Question
Use the graph for both parts of the question. (The graph is a ray, so it's not all real numbers here. It starts at 0,-2.)
Part 1:
Identify the domain of the function.
Part 2:
Identify the range of the function.
Part 1:
Identify the domain of the function.
Part 2:
Identify the range of the function.
Answers
Answer
The ray DOES end somewhere. At (5,0).
Answer
Posh Pups charges $35 for a shampoo and hair cut. The salon charges an additional $10 per hour for each hour a dog stays at the salon.
Part 1:
Write an equation in Slope-Intercept Form to represent the relationship between the hours (x) the dog is at the salon and total cost (y).
Part 2:
Write the equation in Standard Form.
Part 1:
Write an equation in Slope-Intercept Form to represent the relationship between the hours (x) the dog is at the salon and total cost (y).
Part 2:
Write the equation in Standard Form.
Answer
Write the equation for the line that passes through (3, -4) and is perpendicular to the line that passes through (5, 6) and (8, 15).
Part 1:
Write the equation in Point-Slope Form.
Part 2:
Write the equation in Slope-Intercept Form.
Part 1:
Write the equation in Point-Slope Form.
Part 2:
Write the equation in Slope-Intercept Form.
Answer
Dennis is attending the Plant City Strawberry Festival. He plans to park his car in a lot from 6:00pm to 11:00pm. The lot charges $5 for the first hour and $2 for each additional hour.
Part 1:
Create a graph that represents the relationship between the price (y) and hours parked (x).
Part 2:
Identify the Domain of the function.
Part 3:
Identify the Range of the function.
Part 1:
Create a graph that represents the relationship between the price (y) and hours parked (x).
Part 2:
Identify the Domain of the function.
Part 3:
Identify the Range of the function.
Answer
Please do inequalities for the domain and range.
Answered by
GPT-4o mini
To provide accurate answers for the domain and range of the function represented by the ray that starts at the point (0, -2), I will describe the characteristics of a ray.
**Part 1: Domain of the Function**
Since the ray starts at the point (0, -2) and extends indefinitely in one direction, the domain will consist of all \( x \)-values that correspond to the \( y \)-values of the ray. If the ray stretches to the right (which is typical for a ray starting at a given point), the domain would be all \( x \)-values starting from \( 0 \) and going to positive infinity.
So, the domain of the function is:
\[ [0, \infty) \]
**Part 2: Range of the Function**
To find the range, we look at the \( y \)-values that the ray attains. Since the ray starts at the point (0, -2) and extends infinitely in the upward direction, the minimum \( y \)-value is \( -2 \), and it will extend upward indefinitely.
Thus, the range of the function is:
\[ [-2, \infty) \]
In summary:
- Domain: \([0, \infty)\)
- Range: \([-2, \infty)\)
**Part 1: Domain of the Function**
Since the ray starts at the point (0, -2) and extends indefinitely in one direction, the domain will consist of all \( x \)-values that correspond to the \( y \)-values of the ray. If the ray stretches to the right (which is typical for a ray starting at a given point), the domain would be all \( x \)-values starting from \( 0 \) and going to positive infinity.
So, the domain of the function is:
\[ [0, \infty) \]
**Part 2: Range of the Function**
To find the range, we look at the \( y \)-values that the ray attains. Since the ray starts at the point (0, -2) and extends infinitely in the upward direction, the minimum \( y \)-value is \( -2 \), and it will extend upward indefinitely.
Thus, the range of the function is:
\[ [-2, \infty) \]
In summary:
- Domain: \([0, \infty)\)
- Range: \([-2, \infty)\)
Answered by
GPT-4o mini
Thank you for the clarification! The fact that the ray has a definite endpoint at (5, 0) changes the analysis of both the domain and the range. Let's revise the answers accordingly.
**Part 1: Domain of the Function**
The ray starts at (0, -2) and ends at (5, 0). The domain consists of all \( x \)-values from the starting point to the endpoint of the ray, which means it covers the interval from \( 0 \) to \( 5 \).
So, the domain of the function is:
\[ [0, 5] \]
**Part 2: Range of the Function**
The ray starts at \( y = -2 \) when \( x = 0 \) and ends at \( y = 0 \) when \( x = 5 \). The \( y \)-values increase from \( -2 \) to \( 0 \) as \( x \) goes from \( 0 \) to \( 5 \).
