Question
A coordinate plane with 4 lines. The first line is labeled a(x) and passes through (2, negative 4) and (0, 2) and continues past (negative 1, 5). The second line is labeled f(x) and passes through (negative 6, negative 6), (negative 3, 0), and (0, 6). The third line is labeled g(x) and passes through (negative 2, negative 7), (0, negative 3), (1, negative 1) and (2, 1). The fourth line is labeled h(x) and passes through (negative 6, negative 6), (0, negative 3), and (6, 0).When Janelle woke up, it was –3 degrees Fahrenheit outside. As the morning progressed, the temperature rose 2 degrees every hour.Which line on the graph could represent this scenario?a(x)f(x)g(x)h(x)
Answers
GPT-4o mini
To determine which line represents the temperature scenario you described, we first need to understand the given temperature situation:
Janelle woke up to a temperature of -3 degrees Fahrenheit, and the temperature is rising at a rate of 2 degrees per hour. This situation can be modeled by a linear equation.
Starting with an initial temperature of -3 degrees, the temperature after \(t\) hours can be expressed as:
\[ T(t) = -3 + 2t \]
Here, \( T(t) \) is the temperature in degrees Fahrenheit, and \( t \) is the number of hours after Janelle woke up.
Now, we can analyze each line provided to see which one could represent this equation.
1. **Line a(x)**: Given points are (2, -4) and (0, 2), and the line seems to rise. Its slope can be calculated using these points. The slope between (0, 2) and (2, -4) is:
\[
\text{slope} = \frac{-4 - 2}{2 - 0} = \frac{-6}{2} = -3
\]
This line does not seem to match the scenario as it decreases.
2. **Line f(x)**: Given points include (0, 6), (−3, 0), and (−6, −6). This line also appears to decrease after its peak.
3. **Line g(x)**: Given points include (0, -3), and seems to represent temperatures well since it starts at -3. We calculate the slope. The change in temperature per hour based on points (0, -3) and (1, -1):
\[
\text{slope} = \frac{-1 - (-3)}{1 - 0} = \frac{2}{1} = 2
\]
This matches our temperature rise of 2 degrees per hour.
4. **Line h(x)**: The points also indicate a negative slope based on provided coordinates.
From this analysis, **g(x)** is the line that starts at -3 and has a slope of 2, which corresponds perfectly to the scenario of a temperature increasing by 2 degrees each hour starting from -3 degrees.
Therefore, **the correct line that represents Janelle's temperature scenario is g(x)**.
Janelle woke up to a temperature of -3 degrees Fahrenheit, and the temperature is rising at a rate of 2 degrees per hour. This situation can be modeled by a linear equation.
Starting with an initial temperature of -3 degrees, the temperature after \(t\) hours can be expressed as:
\[ T(t) = -3 + 2t \]
Here, \( T(t) \) is the temperature in degrees Fahrenheit, and \( t \) is the number of hours after Janelle woke up.
Now, we can analyze each line provided to see which one could represent this equation.
1. **Line a(x)**: Given points are (2, -4) and (0, 2), and the line seems to rise. Its slope can be calculated using these points. The slope between (0, 2) and (2, -4) is:
\[
\text{slope} = \frac{-4 - 2}{2 - 0} = \frac{-6}{2} = -3
\]
This line does not seem to match the scenario as it decreases.
2. **Line f(x)**: Given points include (0, 6), (−3, 0), and (−6, −6). This line also appears to decrease after its peak.
3. **Line g(x)**: Given points include (0, -3), and seems to represent temperatures well since it starts at -3. We calculate the slope. The change in temperature per hour based on points (0, -3) and (1, -1):
\[
\text{slope} = \frac{-1 - (-3)}{1 - 0} = \frac{2}{1} = 2
\]
This matches our temperature rise of 2 degrees per hour.
4. **Line h(x)**: The points also indicate a negative slope based on provided coordinates.
From this analysis, **g(x)** is the line that starts at -3 and has a slope of 2, which corresponds perfectly to the scenario of a temperature increasing by 2 degrees each hour starting from -3 degrees.
