Question

An interior designer charges customers an initial consultation fee plus an hourly rate. The table shows the linear relationship between x, the number of hours, and y, the total cost of hiring the designer.
hours: 0,2,8,15
cost: 125,235,565,950


What is the y-intercept and what does it represent in this situation? You must show ALL work to earn full credit.

Answer 1

To find the y-intercept of the linear relationship between the number of hours \( x \) and the total cost \( y \), we'll use the given data points to determine the equation of the line. The y-intercept represents the initial consultation fee charged by the interior designer when no hours have been worked (i.e., when \( x = 0 \)).

### Step 1: Identify the points
From the table, we have four data points:
1. \( (0, 125) \)
2. \( (2, 235) \)
3. \( (8, 565) \)
4. \( (15, 950) \)

### Step 2: Determine the slope (\( m \))
To find the slope of the line, we can use two points from the data. Let's use points \( (2, 235) \) and \( (0, 125) \):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{235 - 125}{2 - 0} = \frac{110}{2} = 55
\]

The slope \( m \) is \( 55 \), which represents the hourly rate charged by the designer.

### Step 3: Find the y-intercept (\( b \))
Using the slope-intercept form of the equation of a line \( y = mx + b \), where \( b \) is the y-intercept, we can substitute one of the points we have into this equation to find \( b \).

We can use the point \( (0, 125) \):

\[
y = mx + b
\]
\[
125 = 55(0) + b
\]
\[
b = 125
\]

### Step 4: Interpretation
The y-intercept (the value of \( b \)) is \( 125 \). This value represents the initial consultation fee charged by the interior designer when no hours have been worked, meaning that even if a customer does not hire the designer for any hours, they will still have to pay this fee.

### Conclusion
The y-intercept is \( 125 \), which represents the initial consultation fee charged by the interior designer.