Question
The profit earned by a hot dog stand is a linear function of the number of hot dogs sold. It costs the owner $48 dollars each morning for the day’s supply of hot dogs, buns and mustard, but he earns $2 profit for each hot dog sold. Which equation represents y, the profit earned by the hot dog stand for x hot dogs sold?
y=48x−2
y=48x+2
y=2x−48
y=2x+48
2. Four lines are drawn on a coordinate plane to form trapezoid WXYZ.A coordinate grid with 4 lines. Line X W is drawn with point W at (negative 4, 1) and passes through (0, 2) with and (3, 3). Line Y X is drawn with point Y at (3, 0) and passes through (0, 3). Line Z W is drawn with point Z at (0, negative 3) and point W at (negative 4, 1). LIne Z Y is drawn with point Y at (3, 0) and point Z at (0, negative 3).Which statements are true about the lines? Select three options.Line WZ has the same slope as line XY.Line YX has a greater slope than line ZY.Line XW has a lesser slope than line YZ.Line ZW has the same y-intercept as line YZ.Line XY has a lesser y-intercept than line XW.
3. The graph represents revenue in dollars as a function of greeting cards sold.
A coordinate plane showing Greeting Card Revenue, Number of Cards Sold on the x-axis and Revenue in dollars on the y-axis. A line starts at (0. 0) and passes through (2, 8), (4, 16), and ends at (5, 20).
Which equation represents the function shown on the graph?
y = 1/4 x
y = 1/2 x
y = 2x
y = 4x
4. A coordinate plane with a line drawn passing through the points (0, 3) and (1,1).
Which equation represents the graphed function?
y = –2x + 3
y = 2x + 3
y = 1/2 x + 3
y = –1/2 x + 3
5. What is the slope of the line represented by the equation y = –y equals negative StartFraction one-half EndFraction x plus StartFraction one-fourth EndFraction.x + StartFraction one-fourth EndFraction.?–negative StartFraction one-half EndFraction.–negative StartFraction one-fourth EndFraction.StartFraction one-fourth EndFraction.StartFraction one-half EndFraction.
6. What is the y-intercept of the function f(x) = –negative StartFraction 2 Over 9 EndFraction.x + ?–negative StartFraction 2 Over 9 EndFraction.–negative StartFraction 1 Over 3 EndFraction.StartFraction 1 Over 3 EndFraction.StartFraction 2 Over 9 EndFraction.
7. What is the slope of the line represented by the equation y = y equals StartFraction 4 Over 5 EndFraction x minus 3.x – 3?–3Negative StartFraction 4 Over 5 EndFraction.StartFraction 4 Over 5 EndFraction.3
8. A coordinate plane with a line drawn passing through the points (0, 3) and (3, 2).Which equation represents the graphed function?y = –3x + 3y = 3x – 3y = 3x – negative StartFraction 1 Over 3 EndFraction.y = –x + 3
9. Harita must memorize 90 measures of music for her cello solo at a concert. She plans on memorizing 18 new measures for every 3 days of practice. Which equation can be used to determine m, the number of measures Harita still needs to memorize, as a function of d, the number of days of practice since she began learning the piece?
m = 72 – 15d
m = 90 – 6d
m = 101 – 21d
m = 108 – 3d
10. A coordinate plane with a line passing through (negative 4, 3), (0, 1), and (2, 0).What are the slope and y-intercept of the linear function graphed to the left?slope: –2; y-intercept: 2slope: ; y-intercept: 1slope: ; y-intercept: 2slope: 2; y-intercept: 1
y=48x−2
y=48x+2
y=2x−48
y=2x+48
2. Four lines are drawn on a coordinate plane to form trapezoid WXYZ.A coordinate grid with 4 lines. Line X W is drawn with point W at (negative 4, 1) and passes through (0, 2) with and (3, 3). Line Y X is drawn with point Y at (3, 0) and passes through (0, 3). Line Z W is drawn with point Z at (0, negative 3) and point W at (negative 4, 1). LIne Z Y is drawn with point Y at (3, 0) and point Z at (0, negative 3).Which statements are true about the lines? Select three options.Line WZ has the same slope as line XY.Line YX has a greater slope than line ZY.Line XW has a lesser slope than line YZ.Line ZW has the same y-intercept as line YZ.Line XY has a lesser y-intercept than line XW.
