graphing Tool, graph the two linear functions. Paste a screenshot of the two functions in the space provided. How could you find which puppy had a greater initial weight from the graph? How could you find which puppy gained weight the slowest?

The edmentum graphing tool is opening up I tried it more than once but, the linear graphs would be Camille puppy and olivia's. And I could tell which one had a greater weight by how much they had at week 1 and how much they gained the weeks later. I could find which puppy gained weight the slowest by looking at the weight gained and graphed.

Step-by-step explanation:

1. The correct answer is letter (C) There is no solution since 4 = 5 is a false statement.

2. The equation that has solution is:
6x - (3x + 8) = 16x

6x - 3x - 8 = 16x

3x - 8 = 16x

-13x = 8

x = -8/13

3. The coordinate grids above shows equations y = 3x - 2 and 2x - y = 4. The correct answer is graph letter (A)

4. Select ALL of the functions.

The functions are letter (A) and letter (D).

5. Write the difference in the rates of change for these two functions

The equation in the table is

m = (Y1-Y2) / (X1-X2)

m = (4-12) / (0-2)

m = -8 / -2

m = 4

y = mx+b

4 = 4(0) + b

b = -4

So the equation of the table is y = 4x - 4.

The second equation is y = 7x + 4.

(0 , 4)

(2 , 18)

(4 , 32)

(6 , 46)

The difference rate is (0, 6, 12, 18) is 6n.

6. The function is linear.

7. The function is nonlinear.

8. The function is linear.

9. The function is nonlinear.

10. The correct answer is letter (C).

11. The correct answer is letter D. Linear function with positive slope

Graphing them and see what the difference is