Thus, the range of the function is:
\[ [-2, 0] \]
In summary:
- **Domain:** \([0, 5]\)
- **Range:** \([-2, 0]\)
**Part 1: Domain of the Function**
The ray starts at (0, -2) and ends at (5, 0). The domain consists of all \( x \)-values from the starting point to the endpoint of the ray, which means it covers the interval from \( 0 \) to \( 5 \).
So, the domain of the function is:
\[ [0, 5] \]
**Part 2: Range of the Function**
The ray starts at \( y = -2 \) when \( x = 0 \) and ends at \( y = 0 \) when \( x = 5 \). The \( y \)-values increase from \( -2 \) to \( 0 \) as \( x \) goes from \( 0 \) to \( 5 \).
Thus, the range of the function is:
\[ [-2, 0] \]
In summary:
- **Domain:** \([0, 5]\)
- **Range:** \([-2, 0]\)
Answered by
GPT-4o mini
To determine the equations for the cost of taking a dog to Posh Pups, we start with the information provided.
The base cost for a shampoo and haircut is $35, and there is an additional charge of $10 per hour that the dog stays at the salon. We can use this information to write the equations.
**Part 1: Slope-Intercept Form**
In slope-intercept form, the equation is given as:
\[ y = mx + b \]
Where:
- \( m \) is the slope (the rate of change in cost per hour),
- \( b \) is the y-intercept (the starting cost).
For this scenario:
- The slope \( m \) (cost per hour) is $10.
- The y-intercept \( b \) (starting cost for the haircut) is $35.
So the equation representing the total cost \( y \) based on the hours \( x \) the dog stays at the salon is:
\[ y = 10x + 35 \]
**Part 2: Standard Form**
To write the equation in standard form, we need it in the format \( Ax + By = C \) where \( A \), \( B \), and \( C \) are integers. Starting from the slope-intercept form \( y = 10x + 35 \):
1. Rearrange to get \( -10x + y = 35 \).
2. To convert it into standard form, we typically want \( A \) to be positive. Hence, multiply the entire equation by \(-1\):
\[ 10x - y = -35 \]
Thus, the standard form of the equation is:
\[ 10x - y = -35 \]
In summary:
- **Slope-Intercept Form:** \( y = 10x + 35 \)
- **Standard Form:** \( 10x - y = -35 \)
The base cost for a shampoo and haircut is $35, and there is an additional charge of $10 per hour that the dog stays at the salon. We can use this information to write the equations.
**Part 1: Slope-Intercept Form**
In slope-intercept form, the equation is given as:
\[ y = mx + b \]
Where:
- \( m \) is the slope (the rate of change in cost per hour),
- \( b \) is the y-intercept (the starting cost).
For this scenario:
- The slope \( m \) (cost per hour) is $10.
- The y-intercept \( b \) (starting cost for the haircut) is $35.
So the equation representing the total cost \( y \) based on the hours \( x \) the dog stays at the salon is:
\[ y = 10x + 35 \]
**Part 2: Standard Form**
To write the equation in standard form, we need it in the format \( Ax + By = C \) where \( A \), \( B \), and \( C \) are integers. Starting from the slope-intercept form \( y = 10x + 35 \):
1. Rearrange to get \( -10x + y = 35 \).
2. To convert it into standard form, we typically want \( A \) to be positive. Hence, multiply the entire equation by \(-1\):
\[ 10x - y = -35 \]
Thus, the standard form of the equation is:
\[ 10x - y = -35 \]
In summary:
- **Slope-Intercept Form:** \( y = 10x + 35 \)
- **Standard Form:** \( 10x - y = -35 \)
Answered by
GPT-4o mini
To find the equation of the line that passes through the point (3, -4) and is perpendicular to the line passing through the points (5, 6) and (8, 15), we will follow these steps:
### Part 1: Find the slope of the line through (5, 6) and (8, 15)
1. **Calculate the slope \( m \)** of the line through points \( (5, 6) \) and \( (8, 15) \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15 - 6}{8 - 5} = \frac{9}{3} = 3
\]
2. **Find the slope of the perpendicular line.** The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Thus, if the slope of the original line is \( 3 \):
\[
m_{perpendicular} = -\frac{1}{3}
\]
3. **Use the point-slope form of the equation** for a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (3, -4) \) and \( m = -\frac{1}{3} \).
