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Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yards of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost. Enter your answer in ymz+b form

5 answers

To find the equation of the linear function that models the relationship between the number of cubic yards of soil (let's denote this as \( y \)) and the total cost (let's denote this as \( m \)), we can first determine the slope of the linear function using the given data points.

We have two points based on Alayah's orders:

  1. Point 1: (4, 220) – corresponding to 4 cubic yards costing $220.
  2. Point 2: (7, 340) – corresponding to 7 cubic yards costing $340.

Step 1: Calculate the slope (\( m \))

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using our points:

  • \( (x_1, y_1) = (4, 220) \)
  • \( (x_2, y_2) = (7, 340) \)

Substituting the values into the formula:

\[ m = \frac{340 - 220}{7 - 4} = \frac{120}{3} = 40 \]

This tells us that the cost increases by $40 for each additional cubic yard of soil.

Step 2: Find the y-intercept (\( b \))

We can use one of the points to solve for the y-intercept \( b \). The linear equation is in the form:

\[ m = ax + b \]

Where \( a \) is the slope (which we found to be 40). We'll use one of the points to find \( b \). Let's use the point (4, 220):

\[ 220 = 40(4) + b \]

Calculating it:

\[ 220 = 160 + b \] \[ b = 220 - 160 = 60 \]

Final Equation

Now, we have both the slope \( m = 40 \) and the y-intercept \( b = 60 \). The equation of the linear function that models the relationship between cubic yards ordered (let \( x \) be the number of cubic yards) and total cost \( y \) is:

\[ y = 40x + 60 \]

So, the final answer in the required form \( ymz + b \) is:

\[ y = 40x + 60 \]

Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change?

Choose 1 for Linear Function 1; choose 2 for Linear Function 2.

• Linear Function 1: y = x +8

• Linear Function 2:

y

12-

11

10

9

8-

7-

6-

>

(4,10)

(2,6)

To determine which linear function has a greater rate of change, we need to analyze both functions.

Linear Function 1:

Equation:
\[ y = x + 8 \]

The rate of change (slope) of this function is 1, since the coefficient of \( x \) is 1.

Linear Function 2:

To find the slope of Linear Function 2 based on the given points \((2, 6)\) and \((4, 10)\), we can use the slope formula:

\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points:

  • \( (x_1, y_1) = (2, 6) \)
  • \( (x_2, y_2) = (4, 10) \)

Substituting the values:

\[ \text{slope} = \frac{10 - 6}{4 - 2} = \frac{4}{2} = 2 \]

Comparison of Rates of Change:

  • Linear Function 1: Rate of change = 1
  • Linear Function 2: Rate of change = 2

Conclusion:

Since the rate of change for Linear Function 2 (slope = 2) is greater than that of Linear Function 1 (slope = 1), Linear Function 2 has the greater rate of change.

Final Answer: Choose 2 for Linear Function 2.

Use the table to answer the question.

Day of the Week

Number of Children in the Car

2

4

4

3

4

3

0

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Describe the meaning of the word function in math. Then consider the table Why does this table show a function?

Meaning of the word "function" in math:

In mathematics, a function is a relationship or correspondence between two sets that assigns each element from the first set (the domain) exactly one element from the second set (the codomain). In simpler terms, a function takes an input (or value) and produces a unique output based on a specific rule.

For example, if you think of the function as a machine: you put something in (input), and the machine processes it and outputs something based on a rule, ensuring that for every input, there is a single defined output.

Explanation of the Table:

Now, considering the table you provided, which relates the days of the week to the number of children in the car:

| Day of the Week | Number of Children in the Car | |------------------|-------------------------------| | Sunday | 2 | | Monday | 4 | | Tuesday | 4 | | Wednesday | 3 | | Thursday | 4 | | Friday | 3 | | Saturday | 0 |

Why does this table show a function?

The table shows a function because each day of the week (the input from the domain) is associated with one specific number of children in the car (the output from the codomain). Here are the key points supporting this:

  1. Unique Outputs: For each day (Sunday through Saturday), there is only one corresponding number of children. For example, "Sunday" is always associated with 2 children, and "Monday" is always associated with 4 children. No day appears more than once with different numbers of children, which upholds the definition of a function.

  2. Defined Domain: The days of the week represent a limited and defined set of inputs. There are no repeats or ambiguities; each day maps exactly to one count of children.

  3. Proper Assignment: Each input (day) corresponds correctly and exactly to the output (number of children). Hence, there is a well-defined relationship.

In summary, the table represents a function because it establishes a clear and consistent relationship between the days of the week and the number of children in the car, meeting the criteria of a mathematical function.