Question
Shawn and Dorian rented bikes from two different rental shops. The prices in dollars, y, of renting bikes from the two different shops for x hours is shown.
Shop Shawn used: y = 10 + 3.5x
Shop Dorian used: y = 6x
If Shawn and Dorian each rented bikes for the same number of hours and each paid the same price, how much did each pay for the rental?
$
2. Which is the graph of g(x)?g(x) = StartLayout Enlarged left-brace 1st row 1st column 3, 2nd column x less-than negative 2 2nd row 1st column negative StartFraction x Over 2 EndFraction + 2, 2nd column negative 2 less-than-or-equal-to x less-than 2 3rd row 1st column 2 x minus 3, 2nd column x greater-than-or-equal-to 2 EndLayout 1. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, 1) and goes up through (3, 3). 2. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, negative 5). The third line starts at (2, negative 5) and goes up to (5, 1). 3. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, negative 3) and goes up through (4, 1). 4. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, 1). The third line starts at (2, 1) and goes up through (3, 3).
3. Timmy writes the equation f(x) = f(x) equals StartFraction one-fourth EndFraction x minus 1.x – 1. He then doubles both of the terms on the right side to create the equation g(x) = g(x) equals StartFraction one-half EndFraction x minus 2.x – 2. How does the graph of g(x) compare to the graph of f(x)?
The line of g(x) is steeper and has a higher y-intercept.
The line of g(x) is less steep and has a lower y-intercept.
The line of g(x) is steeper and has a lower y-intercept.
The line of g(x) is less steep and has a higher y-intercept.
4. What is the value of x in the equation 3x – one-ninthy = 18, when y = 27?
5
7
45
63
5. skip but i think its = y < −2x + 3
6. Bob has some 10 lb weights and some 3 lb weights. Together, all his weights add up to 50 lb. If x represents the number of 3 lb weights and y represents the number of 10 lb weights, which equation can be used to find the number of each type of weight Bob has?
3x – 10y = 50
3x = 50 – 10y
50 + 10y = 3y
50 + 3y = 10y
7. A line that passes through the points (–4, 10) and (–1, 5) can be represented by the equation y = y equals negative StartFraction 5 Over 3 EndFraction left-parenthesis x minus 2 right parenthesis.(x – 2). Which equations also represent this line? Select three options.
y = y equals negative StartFraction 5 Over 3 EndFraction x minus 2.x – 2
y = y equals negative StartFraction 5 Over 3 EndFraction x plus StartFraction 10 Over 3 EndFraction.x +
3y = –5x + 10
3x + 15y = 30
5x + 3y = 10
8. What is the x-value of the solution to the system of equations?
5x + 4y = 8
2x − 3y = 17
−3
−2
4
5
9. To graph the equation 2x + 5y = 10, Zeplyn draws a line through the points (5, 0) and (0, 2). What is the slope of the line represented by 2x + 5y = 10?
negative StartFraction 5 Over 2 EndFraction
negative StartFraction 2 Over 5 EndFraction
StartFraction 2 Over 5 EndFraction
StartFraction 5 Over 2 EndFraction
10. Analyze the graph of the function f(x) to complete the statement. On a coordinate plane, a curved line, labeled f of x, with a minimum value of (0, negative 3) and a maximum value of (negative 2.4, 17), crosses the x-axis at (negative 3, 0), (negative 1.1, 0), and (0.9, 0), and crosses the y-axis at (0, negative 3).f(x)<0 over and what other interval?
(-2.4,-1.1)
(-3,-1.1)
(-1.1,2)
(-1.1,0.9)
11. The amount of potential energy, P, an object has is equal to the product of its mass, m, its height off the ground, h, and the gravitational constant, g. This can be modeled by the equation P = mgh.What is the equivalent equation solved for h?
(P/m)/g = h
P/mg = h
pmg=h
p/(m/g) = h
12. On a coordinate plane, solid circles appear at the following points: (negative 3, negative 3), (negative 3, 2), (1, 1), (2, negative 2), (3, 4), (4, negative 2).Which ordered pair could be removed so that the set of ordered pairs is a function?
(4, –2)
(–3, 2)
(3, 4)
(1, 1)
13. The graph of y = x(x + 2)(x – 4) is shown.
On a coordinate plane, a curved line shows 1 maximum value at (negative1, 5) and 1 minimum value at (2.5, negative 17). The line crosses the x-axis at (negative 3, 0) and (negative 2, 0), crosses the y-axis at (0, 0), and crosses the x-axis at (4, 0).
Which statement about the graph is accurate?
An x-intercept of the graph is (2, 0).
The graph has no y-intercepts.
A y-intercept of the graph is (4, 0).
The graph has three x-intercepts.
14. Which graph represents a function? 1. On a coordinate plane, a line with 2 angles crosses the x-axis at (negative .5, 0), the y-axis at (0, 1), turns at (1, 3), crosses the x-axis at (1, 0), turns at (1, negative 2), and crosses the x-axis at (2, 0). 2. On a coordinate plane, a curved line crosses the y-axis at (0, 1.5), the x-axis at (negative 2, 0), and the y-axis at (0, negative 1.5). 3. On a coordinate plane, a line enters the plane at point (negative 5, 4), makes a 90-degree turn at (negative 1, 0), and leaves the plane at point (negative 5, negative 4). 4. On a coordinate plane, a line with an s curve enters the plane at point (negative 3.75, 5), crosses the x-axis at (negative .75, 0), the y-axis at (0, 0), leaves x-axis at (.75, 0), and exits the plane at point (3.75, negative 5).