Substituting in the values:
\[
y - (-4) = -\frac{1}{3}(x - 3)
\]
This simplifies to:
\[
y + 4 = -\frac{1}{3}(x - 3)
\]
### Part 2: Write in Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (\( y = mx + b \)), we need to distribute and isolate \( y \):
1. Distributing the right side:
\[
y + 4 = -\frac{1}{3}x + 1
\]
2. Subtracting 4 from both sides:
\[
y = -\frac{1}{3}x + 1 - 4
\]
\[
y = -\frac{1}{3}x - 3
\]
### Final Results
- **Part 1: Point-Slope Form:**
\[
y + 4 = -\frac{1}{3}(x - 3)
\]
- **Part 2: Slope-Intercept Form:**
\[
y = -\frac{1}{3}x - 3
\]
These are the equations for the line passing through the point (3, -4) and perpendicular to the line connecting (5, 6) and (8, 15).
### Part 1: Find the slope of the line through (5, 6) and (8, 15)
1. **Calculate the slope \( m \)** of the line through points \( (5, 6) \) and \( (8, 15) \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15 - 6}{8 - 5} = \frac{9}{3} = 3
\]
2. **Find the slope of the perpendicular line.** The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Thus, if the slope of the original line is \( 3 \):
\[
m_{perpendicular} = -\frac{1}{3}
\]
3. **Use the point-slope form of the equation** for a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \( (x_1, y_1) = (3, -4) \) and \( m = -\frac{1}{3} \).
Substituting in the values:
\[
y - (-4) = -\frac{1}{3}(x - 3)
\]
This simplifies to:
\[
y + 4 = -\frac{1}{3}(x - 3)
\]
### Part 2: Write in Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (\( y = mx + b \)), we need to distribute and isolate \( y \):
1. Distributing the right side:
\[
y + 4 = -\frac{1}{3}x + 1
\]
2. Subtracting 4 from both sides:
\[
y = -\frac{1}{3}x + 1 - 4
\]
\[
y = -\frac{1}{3}x - 3
\]
### Final Results
- **Part 1: Point-Slope Form:**
\[
y + 4 = -\frac{1}{3}(x - 3)
\]
- **Part 2: Slope-Intercept Form:**
\[
y = -\frac{1}{3}x - 3
\]
These are the equations for the line passing through the point (3, -4) and perpendicular to the line connecting (5, 6) and (8, 15).
Answered by
GPT-4o mini
To solve this problem, let’s first break down the costs associated with Dennis parking his car at the Plant City Strawberry Festival.
### Part 1: Create a Graph
**Calculating the Costs**
- **Time Period**: Dennis parks from 6:00 PM to 11:00 PM, which is a total of 5 hours.
- **Cost Structure**:
- The first hour (6:00 PM to 7:00 PM) costs $5.
- The next four hours (7:00 PM to 11:00 PM) cost $2 each, which totals $8 for the additional hours.
**Total Cost Calculation**:
- Total cost after 5 hours = $5 (first hour) + $8 (next four hours) = **$13**.
**Breaking down the hours and corresponding costs**:
- 0 hours: $0
- 1 hour: $5
- 2 hours: $7 ($5 + $2)
- 3 hours: $9 ($5 + $2 + $2)
- 4 hours: $11 ($5 + $2 + $2 + $2)
- 5 hours: $13 ($5 + $2 + $2 + $2 + $2)
Thus, the relationship can be summarized as:
- \( \begin{align*}
(0, 0), \\
(1, 5), \\
(2, 7), \\
(3, 9), \\
(4, 11), \\
(5, 13)
\end{align*} \)
**Graphing**:
You can plot these points on a Cartesian plane where the \( x \)-axis represents the number of hours parked (from 0 to 5) and the \( y \)-axis represents the cost in dollars.
The graph will be a piecewise function with:
- A constant increase: from (0, 0) to (1, 5), then a slope of 1 for the increase from (1, 5) to (5, 13) where it becomes linear.
### Part 2: Identify the Domain of the Function
The domain represents the possible values of \( x \), which in this context, refers to the number of hours Dennis can park.
- Since Dennis plans to park for a minimum of 0 hours (not parked) to a maximum of 5 hours, the domain of the function is:
\[ [0, 5] \]
### Part 3: Identify the Range of the Function
The range represents the possible values of \( y \), which corresponds to the total cost incurred from parking.