Therefore, **the correct line that represents Janelle's temperature scenario is g(x)**.
A video game arcade offers a yearly membership with reduced rates for game play. A single membership costs $60 per year. Game tokens can be purchased by members at the reduced rate of $1.00 per 10 tokens.
Which statements represent the function of the yearly cost in dollars, y, based on x, the number of game tokens purchased for a member of the arcade? Select three answers.
The slope of the function is $1.00.
The y-intercept of the function is $60.
The function can be represented by the equation y = y equals StartFraction 1 Over 10 EndFraction x plus 60.x + 60.
The domain is all real numbers.
The range is {y| y ≥ 60}.
Which statements represent the function of the yearly cost in dollars, y, based on x, the number of game tokens purchased for a member of the arcade? Select three answers.
The slope of the function is $1.00.
The y-intercept of the function is $60.
The function can be represented by the equation y = y equals StartFraction 1 Over 10 EndFraction x plus 60.x + 60.
The domain is all real numbers.
The range is {y| y ≥ 60}.
GPT-4o mini
To represent the yearly cost in dollars, \(y\), based on the number of game tokens purchased, \(x\), let's break down the costs involved.
1. **Basic Membership Fee**: Each member pays a yearly fee of $60.
2. **Cost of Game Tokens**: Members can buy game tokens for $1.00 for every 10 tokens. Therefore, the cost per token is \(\frac{1}{10}\) dollars.
Putting this together, the total cost \(y\) for a member who purchases \(x\) tokens can be represented mathematically as:
\[
y = 60 + \frac{1}{10}x
\]
Now let's analyze the statements:
1. **The slope of the function is $1.00.**
- **False.** The slope of the function, which represents the cost per token, is \(\frac{1}{10}\) not $1.00 since the cost is $0.10 for each token (which is $1.00 for 10 tokens).
2. **The y-intercept of the function is $60.**
- **True.** The y-intercept is the value of \(y\) when \(x = 0\); at that point, the cost is just the membership fee, which is $60.
3. **The function can be represented by the equation \(y = \frac{1}{10}x + 60\).**
- **True.** This correctly represents the relationship for the yearly cost: $60 for the membership plus \(\frac{1}{10}x\) for the game tokens.
4. **The domain is all real numbers.**
- **False.** The number of tokens purchased, \(x\), must be a non-negative integer (0 or more tokens), so the domain is not all real numbers. The domain is \(x \geq 0\).
5. **The range is \{y| y ≥ 60\}.**
- **True.** The minimum cost occurs when no tokens are purchased (when \(x = 0\)), which gives a cost of $60. As more tokens are purchased, \(y\) increases, so the minimum \(y\) is 60, making the range \(y \geq 60\).
Based on this analysis, the three true statements are:
- **The y-intercept of the function is $60.**
- **The function can be represented by the equation \(y = \frac{1}{10}x + 60\).**
- **The range is {y| y ≥ 60}.**
1. **Basic Membership Fee**: Each member pays a yearly fee of $60.
2. **Cost of Game Tokens**: Members can buy game tokens for $1.00 for every 10 tokens. Therefore, the cost per token is \(\frac{1}{10}\) dollars.
Putting this together, the total cost \(y\) for a member who purchases \(x\) tokens can be represented mathematically as:
\[
y = 60 + \frac{1}{10}x
\]
Now let's analyze the statements:
1. **The slope of the function is $1.00.**
- **False.** The slope of the function, which represents the cost per token, is \(\frac{1}{10}\) not $1.00 since the cost is $0.10 for each token (which is $1.00 for 10 tokens).
2. **The y-intercept of the function is $60.**
- **True.** The y-intercept is the value of \(y\) when \(x = 0\); at that point, the cost is just the membership fee, which is $60.
3. **The function can be represented by the equation \(y = \frac{1}{10}x + 60\).**
- **True.** This correctly represents the relationship for the yearly cost: $60 for the membership plus \(\frac{1}{10}x\) for the game tokens.