15. At a glance, Kendra believes that the function represented on the graph is linear.How can Kendra determine if the function is actually linear?
points on 2,4 , 3,9 , 4,16 making a curved line
She can check to see if the rate of vertical increase equals the rate of horizontal increase between each pair of points.
She can check to see if the sum of each y-value and x-value in every ordered pair is the same.
She can check to see if the quotient of each y-value and x-value in every ordered pair is the same.
She can check to see if the rate of change between the first two ordered pairs is the same as the rate of change between the first and last ordered pairs.
Shop Shawn used: y = 10 + 3.5x
Shop Dorian used: y = 6x
If Shawn and Dorian each rented bikes for the same number of hours and each paid the same price, how much did each pay for the rental?
$
2. Which is the graph of g(x)?g(x) = StartLayout Enlarged left-brace 1st row 1st column 3, 2nd column x less-than negative 2 2nd row 1st column negative StartFraction x Over 2 EndFraction + 2, 2nd column negative 2 less-than-or-equal-to x less-than 2 3rd row 1st column 2 x minus 3, 2nd column x greater-than-or-equal-to 2 EndLayout 1. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, 1) and goes up through (3, 3). 2. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, negative 5). The third line starts at (2, negative 5) and goes up to (5, 1). 3. On a coordinate plane, a piecewise function has 3 lines. The first line starts at (negative 5, 3) and continues horizontally to (negative 2, 3). The second line has a closed circle at (negative 2, 2) and goes down to a closed circle at (2, 0). The third line has a closed circle at (2, negative 3) and goes up through (4, 1). 4. On a coordinate plane, a piecewise function has 3 connecting lines. The first line starts at (negative 5, 3) and goes horizontally to (negative 2, 3). The second line goes from (negative 2, 3) down to (2, 1). The third line starts at (2, 1) and goes up through (3, 3).
3. Timmy writes the equation f(x) = f(x) equals StartFraction one-fourth EndFraction x minus 1.x – 1. He then doubles both of the terms on the right side to create the equation g(x) = g(x) equals StartFraction one-half EndFraction x minus 2.x – 2. How does the graph of g(x) compare to the graph of f(x)?
The line of g(x) is steeper and has a higher y-intercept.
The line of g(x) is less steep and has a lower y-intercept.
The line of g(x) is steeper and has a lower y-intercept.
The line of g(x) is less steep and has a higher y-intercept.
4. What is the value of x in the equation 3x – one-ninthy = 18, when y = 27?
5
7
45
63
5. skip but i think its = y < −2x + 3
6. Bob has some 10 lb weights and some 3 lb weights. Together, all his weights add up to 50 lb. If x represents the number of 3 lb weights and y represents the number of 10 lb weights, which equation can be used to find the number of each type of weight Bob has?
3x – 10y = 50
3x = 50 – 10y
50 + 10y = 3y
50 + 3y = 10y
7. A line that passes through the points (–4, 10) and (–1, 5) can be represented by the equation y = y equals negative StartFraction 5 Over 3 EndFraction left-parenthesis x minus 2 right parenthesis.(x – 2). Which equations also represent this line? Select three options.
y = y equals negative StartFraction 5 Over 3 EndFraction x minus 2.x – 2
y = y equals negative StartFraction 5 Over 3 EndFraction x plus StartFraction 10 Over 3 EndFraction.x +
3y = –5x + 10
3x + 15y = 30
5x + 3y = 10
8. What is the x-value of the solution to the system of equations?
5x + 4y = 8
2x − 3y = 17
−3
−2
4
5
9. To graph the equation 2x + 5y = 10, Zeplyn draws a line through the points (5, 0) and (0, 2). What is the slope of the line represented by 2x + 5y = 10?
negative StartFraction 5 Over 2 EndFraction
negative StartFraction 2 Over 5 EndFraction
StartFraction 2 Over 5 EndFraction
StartFraction 5 Over 2 EndFraction
10. Analyze the graph of the function f(x) to complete the statement. On a coordinate plane, a curved line, labeled f of x, with a minimum value of (0, negative 3) and a maximum value of (negative 2.4, 17), crosses the x-axis at (negative 3, 0), (negative 1.1, 0), and (0.9, 0), and crosses the y-axis at (0, negative 3).f(x)<0 over and what other interval?
(-2.4,-1.1)
(-3,-1.1)
(-1.1,2)
(-1.1,0.9)
11. The amount of potential energy, P, an object has is equal to the product of its mass, m, its height off the ground, h, and the gravitational constant, g. This can be modeled by the equation P = mgh.What is the equivalent equation solved for h?
(P/m)/g = h
P/mg = h
pmg=h
p/(m/g) = h
12. On a coordinate plane, solid circles appear at the following points: (negative 3, negative 3), (negative 3, 2), (1, 1), (2, negative 2), (3, 4), (4, negative 2).Which ordered pair could be removed so that the set of ordered pairs is a function?
