## Envision Math Grade 8 Volume 1 Chapter 3 Use Functions To Model Relationships

**Page 183 Exercise 1 Answer**

We need to determine the properties of the function.

We can use more different representations to help us determine the properties of functions, and those representations are the graphs and table data.

When we use graph we can see more easier whether the function is linear or not but it is more difficult to determine what is the initial value when we have bigger numbers.

The table data should show the initial value much more clearly but at the same time it is more difficult to determine whether the function is linear or not.

Graph help us to determine whether the function is linear easier, but the table data will show us the initial value more clearly.

**Page 184 Question 1 Answer**

Given

How can you use a function to represent a linear relationship?

When we use an equation of a function that looks like y = mx + b,

The m represents the slope or the constant rate of change

The b is the y-intercept or the initial value

Now we know that a function that is in the form y = mx + b represents a linear relationship between x and y.

A function that is written in form y = mx + b represents a linear relationship between x and y.

**Page 184 Exercise 1 Answer**

Given

Height 8 inches

Triangle base is 15 inches

Show the graph of the function?

First we have to graph the new function. The new slope is going to be \(\frac{3}{5}=\frac{1}{5}\)

Since we know that there is no initial value than we know that the equation of the function is \(y=\frac{1}{5} x\).

For the last part we simply have to substitute the x with 110 and calculate from the equation.

\(y=\frac{1}{5} .110\)

y = 22

The height of the ramp is going to be 22 inches when the base length is 110 inches.

New slope \(\frac{1}{5}\)

New equation \(y=\frac{1}{5} x\)

The height of the ramp is going to be 22 inches when the base length is 110 inches

**Page 185 Exercise 2 Answer**

Given

After 2 weeks he feed to dog is \(8 \frac{1}{2}\)

After 5 weeks he feed to dog is \(21 \frac{1}{4}\)

Construct the function in the form of mx + c ?

We are simply going to subtract the amount he used after 3 weeks from the amount that her used after 5 weeks

d = 21.25 − 8.5

d = 12.75

Now we got the amount of food that he used in time period of 3 weeks.

Simply divide the result that we got with 3

d ÷ 3 = 12.75 ÷ 3 = 4.25.

Since we know the slope now, we need to find the initial value to be able to write the equation for the function.

To do this we can simply subtract 4.25 for every week that he fed the dogs from the value of food that he used after 2 weeks.

8.5 − 2.4.25 = 0

Now we know that the initial value is 0

Simply write equation

y = 4.25x

So, function is y = 4.25x

**Page 185 Exercise 3 Answer**

Given

Construct the function in the form of mx + b ?

If we would want to write the function for given graph than we simply have to find what is the slope of the given function, and the initial value of the function.

The initial value of the function is 1.

This means that printer needs 1 minute to warm-up before each printing.

Now to find the slope we can use 2 points that are already marked on the graph. These points are (10,2) and (30,4)

Now that we know the slope and the initial value we can simply write equation for given function

y = 0.1x + 1

y = 0.1x + 1

**Page 184 Exercise 1 Answer**

Given

Find the initial function of all function and proportional relationship?

The initial value of all linear functions that show a proportional relationship. Proportional relationships always start in the origin.

When we calculate y-intercept then we substitute x = 0. Which is same as initial values.

The initial value is same as y-intercept.

**Page 186 Exercise 1 Answer**

Given

How can you use a function to represent a linear relationship?

When we use an equation of a function that looks like y = mx + b,

The m represents the slope or the constant rate of change

The b is the y-intercept or the initial value

Now we know that a function that is in the form y = mx + b represents a linear relationship between x and y.

A function that is written in form y = mx + b represents a linear relationship between x and y.

**Page 186 Exercise 3 Answer**

Given

Find the initial function of all function and proportional relationship?

The initial value of all linear functions that show a proportional relationship is 0.

Proportional relationships always start in the origin.

The initial value is 0.

**Page 186 Exercise 5 Answer**

Given

Fill the missing data?

As we can see from the table both the x and y are increasing linearly. So we know that in the blank box in x row the missing number is 30 because the x is increasing by 10.

Same thing we can do for the y, we can see that y is increasing by 5. We simply need to add 5 to the third value in the y row and we get the last row.

20 + 5 = 25

The missing value of y is 25

The missing value of x is 30, and The missing value of y is 25.

**Page 186 Exercise 6 Answer**

Given

The data is 5

Find the equation form of y = mx + b ?

We can simply use two ordered pairs to find the equation that is described by the data in item 5.

We are going to be using points (10,10) and (20,15)

The way we find the slope is by the formula:

Now we can use the slope that we just calculated to find the initial value. We do this by calculating either one of the points.

10 = 0.5.10 + b

10 = 5 + b

10 − 5 = b

b = 5

Now we have everything we need to write the function for given line.

y = 0.5x + 5

So, equation of linear function is y = 0.5x + 5

**Page 187 Exercise 7 Answer**

Given

The points is (4,19) and (9,24)

Find the equation form of y = mx + b ?

We are going to be using points (4,19) and (9,24)

The way we find the slope is by the formula:

Now we can use the slope that we just calculated to find the initial value. We do this by calculating either one of the points.

19 = 4 + b

19 − 4 = b

b = 15

Now we have everything we need to write the function for given line.

y = x + 15

So, function is y = x + 15

**Page 187 Exercise 8 Answer**

Given

The line passing through (4.5,4.25) with y-intercept 2.5

Find the equation form of y = mx + b ?

