Question
A gym in Beaufort, South Carolina, charges its members $32.50 a month and a one-time initial fee of $65. (1 point)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Part A: What is the equation for the cost of membership if y
is the total cost (in dollars) for a membership for a certain number of months,x
y=
Response area
Part B: Graph the equation using Desmos. (Link to Desmos)
Part C: Determine the key feature of the function
x-intercept Response area
y-intercept Response area
slope Response area
domain Response area
range Response area
32.50x+65, -2, 65, all real numbers, y>65, 32.5, y<65, x<65, x,>65, only positive values, 65x+32.50
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Part A: What is the equation for the cost of membership if y
is the total cost (in dollars) for a membership for a certain number of months,x
y=
Response area
Part B: Graph the equation using Desmos. (Link to Desmos)
Part C: Determine the key feature of the function
x-intercept Response area
y-intercept Response area
slope Response area
domain Response area
range Response area
32.50x+65, -2, 65, all real numbers, y>65, 32.5, y<65, x<65, x,>65, only positive values, 65x+32.50
Answers
GPT-4o mini
Let's break down the parts of the question:
### Part A: Equation for Cost of Membership
The total cost \( y \) for a membership for \( x \) months would include the one-time initial fee of $65 and the monthly fee of $32.50 for each month. Therefore, the equation representing the total cost can be expressed as:
\[
y = 32.50x + 65
\]
### Part B: Graph the Equation on Desmos
You can enter the equation \( y = 32.50x + 65 \) in Desmos to graph it. The graph will be a straight line with a positive slope.
### Part C: Key Features of the Function
1. **x-intercept**: This is where the graph crosses the x-axis (where \( y = 0 \)). To find the x-intercept, set \( y \) to \( 0 \):
\[
0 = 32.50x + 65
\]
\[
32.50x = -65
\]
\[
x = -\frac{65}{32.50} \approx -2
\]
Thus, the **x-intercept** is approximately **-2**.
2. **y-intercept**: This is where the graph crosses the y-axis (where \( x = 0 \)). Substituting \( x = 0 \) into the equation gives:
\[
y = 32.50(0) + 65 = 65
\]
Thus, the **y-intercept** is **65**.
3. **Slope**: The slope \( m \) of the line represented by the equation is the coefficient of \( x \), which is **32.50**.
4. **Domain**: Since the number of months \( x \) cannot be negative, the domain is:
\[
\text{Domain: } x \geq 0 \text{ (only positive values)}
\]
5. **Range**: Since \( y \) represents total cost and cannot be lower than the initial fee, the minimum value of \( y \) occurs at \( x = 0 \):
\[
\text{Range: } y \geq 65
\]
### Summary of Responses
- **Part A:** \( y = 32.50x + 65 \)
- **x-intercept:** -2
- **y-intercept:** 65
- **slope:** 32.50
- **domain:** \( x \geq 0 \) (only positive values)
- **range:** \( y \geq 65 \)
### Part A: Equation for Cost of Membership
The total cost \( y \) for a membership for \( x \) months would include the one-time initial fee of $65 and the monthly fee of $32.50 for each month. Therefore, the equation representing the total cost can be expressed as:
\[
y = 32.50x + 65
\]
### Part B: Graph the Equation on Desmos
You can enter the equation \( y = 32.50x + 65 \) in Desmos to graph it. The graph will be a straight line with a positive slope.
### Part C: Key Features of the Function
1. **x-intercept**: This is where the graph crosses the x-axis (where \( y = 0 \)). To find the x-intercept, set \( y \) to \( 0 \):
\[
0 = 32.50x + 65
\]
\[
32.50x = -65
\]
\[
x = -\frac{65}{32.50} \approx -2
\]
Thus, the **x-intercept** is approximately **-2**.
2. **y-intercept**: This is where the graph crosses the y-axis (where \( x = 0 \)). Substituting \( x = 0 \) into the equation gives:
\[
y = 32.50(0) + 65 = 65
\]
Thus, the **y-intercept** is **65**.
3. **Slope**: The slope \( m \) of the line represented by the equation is the coefficient of \( x \), which is **32.50**.
4. **Domain**: Since the number of months \( x \) cannot be negative, the domain is:
\[
\text{Domain: } x \geq 0 \text{ (only positive values)}
\]
5. **Range**: Since \( y \) represents total cost and cannot be lower than the initial fee, the minimum value of \( y \) occurs at \( x = 0 \):
\[
\text{Range: } y \geq 65
\]
### Summary of Responses
- **Part A:** \( y = 32.50x + 65 \)
- **x-intercept:** -2
- **y-intercept:** 65
- **slope:** 32.50
- **domain:** \( x \geq 0 \) (only positive values)
- **range:** \( y \geq 65 \)