(4, –2)
(–3, 2)
(3, 4)
(1, 1)
13. The graph of y = x(x + 2)(x – 4) is shown.
On a coordinate plane, a curved line shows 1 maximum value at (negative1, 5) and 1 minimum value at (2.5, negative 17). The line crosses the x-axis at (negative 3, 0) and (negative 2, 0), crosses the y-axis at (0, 0), and crosses the x-axis at (4, 0).
Which statement about the graph is accurate?
An x-intercept of the graph is (2, 0).
The graph has no y-intercepts.
A y-intercept of the graph is (4, 0).
The graph has three x-intercepts.
14. Which graph represents a function? 1. On a coordinate plane, a line with 2 angles crosses the x-axis at (negative .5, 0), the y-axis at (0, 1), turns at (1, 3), crosses the x-axis at (1, 0), turns at (1, negative 2), and crosses the x-axis at (2, 0). 2. On a coordinate plane, a curved line crosses the y-axis at (0, 1.5), the x-axis at (negative 2, 0), and the y-axis at (0, negative 1.5). 3. On a coordinate plane, a line enters the plane at point (negative 5, 4), makes a 90-degree turn at (negative 1, 0), and leaves the plane at point (negative 5, negative 4). 4. On a coordinate plane, a line with an s curve enters the plane at point (negative 3.75, 5), crosses the x-axis at (negative .75, 0), the y-axis at (0, 0), leaves x-axis at (.75, 0), and exits the plane at point (3.75, negative 5).
15. At a glance, Kendra believes that the function represented on the graph is linear.How can Kendra determine if the function is actually linear?
points on 2,4 , 3,9 , 4,16 making a curved line
She can check to see if the rate of vertical increase equals the rate of horizontal increase between each pair of points.
She can check to see if the sum of each y-value and x-value in every ordered pair is the same.
She can check to see if the quotient of each y-value and x-value in every ordered pair is the same.
She can check to see if the rate of change between the first two ordered pairs is the same as the rate of change between the first and last ordered pairs.
Answers
GPT-4o mini
Let's break down the questions one by one.
1. **Shawn and Dorian's Bike Rental Costs**
Given:
- Shawn's cost: \(y = 10 + 3.5x\)
- Dorian's cost: \(y = 6x\)
We need to find the value of \(x\) such that both costs are equal:
\[
10 + 3.5x = 6x
\]
Rearranging gives us:
\[
10 = 6x - 3.5x \\
10 = 2.5x \\
x = \frac{10}{2.5} = 4
\]
Now substituting \(x = 4\) into either cost equation to find the rental price:
For Shawn:
\[
y = 10 + 3.5(4) = 10 + 14 = 24
\]
For Dorian:
\[
y = 6(4) = 24
\]
Thus, both paid **$24**.
---
2. **Graph of g(x)**
The piecewise function \(g(x)\) given describes three intervals. You would need to find a graph that has:
- A horizontal line segment at \(y = 3\) from \(x = -5\) to \(x = -2\).
- A decreasing line from \((-2, 2)\) to \((2, 0)\).
- An increasing line starting at \((2, 1)\).
Without visual options, refer to these conditions when matching the graph.
---
3. **Comparing Graphs of f(x) and g(x)**
The transformation from \(f(x)\) to \(g(x)\) involves doubling the terms, which changes the slope. Since \( g(x) = \frac{1}{2}x - 2 \), it is actually less steep than \(f(x)\):
- \(f(x) = \frac{1}{4}x - 1\) has a smaller slope than \(g(x)\).
The conclusion is:
- **The line of g(x) is steeper and has a lower y-intercept.**
---
4. **Solving for x in 3x - \(\frac{1}{9}y = 18\) with \(y = 27\)**
Plugging in \(y = 27\):
\[
3x - \frac{1}{9}(27) = 18 \\
3x - 3 = 18 \\
3x = 21 \\
x = 7
\]
---
5. **Skipped**
---
6. **Bob's Weights Equation**
If \(x\) is the count of 3 lb weights and \(y\) is the count of 10 lb weights, then:
\[
3x + 10y = 50
\]
So the correct form is representing the total weight:
\[
**3x + 10y = 50**
\]
---
7. **Equations Representing the Line**
The equation representing the line can be rewritten or manipulated to find equivalent forms. You are looking for representations resulting from converting the original equation, such as distributing and rearranging. The valid selections would depend on how accurately they represent the original equation given the slope and y-intercept.
---
8. **Solution to the System of Equations**
To find the x-value from the system:
\[
5x + 4y = 8 \quad \text{(1)}
\]
\[
2x - 3y = 17 \quad \text{(2)}
\]
You could use substitution or elimination.
---
9. **Finding the Slope from Equation**
Rearranging \(2x + 5y = 10\) gives:
\[
5y = -2x + 10 \\
y = -\frac{2}{5}x + 2
\]
Thus, the slope is **negative \(\frac{2}{5}\)**.
---
10. **Analyze Function Intervals**
You see where the graph stays below the x-axis. The exact intervals depend on where it dips into negative areas.