We are going to be using points (4.5,4.25) with y-intercept 2.5

Since we know b in given task we can simply use the points through which the line passes and put the values into the equation to get the slope

−4.25 = 4.5m + 2.5

Subtract 2.5 from both sides of the equation

−4.25 − 2.5 = 4.5m

−6.75 = 4.5m

Now simply divide both sides of the equation with 4.5.

m = −1.5

The linear function we are looking for is y = −1.5x + 2.5

So, equation is y = -1.5x + 2.5

**Page 187 Exercise 9 Answer**

Given

t = 0

at 8 Seconds

840 feet

Find the equation form of y = mx + b ?

We simply have to divide the distance that car has traveled with the time has passed which is 8 seconds.

840 ÷ 8 = 105

Now we know how much distance can the car pass in 1 seconds.

We can now simply write the equation because we know that the initial value is 0

d = 105t

Linear function is d = 105t

**Page 187 Exercise 10 Answer**

Given

t = 0

after 56 minutes

8 inches

Find the equation form of y = mx + b?

In this task we must simply find how much time is needed for 1 inch of water to get into the bucket. We do this by dividing the amount of inches in the bucket.

56 ÷ 8 = 7

Now we know that it is needed 7 minutes for each inch that drips into the bucket.

Simply write the equation because we know that there is no initial value, in other words the initial value is 0.

\(w=\frac{1}{7} t\) \(w=\frac{1}{7} t\)

**Page 187 Exercise 11 Answer**

Given

Find the equation form of y = mx + b ?

To find the linear function in the form of y = mx + b we needed to read two points from the graph and calculate the slope so we can find the initial value.

We are going to be using points (1,10) and (4,16)

The way we find the slope is by the formula:

Now we can use the slope that we just calculated to find the initial value. We do this by calculating either one of the points.

10 = 2 + b

10 − 2 = b

b = 8

Now we have everything we need to write the function for given line.

y = 2x + 8

y = 2x + 8

**Page 187 Exercise 12 Answer**

Given

Company charges $6.50

Flat fee $3.99

Find the equation form of y = mx + b ?

We can simply write the linear function from the data that we got in the task.

The sweatshirts are going to be our Variable x

And the shipping fee is going to be b

y = 6.5x + 3.99

y = 6.5x + 3.99

We need to describe how the linear function would change the shipping charge.

If the shipping charge would apply to each sweatshirt then we would have to change our linear function. If the shipping charge is applied to each sweatshirt than we would simply have to add 3.99 to the factor next to x, so that the shipping fee is applied to every single sweatshirt.

y = (6.5 + 3.99) × x

The linear function would change from y = 6.5x + 3.99 to y = (6.5 + 3.99)x

**Page 187 Exercise 13 Answer**

Given:

1 poster + 6 comics for $12.75 and 1 poster + 13 comics for $19.75

We consider:

y as the total cost

x as the cost of comic books

m as the number of comic books.

b as the cost of posters.

We get two functions from the given data:

12.75 = 6m + 1

And

19.75 = 13m + 1

Subtracting both the equations we get:

7x = 7

x = 1

The cost of one comic book is $1.

Substituting the value of x in first equation we get:

b = 6.75

Therefore the required equation is:

y = m + 6.75

The required linear function is y = m + 6.75

Given:

The initial value of the package sold by another seller is $7.99.

We consider:

y as the total cost

x as the cost of comic books

m as the number of comic books

b as the cost of posters.

Since the shop sells poster with a comic book, initial value is the cost of one comic book plus one poster.

7.99 = x + b

We get:

x = 1

In a, the cost of one book plus poster = $1 + 6.75 = $7.75 which is lesser than $7.99.

Therefore, the seller B has the best deal.

In a, the cost of one book plus poster = $1 + 6.75 = $7.75 which is lesser than $7.99. Therefore, the seller B has the best deal.

**Page 188 Exercise 15 Answer**

Given:

To find the constant rate of change we simply have to use two points that we can read from the graph.

The constant rate of change is 25. This means that one cubic yard of mulch costs $25.

The constant rate of change is 25 and that means that one cubic yard of mulch costs per 25$.

Given that, the graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.

We need to find the initial value. Also, we need to explain what it represents.

The given graph is,

The given graph is,

It is visible from the graph that the initial point on the graph is (0,50)

The initial value is nothing but the starting point of the graph.

Therefore, the initial value is found to be $50 for 0 mulch ordered.

Thus, this initial value represents the shipping fee incurs for each shipment done.

The initial value is $50. This initial value represents the shipping fee for each shipment.

**Page 188 Exercise 17 Answer**

Given that, some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.

We need to write a linear function in the form y = mx + b that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x

Also, we need to find how many additional hours would the students spend to make 640 muffins.

The linear function is of the form y = mx + b

Here, b = 400 since they have already made those muffins.

Also, they can make 40 muffins in an hour.

Thus, the value of m = 40

Substituting this in the equation, we get,

y = 40x + 200

Finding the additional hours would the students spend to make 640 muffins, we get,

Thus, it took 11 more hours to make 640 muffins.

The linear equation will be y = 40x + 200

11 additional hours are needed to make 640 muffins.

**Page 188 Exercise 17 Answer**

Given:

Eight graders made 200 muffins. They can make 40 muffins per hour.

We consider:

Initial value as 200 muffins.

Let, x is the estimate, 40 muffins per hour.

Now we simply write linear function:

y = 40x + 200

Therefore the required linear function is y = 40x + 200

y = 40x + 200 From(A)

We consider:

Let y= 640

Putting this value in the linear function obtained from Part (A):

640 = 40x + 200

640 − 200 = 40x

40x = 440

x = 11

The students are going to need 11 additional hours if they want to make 640 muffins.

The students are going to need 11 additional hours if they want to make 640 muffins.