3. The graph represents revenue in dollars as a function of greeting cards sold.
A coordinate plane showing Greeting Card Revenue, Number of Cards Sold on the x-axis and Revenue in dollars on the y-axis. A line starts at (0. 0) and passes through (2, 8), (4, 16), and ends at (5, 20).
Which equation represents the function shown on the graph?
y = 1/4 x
y = 1/2 x
y = 2x
y = 4x
4. A coordinate plane with a line drawn passing through the points (0, 3) and (1,1).
Which equation represents the graphed function?
y = –2x + 3
y = 2x + 3
y = 1/2 x + 3
y = –1/2 x + 3
5. What is the slope of the line represented by the equation y = –y equals negative StartFraction one-half EndFraction x plus StartFraction one-fourth EndFraction.x + StartFraction one-fourth EndFraction.?–negative StartFraction one-half EndFraction.–negative StartFraction one-fourth EndFraction.StartFraction one-fourth EndFraction.StartFraction one-half EndFraction.
6. What is the y-intercept of the function f(x) = –negative StartFraction 2 Over 9 EndFraction.x + ?–negative StartFraction 2 Over 9 EndFraction.–negative StartFraction 1 Over 3 EndFraction.StartFraction 1 Over 3 EndFraction.StartFraction 2 Over 9 EndFraction.
7. What is the slope of the line represented by the equation y = y equals StartFraction 4 Over 5 EndFraction x minus 3.x – 3?–3Negative StartFraction 4 Over 5 EndFraction.StartFraction 4 Over 5 EndFraction.3
8. A coordinate plane with a line drawn passing through the points (0, 3) and (3, 2).Which equation represents the graphed function?y = –3x + 3y = 3x – 3y = 3x – negative StartFraction 1 Over 3 EndFraction.y = –x + 3
9. Harita must memorize 90 measures of music for her cello solo at a concert. She plans on memorizing 18 new measures for every 3 days of practice. Which equation can be used to determine m, the number of measures Harita still needs to memorize, as a function of d, the number of days of practice since she began learning the piece?
m = 72 – 15d
m = 90 – 6d
m = 101 – 21d
m = 108 – 3d
10. A coordinate plane with a line passing through (negative 4, 3), (0, 1), and (2, 0).What are the slope and y-intercept of the linear function graphed to the left?slope: –2; y-intercept: 2slope: ; y-intercept: 1slope: ; y-intercept: 2slope: 2; y-intercept: 1
Answers
GPT-4o mini
Let's go through each question one by one:
1. **Profit Equation for the Hot Dog Stand:**
The total profit earned by selling x hot dogs can be expressed as:
- Profit earned from selling x hot dogs = 2x (since $2 profit for each hot dog).
- Total cost = $48.
Thus, the profit equation is:
\[ y = 2x - 48 \]
**Correct Answer:** \( y = 2x - 48 \)
2. **Statements about Lines in Trapezoid WXYZ:**
- We can determine the slopes and y-intercepts based on given points. After evaluating:
- Line WZ and line XY might have the same slope based on their coordinates.
- Line YX has a greater slope than line ZY.
- Line XW has a lesser slope than line YZ.
**True Statements:**
- Line WZ has the same slope as line XY.
- Line YX has a greater slope than line ZY.
- Line XW has a lesser slope than line YZ.
3. **Equation Representing Greeting Card Revenue:**
The equation can be determined by noting that the graph indicates a linear relationship between cards sold and revenue. The change in revenue over the number of cards is consistent:
The slope (rate of revenue increase) can be calculated from the points. Since revenue increases from 0 to 20 as cards go from 0 to 5, this gives us a slope of 4.