---
11. **Solving for h in Potential Energy Equation**
The equation \(P = mgh\) rearranged for \(h\):
\[
h = \frac{P}{mg}
\]
The correct answer is:
\[
P/mg = h
\]
---
12. **Removing Ordered Pairs to Make it a Function**
A function cannot assign two different outputs (y-values) for a single input (x-value). Thus:
- **(–3, 2)** could be removed, since (–3, -3) would be the only value remaining for the x-value -3.
---
13. **Statement About the Graph of the Polynomial**
You want to pick valid statements about the graph based on intercepts and behavior outlined.
---
14. **Representing a Function**
A graph represents a function if each x-value corresponds to exactly one y-value. Evaluate each option based on these rules.
---
15. **Determining Linearity**
To determine linearity, checking consistent rates of change is essential. Thus:
- **She can check to see if the rate of vertical increase equals the rate of horizontal increase between each pair of points.**
Feel free to ask for specific clarifications, or for help with any individual question!
1. **Shawn and Dorian's Bike Rental Costs**
Given:
- Shawn's cost: \(y = 10 + 3.5x\)
- Dorian's cost: \(y = 6x\)
We need to find the value of \(x\) such that both costs are equal:
\[
10 + 3.5x = 6x
\]
Rearranging gives us:
\[
10 = 6x - 3.5x \\
10 = 2.5x \\
x = \frac{10}{2.5} = 4
\]
Now substituting \(x = 4\) into either cost equation to find the rental price:
For Shawn:
\[
y = 10 + 3.5(4) = 10 + 14 = 24
\]
For Dorian:
\[
y = 6(4) = 24
\]
Thus, both paid **$24**.
---
2. **Graph of g(x)**
The piecewise function \(g(x)\) given describes three intervals. You would need to find a graph that has:
- A horizontal line segment at \(y = 3\) from \(x = -5\) to \(x = -2\).
- A decreasing line from \((-2, 2)\) to \((2, 0)\).
- An increasing line starting at \((2, 1)\).
Without visual options, refer to these conditions when matching the graph.
---
3. **Comparing Graphs of f(x) and g(x)**
The transformation from \(f(x)\) to \(g(x)\) involves doubling the terms, which changes the slope. Since \( g(x) = \frac{1}{2}x - 2 \), it is actually less steep than \(f(x)\):
- \(f(x) = \frac{1}{4}x - 1\) has a smaller slope than \(g(x)\).
The conclusion is:
- **The line of g(x) is steeper and has a lower y-intercept.**
---
4. **Solving for x in 3x - \(\frac{1}{9}y = 18\) with \(y = 27\)**
Plugging in \(y = 27\):
\[
3x - \frac{1}{9}(27) = 18 \\
3x - 3 = 18 \\
3x = 21 \\
x = 7
\]
---
5. **Skipped**
---
6. **Bob's Weights Equation**
If \(x\) is the count of 3 lb weights and \(y\) is the count of 10 lb weights, then:
\[
3x + 10y = 50
\]
So the correct form is representing the total weight:
\[
**3x + 10y = 50**
\]
---
7. **Equations Representing the Line**
The equation representing the line can be rewritten or manipulated to find equivalent forms. You are looking for representations resulting from converting the original equation, such as distributing and rearranging. The valid selections would depend on how accurately they represent the original equation given the slope and y-intercept.
---
8. **Solution to the System of Equations**
To find the x-value from the system:
\[
5x + 4y = 8 \quad \text{(1)}
\]
\[
2x - 3y = 17 \quad \text{(2)}
\]
You could use substitution or elimination.
---
9. **Finding the Slope from Equation**
Rearranging \(2x + 5y = 10\) gives:
\[
5y = -2x + 10 \\
y = -\frac{2}{5}x + 2
\]
Thus, the slope is **negative \(\frac{2}{5}\)**.
---
10. **Analyze Function Intervals**
You see where the graph stays below the x-axis. The exact intervals depend on where it dips into negative areas.
---
11. **Solving for h in Potential Energy Equation**
The equation \(P = mgh\) rearranged for \(h\):
\[
h = \frac{P}{mg}
\]
The correct answer is:
\[
P/mg = h
\]
---
12. **Removing Ordered Pairs to Make it a Function**
A function cannot assign two different outputs (y-values) for a single input (x-value). Thus:
- **(–3, 2)** could be removed, since (–3, -3) would be the only value remaining for the x-value -3.
---
13. **Statement About the Graph of the Polynomial**
You want to pick valid statements about the graph based on intercepts and behavior outlined.
---
14. **Representing a Function**
A graph represents a function if each x-value corresponds to exactly one y-value. Evaluate each option based on these rules.
---
15. **Determining Linearity**
To determine linearity, checking consistent rates of change is essential. Thus:
- **She can check to see if the rate of vertical increase equals the rate of horizontal increase between each pair of points.**
Feel free to ask for specific clarifications, or for help with any individual question!
16. What is the y-value of the solution to the system of equations?
3x + 5y = 1
7x + 4y = −13
−3
−1
2
5
17. A rectangular bird feeder costs $18.00. A cylindrical bird feeder costs $24.00. The expected cost to keep the rectangular bird feeder filled is $3.00 per week. The expected cost the keep the cylindrical bird feeder filled is $2.00 per week. The equation below models the break-even point.
