Question

1. the table models the total cost for the flat delivery fee and purchase of mulch (based on cubic yards).

find and interpret the rate of change (aka slope).
find and interpret the initial value (aka y-intercept).

number of cubic yards, x	total cost ($), y
3	145
4	180
5	215

Explanation:

Rate of Change (Slope)

Step 1: Recall the slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the first two points \( (2, 110) \) and \( (3, 145) \).

Step 2: Substitute the values into the formula

Here, \( x_1 = 2 \), \( y_1 = 110 \), \( x_2 = 3 \), \( y_2 = 145 \). So, \( m=\frac{145 - 110}{3 - 2}=\frac{35}{1} = 35 \). We can check with other points (e.g., \( (3,145) \) and \( (4,180) \)): \( \frac{180 - 145}{4 - 3}=\frac{35}{1}=35 \), and \( (4,180) \) and \( (5,215) \): \( \frac{215 - 180}{5 - 4}=\frac{35}{1}=35 \).

Interpretation of Slope

The slope of 35 means that for each additional cubic yard of mulch, the total cost increases by $35. This represents the cost per cubic yard of mulch (excluding the flat delivery fee).

Initial Value (y - intercept)

Step 1: Recall the slope - intercept form of a line

The equation of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. We know \( m = 35 \), and we can use one of the points (let's use \( (2, 110) \)) to find \( b \).

Step 2: Substitute into the equation

Substitute \( x = 2 \), \( y = 110 \), and \( m = 35 \) into \( y=mx + b \): \( 110=35\times2 + b \).

Step 3: Solve for \( b \)

First, calculate \( 35\times2=70 \). Then, \( 110=70 + b \). Subtract 70 from both sides: \( b=110 - 70=40 \).

Interpretation of y - intercept

The y - intercept \( b = 40 \) represents the flat delivery fee. When \( x = 0 \) (i.e., when 0 cubic yards of mulch are purchased), the total cost is $40, which is the cost of just the delivery.

Final Answers

• Rate of change (slope): The slope is \( \boldsymbol{35} \). It means the cost increases by $35 per cubic yard of mulch.
• Initial value (y - intercept): The y - intercept is \( \boldsymbol{40} \). It represents the $40 flat delivery fee.

Answer:

Rate of Change (Slope)

Step 1: Recall the slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the first two points \( (2, 110) \) and \( (3, 145) \).

Step 2: Substitute the values into the formula

Here, \( x_1 = 2 \), \( y_1 = 110 \), \( x_2 = 3 \), \( y_2 = 145 \). So, \( m=\frac{145 - 110}{3 - 2}=\frac{35}{1} = 35 \). We can check with other points (e.g., \( (3,145) \) and \( (4,180) \)): \( \frac{180 - 145}{4 - 3}=\frac{35}{1}=35 \), and \( (4,180) \) and \( (5,215) \): \( \frac{215 - 180}{5 - 4}=\frac{35}{1}=35 \).

Interpretation of Slope

The slope of 35 means that for each additional cubic yard of mulch, the total cost increases by $35. This represents the cost per cubic yard of mulch (excluding the flat delivery fee).

Initial Value (y - intercept)

Step 1: Recall the slope - intercept form of a line

The equation of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. We know \( m = 35 \), and we can use one of the points (let's use \( (2, 110) \)) to find \( b \).

Step 2: Substitute into the equation

Substitute \( x = 2 \), \( y = 110 \), and \( m = 35 \) into \( y=mx + b \): \( 110=35\times2 + b \).

Step 3: Solve for \( b \)

First, calculate \( 35\times2=70 \). Then, \( 110=70 + b \). Subtract 70 from both sides: \( b=110 - 70=40 \).

Interpretation of y - intercept

The y - intercept \( b = 40 \) represents the flat delivery fee. When \( x = 0 \) (i.e., when 0 cubic yards of mulch are purchased), the total cost is $40, which is the cost of just the delivery.

Final Answers

• Rate of change (slope): The slope is \( \boldsymbol{35} \). It means the cost increases by $35 per cubic yard of mulch.
• Initial value (y - intercept): The y - intercept is \( \boldsymbol{40} \). It represents the $40 flat delivery fee.