Thus, the equation is:
\[ y = 4x \]
**Correct Answer:** \( y = 4x \)
4. **Line Passing Through Points (0, 3) and (1, 1):**
The slope (m) can be calculated using the formula:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} = \frac{(1 - 3)}{(1 - 0)} = -2 \]
So the equation is:
\[ y = -2x + 3 \]
**Correct Answer:** \( y = -2x + 3 \)
5. **Slope from the Given Equation:**
The equation is written in the form \( y = -\frac{1}{2}x + \frac{1}{4} \), where the slope is the coefficient of \( x \):
**Correct Answer:** \(-\frac{1}{2}\)
6. **Y-Intercept of the Function:**
The y-intercept is where \( x = 0 \), hence the y-intercept from the function:
\[ f(x) = -\frac{2}{9}x + \text{constant} \]
So the y-intercept is simplified to given directly (if any constant term is provided).
**Correct Answer:** (based on context)
7. **Slope of the Equation:**
The equation \( y = \frac{4}{5}x - 3 \) has a slope of:
**Correct Answer:** \(\frac{4}{5}\)
8. **Equation from Points (0, 3) and (3, 2):**
Change in y is -1 over a change in x of 3:
\[ m = \frac{(2 - 3)}{(3 - 0)} = -\frac{1}{3} \]
Hence, the equation is:
\[ y = -\frac{1}{3}x + 3 \]
**Correct Answer:** \( y = -\frac{1}{3}x + 3 \)
9. **Equation for Harita's Memorizing Measures:**
Harita memorizes 18 measures every 3 days, thus:
Rate = \( 18/3 = 6 \) measures per day.
Therefore:
\[ m = 90 - 6d \]
**Correct Answer:** \( m = 90 - 6d \)
10. **Slope and Y-Intercept of the Line:**
The slope can be determined using two points on the line:
From points, slope calculation provides a slope of -1/2.
The y-intercept is directly observed from the graph for \( x = 0 \) (1).
Therefore:
**Correct Answer:** Slope: -2; y-intercept: 1.
Feel free to let me know if you need further explanations on any of these solutions!
1. **Profit Equation for the Hot Dog Stand:**
The total profit earned by selling x hot dogs can be expressed as:
- Profit earned from selling x hot dogs = 2x (since $2 profit for each hot dog).
- Total cost = $48.
Thus, the profit equation is:
\[ y = 2x - 48 \]
**Correct Answer:** \( y = 2x - 48 \)
2. **Statements about Lines in Trapezoid WXYZ:**
- We can determine the slopes and y-intercepts based on given points. After evaluating:
- Line WZ and line XY might have the same slope based on their coordinates.
- Line YX has a greater slope than line ZY.
- Line XW has a lesser slope than line YZ.
**True Statements:**
- Line WZ has the same slope as line XY.
- Line YX has a greater slope than line ZY.
- Line XW has a lesser slope than line YZ.
3. **Equation Representing Greeting Card Revenue:**
The equation can be determined by noting that the graph indicates a linear relationship between cards sold and revenue. The change in revenue over the number of cards is consistent:
The slope (rate of revenue increase) can be calculated from the points. Since revenue increases from 0 to 20 as cards go from 0 to 5, this gives us a slope of 4.
Thus, the equation is:
\[ y = 4x \]
**Correct Answer:** \( y = 4x \)
4. **Line Passing Through Points (0, 3) and (1, 1):**
The slope (m) can be calculated using the formula:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} = \frac{(1 - 3)}{(1 - 0)} = -2 \]
So the equation is:
\[ y = -2x + 3 \]
**Correct Answer:** \( y = -2x + 3 \)
5. **Slope from the Given Equation:**
The equation is written in the form \( y = -\frac{1}{2}x + \frac{1}{4} \), where the slope is the coefficient of \( x \):
**Correct Answer:** \(-\frac{1}{2}\)
6. **Y-Intercept of the Function:**
The y-intercept is where \( x = 0 \), hence the y-intercept from the function:
\[ f(x) = -\frac{2}{9}x + \text{constant} \]
So the y-intercept is simplified to given directly (if any constant term is provided).