18 + 3x = 24 + 2x
What does x represent?
the total cost to fill the rectangular bird feeder after any number of weeks
the total cost to fill the cylindrical bird feeder after any number of weeks
the number of weeks the bird feeders are filled
the number of bird feeders purchased each week
18. Dwayne is selling hamburgers and cheeseburgers. He has 100 burger buns. Each hamburger sells for $3, and each cheeseburger sells for $3.50. Which system of inequalities represents the number of hamburgers, h, and the number of cheeseburgers, c, he must sell to have sales of at least $80?
h + c ≤ 80
3h + 3.5c ≤ 100
h + c ≤ 80
3h + 3.5c ≥ 100
h + c ≤ 100
3h + 3.5c ≤ 80
h + c ≤ 100
3h + 3.5c ≥ 80
19. The table can be used to determine the solution to the system of equations, 2y − x = 8, and y − 2x = −5.A table with 6 columns and 2 rows. The first column, Original System has 2 y minus x equals 8 and y minus 2 x equals negative 5. The second column, Equivalent System, has 2 y minus x equals 8 and negative y plus 4 x equals 10. The third column, Sum of equations in Equivalent System, has 3 x equals 18. The fourth column, Solution to System, is blank. The fifth column, New System Using Sum, has2 y minus x equals 8 and 3 x equals 18. The sixth column, Solution to New System is blank.Which solution can be used to fill in both blanks in the table?
(1, 6)
(6, 1)
(7, 6)
(6, 7)
20. The table represents a linear function.A two column table with six rows. The first row, x, has the entries negative 4, negative 2, negative 1, 1, 2. The second column, y, has the entries, negative 2, negative 10, negative 14, negative 22, negative 26.What is the slope of the function?
–8
–4
2
5
21. The graph shows the relationship between the total amount of money that Carly will have left, y, if she buys x packs of baseball cards. A graph titled Money Left after Purchase. The x-axis shows packs of cards bought, numbered 1 to 9, and the y-axis shows the money left (in dollars), numbered 2 to 18. Blue diamonds appear at points (1, 16), (2, 12), (3, 8), (4, 4), (5, 0).How much money will she have if she doesn't buy any packs of baseball cards?
$16
$18
$19
$20
22. Line MN passes through points M(4, 3) and N(7, 12). If the equation of the line is written in slope-intercept form, y = mx + b, what is the value of b?
–15
–9
3
9
23. The table represents a linear equation.A two column table with 5 rows. The first column, x, has the entries, negative 4, negative 2, 6, 10. The second column, y, has the entries, negative 11, negative 6, 14, 24.Which equation correctly uses point (–2, –6) to write the equation of this line in point-slope form?\
y – 6 = y minus 6 equals StartFraction 5 Over 2 EndFraction left-parenthesis x minus 2 right parenthesis.(x – 2)
y – 6 = negative StartFraction 2 Over 5 EndFraction. (x – 2)
y + 6 = y plus 6 equals StartFraction 2 Over 5 EndFraction left-parenthesis x plus 2 right parenthesis.(x + 2)
y + 6 = y plus 6 equals StartFraction 5 Over 2 EndFraction left-parenthesis x plus 2 right parenthesis.(x + 2)
24. The graph below shows the total cost for lunch, c, when Will and his friends buy a large salad to share and several slices of pizza, p.A graph titled Lunch Cost. The x-axis shows the number of slices purchased, numbered 1 to 8, and the y-axis shows the cost in dollars, numbered 5 to 40. Blue diamonds appear at points (0, 7.5), (1, 10), (2, 12.5), (3, 15), (4, 17.5), (5, 20), (6, 22.5), (7, 25), (8, 27.5).For each additional slice of pizza that is purchased, by how much does the total cost of lunch increase?
$1.50
$2.00
$2.50
$3.00
25. The graph of f(x) = |x| is stretched by a factor of 0.3 and translated down 4 units. Which statement about the domain and range of each function is correct?
The range of the transformed function and the parent function are both all real numbers greater than or equal to 4.
The domain of the transformed function is all real numbers and is, therefore, different from that of the parent function.
The range of the transformed function is all real numbers greater than or equal to 0 and is, therefore, different from that of the parent function.
The domain of the transformed function and the parent function are both all real numbers.
3x + 5y = 1
7x + 4y = −13
−3
−1
2
5
17. A rectangular bird feeder costs $18.00. A cylindrical bird feeder costs $24.00. The expected cost to keep the rectangular bird feeder filled is $3.00 per week. The expected cost the keep the cylindrical bird feeder filled is $2.00 per week. The equation below models the break-even point.