4. **The domain is all real numbers.**
- **False.** The number of tokens purchased, \(x\), must be a non-negative integer (0 or more tokens), so the domain is not all real numbers. The domain is \(x \geq 0\).
5. **The range is \{y| y ≥ 60\}.**
- **True.** The minimum cost occurs when no tokens are purchased (when \(x = 0\)), which gives a cost of $60. As more tokens are purchased, \(y\) increases, so the minimum \(y\) is 60, making the range \(y \geq 60\).
Based on this analysis, the three true statements are:
- **The y-intercept of the function is $60.**
- **The function can be represented by the equation \(y = \frac{1}{10}x + 60\).**
- **The range is {y| y ≥ 60}.**
The yearly cost in dollars, y, for a member at a video game arcade based on total game tokens purchased, x, is y =y equals StartFraction 1 Over 10 EndFraction x plus 60.x + 60.
A nonmember pays $0.20 per game token, but no yearly fee.
Which equation represents the yearly cost for a nonmember at the same video game arcade based on the total number of game tokens purchased in a year?
y = y equals StartFraction one-half EndFraction x.x
y = y equals StartFraction 1 Over 10 EndFraction x plus 60.x + 60
y = y equals StartFraction 1 Over 5 EndFraction x. x
y = y equals StartFraction 1 Over 20 EndFraction x plus 60.x + 60
A nonmember pays $0.20 per game token, but no yearly fee.
Which equation represents the yearly cost for a nonmember at the same video game arcade based on the total number of game tokens purchased in a year?
y = y equals StartFraction one-half EndFraction x.x
y = y equals StartFraction 1 Over 10 EndFraction x plus 60.x + 60
y = y equals StartFraction 1 Over 5 EndFraction x. x
y = y equals StartFraction 1 Over 20 EndFraction x plus 60.x + 60
GPT-4o mini
To find the yearly cost \(y\) for a nonmember at the video game arcade based on the total number of game tokens purchased \(x\), we need to consider the cost per game token.
A nonmember pays $0.20 per game token. Therefore, the cost can be expressed directly as a function of \(x\) (the number of tokens purchased):
\[
y = 0.20x
\]
Now, we can represent $0.20 as a fraction:
\[
y = \frac{20}{100}x = \frac{1}{5}x
\]
So the equation representing the yearly cost for a nonmember in terms of the number of tokens \(x\) is:
\[
y = \frac{1}{5}x
\]
Now, let’s evaluate the options given:
1. **\(y = \frac{1}{2}x\)**
- Not correct.
2. **\(y = \frac{1}{10}x + 60\)**
- This represents the cost for a member, not a nonmember.
3. **\(y = \frac{1}{5}x\)**
- **Correct!** This represents the cost for a nonmember.
4. **\(y = \frac{1}{20}x + 60\)**
- Not correct.
Thus, the correct equation representing the yearly cost for a nonmember based on the total number of game tokens purchased is:
**\(y = \frac{1}{5}x\)**.
A nonmember pays $0.20 per game token. Therefore, the cost can be expressed directly as a function of \(x\) (the number of tokens purchased):
\[
y = 0.20x
\]
Now, we can represent $0.20 as a fraction:
\[
y = \frac{20}{100}x = \frac{1}{5}x
\]
So the equation representing the yearly cost for a nonmember in terms of the number of tokens \(x\) is:
\[
y = \frac{1}{5}x
\]
Now, let’s evaluate the options given:
1. **\(y = \frac{1}{2}x\)**
- Not correct.
2. **\(y = \frac{1}{10}x + 60\)**
- This represents the cost for a member, not a nonmember.
3. **\(y = \frac{1}{5}x\)**
- **Correct!** This represents the cost for a nonmember.
4. **\(y = \frac{1}{20}x + 60\)**
- Not correct.
Thus, the correct equation representing the yearly cost for a nonmember based on the total number of game tokens purchased is:
**\(y = \frac{1}{5}x\)**.
Consider the graph of the linear function h(x) = –6 + h(x) equals negative 6 plus StartFraction 2 Over 3 EndFraction x. x. Which quadrant will the graph not go through and why?