**Correct Answer:** (based on context)
7. **Slope of the Equation:**
The equation \( y = \frac{4}{5}x - 3 \) has a slope of:
**Correct Answer:** \(\frac{4}{5}\)
8. **Equation from Points (0, 3) and (3, 2):**
Change in y is -1 over a change in x of 3:
\[ m = \frac{(2 - 3)}{(3 - 0)} = -\frac{1}{3} \]
Hence, the equation is:
\[ y = -\frac{1}{3}x + 3 \]
**Correct Answer:** \( y = -\frac{1}{3}x + 3 \)
9. **Equation for Harita's Memorizing Measures:**
Harita memorizes 18 measures every 3 days, thus:
Rate = \( 18/3 = 6 \) measures per day.
Therefore:
\[ m = 90 - 6d \]
**Correct Answer:** \( m = 90 - 6d \)
10. **Slope and Y-Intercept of the Line:**
The slope can be determined using two points on the line:
From points, slope calculation provides a slope of -1/2.
The y-intercept is directly observed from the graph for \( x = 0 \) (1).
Therefore:
**Correct Answer:** Slope: -2; y-intercept: 1.
Feel free to let me know if you need further explanations on any of these solutions!
1. y=2x−48 good!
2. Line WZ has the same slope as line XY. good!
Line XW has a lesser slope than line YZ. good!
Line ZW has the same y-intercept as line YZ. +
3. y = 4x good!
4. y = –2x + 3 good!
5. –1/2 good!
6. 1/3 +
7. 4/5 good!
8. y = –1/3x + 3 good!
9. m = 90 – 6d good!
10. slope: -1/2; y-intercept: 1 +
2. Line WZ has the same slope as line XY. good!
Line XW has a lesser slope than line YZ. good!
Line ZW has the same y-intercept as line YZ. +
3. y = 4x good!
4. y = –2x + 3 good!
5. –1/2 good!
6. 1/3 +
7. 4/5 good!
8. y = –1/3x + 3 good!
9. m = 90 – 6d good!
10. slope: -1/2; y-intercept: 1 +
GPT-4o mini
Great! It looks like you've validated most of my previous responses, but there were a couple of areas where I got the details slightly off. Let’s clarify those points:
6. **Y-Intercept of the function \( f(x) = -\frac{2}{9}x - \frac{1}{3} \):**
The y-intercept is the constant term when \( x = 0 \). If we write the function in slope-intercept form \( y = mx + b \), then the y-intercept (b) is indeed \( -\frac{1}{3} \). If you're interpreting the question differently and you mean that the y-intercept is \( \frac{1}{3} \) based on some visual aspect, please clarify.
10. **Slope and Y-Intercept:**
For the line passing through the points \( (0, 1) \) and \( (2, 0) \):
- The slope calculated would be \( m = \frac{0 - 1}{2 - 0} = -\frac{1}{2} \).
- The y-intercept was at \( (0, 1) \).
Thus:
**Slope: \(-\frac{1}{2}\); y-intercept: 1** (This was consistent with your correction.)
If you have further queries or need additional clarifications on these problems, feel free to ask!
6. **Y-Intercept of the function \( f(x) = -\frac{2}{9}x - \frac{1}{3} \):**
The y-intercept is the constant term when \( x = 0 \). If we write the function in slope-intercept form \( y = mx + b \), then the y-intercept (b) is indeed \( -\frac{1}{3} \). If you're interpreting the question differently and you mean that the y-intercept is \( \frac{1}{3} \) based on some visual aspect, please clarify.
10. **Slope and Y-Intercept:**
For the line passing through the points \( (0, 1) \) and \( (2, 0) \):
- The slope calculated would be \( m = \frac{0 - 1}{2 - 0} = -\frac{1}{2} \).
- The y-intercept was at \( (0, 1) \).
Thus:
**Slope: \(-\frac{1}{2}\); y-intercept: 1** (This was consistent with your correction.)
If you have further queries or need additional clarifications on these problems, feel free to ask!