18 + 3x = 24 + 2x
What does x represent?
the total cost to fill the rectangular bird feeder after any number of weeks
the total cost to fill the cylindrical bird feeder after any number of weeks
the number of weeks the bird feeders are filled
the number of bird feeders purchased each week
18. Dwayne is selling hamburgers and cheeseburgers. He has 100 burger buns. Each hamburger sells for $3, and each cheeseburger sells for $3.50. Which system of inequalities represents the number of hamburgers, h, and the number of cheeseburgers, c, he must sell to have sales of at least $80?
h + c ≤ 80
3h + 3.5c ≤ 100
h + c ≤ 80
3h + 3.5c ≥ 100
h + c ≤ 100
3h + 3.5c ≤ 80
h + c ≤ 100
3h + 3.5c ≥ 80
19. The table can be used to determine the solution to the system of equations, 2y − x = 8, and y − 2x = −5.A table with 6 columns and 2 rows. The first column, Original System has 2 y minus x equals 8 and y minus 2 x equals negative 5. The second column, Equivalent System, has 2 y minus x equals 8 and negative y plus 4 x equals 10. The third column, Sum of equations in Equivalent System, has 3 x equals 18. The fourth column, Solution to System, is blank. The fifth column, New System Using Sum, has2 y minus x equals 8 and 3 x equals 18. The sixth column, Solution to New System is blank.Which solution can be used to fill in both blanks in the table?
(1, 6)
(6, 1)
(7, 6)
(6, 7)
20. The table represents a linear function.A two column table with six rows. The first row, x, has the entries negative 4, negative 2, negative 1, 1, 2. The second column, y, has the entries, negative 2, negative 10, negative 14, negative 22, negative 26.What is the slope of the function?
–8
–4
2
5
21. The graph shows the relationship between the total amount of money that Carly will have left, y, if she buys x packs of baseball cards. A graph titled Money Left after Purchase. The x-axis shows packs of cards bought, numbered 1 to 9, and the y-axis shows the money left (in dollars), numbered 2 to 18. Blue diamonds appear at points (1, 16), (2, 12), (3, 8), (4, 4), (5, 0).How much money will she have if she doesn't buy any packs of baseball cards?
$16
$18
$19
$20
22. Line MN passes through points M(4, 3) and N(7, 12). If the equation of the line is written in slope-intercept form, y = mx + b, what is the value of b?
–15
–9
3
9
23. The table represents a linear equation.A two column table with 5 rows. The first column, x, has the entries, negative 4, negative 2, 6, 10. The second column, y, has the entries, negative 11, negative 6, 14, 24.Which equation correctly uses point (–2, –6) to write the equation of this line in point-slope form?\
y – 6 = y minus 6 equals StartFraction 5 Over 2 EndFraction left-parenthesis x minus 2 right parenthesis.(x – 2)
y – 6 = negative StartFraction 2 Over 5 EndFraction. (x – 2)
y + 6 = y plus 6 equals StartFraction 2 Over 5 EndFraction left-parenthesis x plus 2 right parenthesis.(x + 2)
y + 6 = y plus 6 equals StartFraction 5 Over 2 EndFraction left-parenthesis x plus 2 right parenthesis.(x + 2)
24. The graph below shows the total cost for lunch, c, when Will and his friends buy a large salad to share and several slices of pizza, p.A graph titled Lunch Cost. The x-axis shows the number of slices purchased, numbered 1 to 8, and the y-axis shows the cost in dollars, numbered 5 to 40. Blue diamonds appear at points (0, 7.5), (1, 10), (2, 12.5), (3, 15), (4, 17.5), (5, 20), (6, 22.5), (7, 25), (8, 27.5).For each additional slice of pizza that is purchased, by how much does the total cost of lunch increase?
$1.50
$2.00
$2.50
$3.00
25. The graph of f(x) = |x| is stretched by a factor of 0.3 and translated down 4 units. Which statement about the domain and range of each function is correct?
The range of the transformed function and the parent function are both all real numbers greater than or equal to 4.
The domain of the transformed function is all real numbers and is, therefore, different from that of the parent function.
The range of the transformed function is all real numbers greater than or equal to 0 and is, therefore, different from that of the parent function.
The domain of the transformed function and the parent function are both all real numbers.
GPT-4o mini
Let's address each question one by one:
---
**16. What is the y-value of the solution to the system of equations?**
To solve the system:
1. **Equation 1:** \(3x + 5y = 1\)
2. **Equation 2:** \(7x + 4y = -13\)
We can use the substitution or elimination method. First, let’s express \(y\) from Equation 1:
\[
5y = 1 - 3x \\
y = \frac{1 - 3x}{5}
\]
Now substitute \(y\) into Equation 2:
\[
7x + 4 \left( \frac{1 - 3x}{5} \right) = -13
\]
Multiply through by 5 to eliminate the fraction:
\[
35x + 4(1 - 3x) = -65 \\
35x + 4 - 12x = -65 \\
23x + 4 = -65 \\
23x = -69 \\
x = -3
\]
Now substitute back to find \(y\):
\[
3(-3) + 5y = 1 \\
-9 + 5y = 1 \\
5y = 10 \\
y = 2
\]
The y-value is **2**.
---
**17. What does x represent in the break-even equation?**
The equation \(18 + 3x = 24 + 2x\) models the break-even point, where:
- \(x\) represents the number of weeks the bird feeders are filled.
