A cleaning service charges an initial fee plus an hourly rate The total cost for differenthumbers of hours, including the initial fee, is shown on the graph. Find and interpret the rate of change and initial value. Then write the equation of the function in the form \( y=m x+b \). Example 1 ?

Certainly! Let's work through **Example 1** based on the scenario you've provided. --- ### **Example 1** A cleaning service charges an **initial fee** plus an **hourly rate**. The total cost for different numbers of hours, including the initial fee, is shown in the table below: | **Number of Hours (x)** | **Total Cost ($) (y)** | |-------------------------|------------------------| | 0 | 50 | | 1 | 80 | | 2 | 110 | | 3 | 140 | | 4 | 170 | | 5 | 200 | **Tasks:** 1. **Find and interpret the rate of change.** 2. **Find and interpret the initial value.** 3. **Write the equation of the function in the form \( y = mx + b \).** --- ### **1. Finding and Interpreting the Rate of Change (\( m \))** **Definition:** The **rate of change** in this context represents the **hourly rate** the cleaning service charges. It tells us how much the total cost increases for each additional hour of service. **Calculation:** To find the rate of change (\( m \)), we can use any two points from the table. Let's use the first two points where \( x = 0 \) and \( x = 1 \): \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{80\,\text{\$} - 50\,\text{\$}}{1\,\text{hour} - 0\,\text{hours}} = \frac{30\,\text{\$}}{1\,\text{hour}} = 30\,\text{\$/hour} \] **Interpretation:** The cleaning service charges **\$30 per hour**. This means for each additional hour of cleaning, the total cost increases by \$30. --- ### **2. Finding and Interpreting the Initial Value (\( b \))** **Definition:** The **initial value** (\( b \)) represents the **initial fee** charged by the cleaning service before any hours are accounted for. It's the fixed cost regardless of the number of hours worked. **Calculation:** From the table, when \( x = 0 \) hours, the total cost \( y = \$50 \). Therefore, the initial value \( b = \$50 \). **Interpretation:** The cleaning service has an **initial fee of \$50**. This fee is charged even if no cleaning hours are purchased. --- ### **3. Writing the Equation of the Function (\( y = mx + b \))** Using the values obtained: - **Rate of Change (\( m \))**: \$30/hour - **Initial Value (\( b \))**: \$50 **Equation:** \[ y = 30x + 50 \] **Where:** - \( y \) = Total Cost in dollars - \( x \) = Number of Hours **Interpretation of the Equation:** To calculate the total cost (\( y \)) for any number of cleaning hours (\( x \)), multiply the number of hours by \$30 and then add the initial fee of \$50. **Example Usage:** If a customer wants 3 hours of cleaning: \[ y = 30(3) + 50 = 90 + 50 = \$140 \] This matches the data provided in the table. --- ### **Visual Representation** While we don't have an actual graph here, the relationship described by the equation \( y = 30x + 50 \) is a **linear function**. If you were to plot this on a graph: - The **y-intercept** (\( b = 50 \)) would be the point where the line crosses the **y-axis** (when \( x = 0 \)). - The **slope** (\( m = 30 \)) indicates that for each additional hour (\( x \)), the total cost (\( y \)) increases by \$30. --- ### **Summary** - **Rate of Change (\( m \))**: \$30/hour *Each additional hour of cleaning costs \$30.* - **Initial Value (\( b \))**: \$50 *There is a fixed initial fee of \$50 regardless of the number of hours.* - **Equation**: \( y = 30x + 50 \) *Total Cost = (30 × Number of Hours) + 50* --- This example demonstrates how to interpret and construct a linear function based on a real-world scenario involving an initial fee and a variable rate.