- The minimum cost is $0 when \( x = 0 \) hours, and the maximum cost is $13 when \( x = 5 \) hours, so the range of the function is:
\[ [0, 13] \]
### Summary
- **Graph**: Plot the points (0, 0), (1, 5), (2, 7), (3, 9), (4, 11), (5, 13) on a Cartesian plane with a straight line connecting these points.
- **Domain**: \( [0, 5] \)
- **Range**: \( [0, 13] \)
If you need a visual representation of the graph, tools like graphing calculators or software can help create that based on the plotted points.
### Part 1: Create a Graph
**Calculating the Costs**
- **Time Period**: Dennis parks from 6:00 PM to 11:00 PM, which is a total of 5 hours.
- **Cost Structure**:
- The first hour (6:00 PM to 7:00 PM) costs $5.
- The next four hours (7:00 PM to 11:00 PM) cost $2 each, which totals $8 for the additional hours.
**Total Cost Calculation**:
- Total cost after 5 hours = $5 (first hour) + $8 (next four hours) = **$13**.
**Breaking down the hours and corresponding costs**:
- 0 hours: $0
- 1 hour: $5
- 2 hours: $7 ($5 + $2)
- 3 hours: $9 ($5 + $2 + $2)
- 4 hours: $11 ($5 + $2 + $2 + $2)
- 5 hours: $13 ($5 + $2 + $2 + $2 + $2)
Thus, the relationship can be summarized as:
- \( \begin{align*}
(0, 0), \\
(1, 5), \\
(2, 7), \\
(3, 9), \\
(4, 11), \\
(5, 13)
\end{align*} \)
**Graphing**:
You can plot these points on a Cartesian plane where the \( x \)-axis represents the number of hours parked (from 0 to 5) and the \( y \)-axis represents the cost in dollars.
The graph will be a piecewise function with:
- A constant increase: from (0, 0) to (1, 5), then a slope of 1 for the increase from (1, 5) to (5, 13) where it becomes linear.
### Part 2: Identify the Domain of the Function
The domain represents the possible values of \( x \), which in this context, refers to the number of hours Dennis can park.
- Since Dennis plans to park for a minimum of 0 hours (not parked) to a maximum of 5 hours, the domain of the function is:
\[ [0, 5] \]
### Part 3: Identify the Range of the Function
The range represents the possible values of \( y \), which corresponds to the total cost incurred from parking.
- The minimum cost is $0 when \( x = 0 \) hours, and the maximum cost is $13 when \( x = 5 \) hours, so the range of the function is:
\[ [0, 13] \]
### Summary
- **Graph**: Plot the points (0, 0), (1, 5), (2, 7), (3, 9), (4, 11), (5, 13) on a Cartesian plane with a straight line connecting these points.
- **Domain**: \( [0, 5] \)
- **Range**: \( [0, 13] \)
If you need a visual representation of the graph, tools like graphing calculators or software can help create that based on the plotted points.
Answered by
GPT-4o mini
Certainly! To represent the domain and range using inequalities, we will express each one accordingly.
### Domain
The domain refers to the possible values of \( x \), which represents the number of hours parked.
- From the previous discussion, the domain is \( [0, 5] \).
- Using inequalities, this can be expressed as:
\[
0 \leq x \leq 5
\]
### Range
The range refers to the possible values of \( y \), which represents the total cost.
- From the previous discussion, the range is \( [0, 13] \).
- Using inequalities, this can be expressed as:
\[
0 \leq y \leq 13
\]
### Summary of Inequalities
- **Domain:** \( 0 \leq x \leq 5 \)
- **Range:** \( 0 \leq y \leq 13 \)
These inequalities clearly state the valid intervals for the number of hours parked and the total cost associated with those hours.
### Domain
The domain refers to the possible values of \( x \), which represents the number of hours parked.
- From the previous discussion, the domain is \( [0, 5] \).
- Using inequalities, this can be expressed as:
\[
0 \leq x \leq 5
\]
### Range
The range refers to the possible values of \( y \), which represents the total cost.
- From the previous discussion, the range is \( [0, 13] \).
- Using inequalities, this can be expressed as:
\[
0 \leq y \leq 13
\]
### Summary of Inequalities
- **Domain:** \( 0 \leq x \leq 5 \)
- **Range:** \( 0 \leq y \leq 13 \)
These inequalities clearly state the valid intervals for the number of hours parked and the total cost associated with those hours.
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