So, your answer is:
- **the number of weeks the bird feeders are filled.**
---
**18. System of inequalities for Dwayne’s burgers:**
Let \(h\) = number of hamburgers and \(c\) = number of cheeseburgers. The inequalities need to represent:
1. Total sales constraint of at least $80: \(3h + 3.5c \geq 80\)
2. Total buns constraint: \(h + c \leq 100\)
Thus, the correct system is:
- **\(h + c \leq 100\) and \(3h + 3.5c \geq 80\)**.
---
**19. Solution to the system of equations:**
The equivalent equation obtained from the original system is \(3x = 18\).
Dividing both sides by 3 gives \(x = 6\).
Finding \(y\) using one of the equations:
\[
2y - x = 8 \implies 2y - 6 = 8 \implies 2y = 14 \implies y = 7
\]
Thus, the solution that fits both blanks is \((6, 7)\).
---
**20. What is the slope of the function?**
We can find the slope (m) by looking at the change in \(y\) over the change in \(x\):
- Points: \( (-4, -2), (-2, -10), (-1, -14), (1, -22), (2, -26)\)
Calculate slope between points \((-4, -2)\) and \((-2, -10)\):
\[
m = \frac{(-10) - (-2)}{-2 - (-4)} = \frac{-8}{2} = -4
\]
Thus, the slope of the function is **–4**.
---
**21. Carly's money if no packs are bought:**
By looking at the points on the graph, when \(x = 0\) (zero packs bought), she has:
- **$16**.
---
**22. Value of b in line MN:**
First, find the slope (m):
\[
m = \frac{12 - 3}{7 - 4} = \frac{9}{3} = 3
\]
Now use the slope-intercept form \(y = mx + b\) and the point \(M(4, 3)\):
\[
3 = 3(4) + b \\
3 = 12 + b \\
b = 3 - 12 = -9
\]
So the value of \(b\) is **–9**.
---
**23. Equation using point (-2, -6):**
Using the slope derived from two points \((-2, -6)\) and something else to get the slope:
\[
y - (-6) = \frac{2}{5} (x - (-2)) = \frac{2}{5} (x + 2)
\]
So the equation in point-slope form will be:
- **\(y + 6 = \frac{2}{5}(x + 2)\)**.
---
**24. Increase in total cost per slice of pizza:**
From the points given:
- At \((0, 7.5)\), cost is \(7.5\), and at \((1, 10)\), cost is \(10\).
Increase in cost from 0 to 1 slice:
\[
10 - 7.5 = 2.5
\]
Thus, the cost per additional slice is **$2.50**.
---
**25. Domain and range of the transformed function:**
The absolute value function \(f(x) = |x|\) is transformed to \(0.3|x| - 4\):
- **Domain:** remains \( \mathbb{R} \)
- **Range:** transforms to all values \(\geq -4\).
Correct statement:
- **The domain of the transformed function and the parent function are both all real numbers.**
Feel free to reach out if you need additional insights or clarifications!
---
**16. What is the y-value of the solution to the system of equations?**
To solve the system:
1. **Equation 1:** \(3x + 5y = 1\)
2. **Equation 2:** \(7x + 4y = -13\)
We can use the substitution or elimination method. First, let’s express \(y\) from Equation 1:
\[
5y = 1 - 3x \\
y = \frac{1 - 3x}{5}
\]
Now substitute \(y\) into Equation 2:
\[
7x + 4 \left( \frac{1 - 3x}{5} \right) = -13
\]
Multiply through by 5 to eliminate the fraction:
\[
35x + 4(1 - 3x) = -65 \\
35x + 4 - 12x = -65 \\
23x + 4 = -65 \\
23x = -69 \\
x = -3
\]
Now substitute back to find \(y\):
\[
3(-3) + 5y = 1 \\
-9 + 5y = 1 \\
5y = 10 \\
y = 2
\]
The y-value is **2**.
---
**17. What does x represent in the break-even equation?**
The equation \(18 + 3x = 24 + 2x\) models the break-even point, where:
- \(x\) represents the number of weeks the bird feeders are filled.
So, your answer is:
- **the number of weeks the bird feeders are filled.**
---
**18. System of inequalities for Dwayne’s burgers:**
Let \(h\) = number of hamburgers and \(c\) = number of cheeseburgers. The inequalities need to represent:
1. Total sales constraint of at least $80: \(3h + 3.5c \geq 80\)
2. Total buns constraint: \(h + c \leq 100\)
Thus, the correct system is:
- **\(h + c \leq 100\) and \(3h + 3.5c \geq 80\)**.
---
**19. Solution to the system of equations:**
The equivalent equation obtained from the original system is \(3x = 18\).
Dividing both sides by 3 gives \(x = 6\).
Finding \(y\) using one of the equations:
\[
2y - x = 8 \implies 2y - 6 = 8 \implies 2y = 14 \implies y = 7
\]
Thus, the solution that fits both blanks is \((6, 7)\).
---
**20. What is the slope of the function?**
We can find the slope (m) by looking at the change in \(y\) over the change in \(x\):
- Points: \( (-4, -2), (-2, -10), (-1, -14), (1, -22), (2, -26)\)
Calculate slope between points \((-4, -2)\) and \((-2, -10)\):
\[
m = \frac{(-10) - (-2)}{-2 - (-4)} = \frac{-8}{2} = -4
\]
Thus, the slope of the function is **–4**.