Quadrant I, because the slope is negative and the y-intercept is positive
Quadrant II, because the slope is positive and the y-intercept is negative
Quadrant III, because the slope is negative and the y-intercept is positive
Quadrant IV, because the slope is positive and the y-intercept is negative
Quadrant I, because the slope is negative and the y-intercept is positive
Quadrant II, because the slope is positive and the y-intercept is negative
Quadrant III, because the slope is negative and the y-intercept is positive
Quadrant IV, because the slope is positive and the y-intercept is negative
GPT-4o mini
To analyze the graph of the linear function \( h(x) = -6 + \frac{2}{3}x \), we can identify its properties by looking at the slope and y-intercept.
1. **Y-Intercept**: The y-intercept occurs when \( x = 0 \):
\[
h(0) = -6 + \frac{2}{3}(0) = -6
\]
So, the y-intercept is at the point (0, -6). This means the graph crosses the y-axis at -6, which is below the x-axis.
2. **Slope**: The slope of the line is \( \frac{2}{3} \), which is positive. This means that as \( x \) increases, \( h(x) \) will also increase.
### Analyzing Quadrants
1. **Quadrant I**: \( (x > 0, y > 0) \)
Since the slope is positive, as \( x \) increases from the y-intercept (-6), the function will eventually rise above 0, meaning it will intersect Quadrant I.
2. **Quadrant II**: \( (x < 0, y > 0) \)
In this quadrant, \( y \) is positive and \( x \) is negative. The graph starts at (0, -6) and will not be above the x-axis for negative \( x \) values, thus it cannot be in Quadrant II.
3. **Quadrant III**: \( (x < 0, y < 0) \)
For negative \( x \) values, since the y-intercept is -6, the function will also go through Quadrant III as it will remain below the x-axis but still fall into this quadrant.
4. **Quadrant IV**: \( (x > 0, y < 0) \)
The graph will enter Quadrant IV when \( x \) is positive but \( h(x) \) has not yet risen above -6 for small positive values of \( x \). Therefore, it will temporarily touch Quadrant IV before eventually moving into Quadrant I.
### Conclusion
Since the function opens upwards due to the positive slope and starts below the x-axis, it will not cross into **Quadrant II**, where the graph would need to be above the x-axis while \( x \) is negative.
Therefore, the correct conclusion is:
**Quadrant II, because the slope is positive and the y-intercept is negative.**
1. **Y-Intercept**: The y-intercept occurs when \( x = 0 \):
\[
h(0) = -6 + \frac{2}{3}(0) = -6
\]
So, the y-intercept is at the point (0, -6). This means the graph crosses the y-axis at -6, which is below the x-axis.
2. **Slope**: The slope of the line is \( \frac{2}{3} \), which is positive. This means that as \( x \) increases, \( h(x) \) will also increase.
### Analyzing Quadrants
1. **Quadrant I**: \( (x > 0, y > 0) \)
Since the slope is positive, as \( x \) increases from the y-intercept (-6), the function will eventually rise above 0, meaning it will intersect Quadrant I.
2. **Quadrant II**: \( (x < 0, y > 0) \)
In this quadrant, \( y \) is positive and \( x \) is negative. The graph starts at (0, -6) and will not be above the x-axis for negative \( x \) values, thus it cannot be in Quadrant II.
3. **Quadrant III**: \( (x < 0, y < 0) \)
For negative \( x \) values, since the y-intercept is -6, the function will also go through Quadrant III as it will remain below the x-axis but still fall into this quadrant.
4. **Quadrant IV**: \( (x > 0, y < 0) \)
The graph will enter Quadrant IV when \( x \) is positive but \( h(x) \) has not yet risen above -6 for small positive values of \( x \). Therefore, it will temporarily touch Quadrant IV before eventually moving into Quadrant I.
### Conclusion
Since the function opens upwards due to the positive slope and starts below the x-axis, it will not cross into **Quadrant II**, where the graph would need to be above the x-axis while \( x \) is negative.
Therefore, the correct conclusion is:
**Quadrant II, because the slope is positive and the y-intercept is negative.**