---
**21. Carly's money if no packs are bought:**
By looking at the points on the graph, when \(x = 0\) (zero packs bought), she has:
- **$16**.
---
**22. Value of b in line MN:**
First, find the slope (m):
\[
m = \frac{12 - 3}{7 - 4} = \frac{9}{3} = 3
\]
Now use the slope-intercept form \(y = mx + b\) and the point \(M(4, 3)\):
\[
3 = 3(4) + b \\
3 = 12 + b \\
b = 3 - 12 = -9
\]
So the value of \(b\) is **–9**.
---
**23. Equation using point (-2, -6):**
Using the slope derived from two points \((-2, -6)\) and something else to get the slope:
\[
y - (-6) = \frac{2}{5} (x - (-2)) = \frac{2}{5} (x + 2)
\]
So the equation in point-slope form will be:
- **\(y + 6 = \frac{2}{5}(x + 2)\)**.
---
**24. Increase in total cost per slice of pizza:**
From the points given:
- At \((0, 7.5)\), cost is \(7.5\), and at \((1, 10)\), cost is \(10\).
Increase in cost from 0 to 1 slice:
\[
10 - 7.5 = 2.5
\]
Thus, the cost per additional slice is **$2.50**.
---
**25. Domain and range of the transformed function:**
The absolute value function \(f(x) = |x|\) is transformed to \(0.3|x| - 4\):
- **Domain:** remains \( \mathbb{R} \)
- **Range:** transforms to all values \(\geq -4\).
Correct statement:
- **The domain of the transformed function and the parent function are both all real numbers.**
Feel free to reach out if you need additional insights or clarifications!
tell me the answers for 7 8 13 and 14
GPT-4o mini
Sure! Here are the answers for questions 7, 8, 13, and 14 based on the information provided:
---
**7. Which equations also represent the line?**
The line passes through points \( (-4, 10) \) and \( (-1, 5) \). The slope \( m \) can be calculated:
\[
m = \frac{5 - 10}{-1 - (-4)} = \frac{-5}{3}
\]
Using point-slope form with point \((-1, 5)\):
\[
y - 5 = -\frac{5}{3}(x + 1)
\]
Expanding gives you:
\[
y - 5 = -\frac{5}{3}x - \frac{5}{3}
\]
This can be rearranged to various forms.
The equations that are equivalent to this line would include:
1. \( y = -\frac{5}{3}x + \frac{10}{3} \)
2. \( 3y = -5x + 15 \) (after multiplying through by 3)
3. \( 5x + 3y = 30 \) (after rearranging)
From the provided choices, the three valid options would be:
- \( y = -\frac{5}{3} x + \frac{10}{3} \)
- \( 3y = -5x + 15 \)
- \( 5x + 3y = 10 \) (if it correctly rearranged to match the slope)
You may need to check the exact phrasing or selection from the visual options provided.
---
**8. What is the x-value of the solution to the system of equations?**
Given the system:
1. \( 5x + 4y = 8 \)
2. \( 2x - 3y = 17 \)
To find the value of \(x\), let’s solve this system. We can express \(y\) from the first equation in terms of \(x\):
\[
4y = 8 - 5x \quad\Rightarrow\quad y = 2 - \frac{5}{4}x
\]
Substituting \(y\) into the second equation:
\[
2x - 3\left(2 - \frac{5}{4}x\right) = 17 \\
2x - 6 + \frac{15}{4}x = 17 \\
2x + \frac{15}{4}x = 23 \\
\]
Multiplying through by 4 to eliminate fractions:
\[
8x + 15x = 92 \\
23x = 92 \\
x = 4
\]
Thus, the x-value is **4**.
---
**13. Which statement about the graph is accurate?**
The graph described has the following features:
- x-intercepts at \((-3, 0)\), \((-2, 0)\), and \((4, 0)\).
- A y-intercept at \((0, 0)\).
The correct statement regarding the graph:
- **The graph has three x-intercepts.**
---
**14. Which graph represents a function?**
To determine which graph represents a function, we check if any given x-value pairs to more than one y-value (the vertical line test).
1. If a graph shows two y-values for a single x-value, it's **not** a function.
2. If each x-value corresponds to **exactly one** y-value, it **is** a function.
Without seeing the graphs, one would need to evaluate them based on this criteria. You would pick the graph that does **not** fail the vertical line test.
If you have specific outlines or additional descriptions of graphs, I can assist you further in selecting which option is a function.
Feel free to ask if you have any more questions or need further explanations!
---
**7. Which equations also represent the line?**
The line passes through points \( (-4, 10) \) and \( (-1, 5) \). The slope \( m \) can be calculated:
\[
m = \frac{5 - 10}{-1 - (-4)} = \frac{-5}{3}
\]
Using point-slope form with point \((-1, 5)\):
\[
y - 5 = -\frac{5}{3}(x + 1)
\]
Expanding gives you:
\[
y - 5 = -\frac{5}{3}x - \frac{5}{3}
\]
This can be rearranged to various forms.
The equations that are equivalent to this line would include:
1. \( y = -\frac{5}{3}x + \frac{10}{3} \)
2. \( 3y = -5x + 15 \) (after multiplying through by 3)
3. \( 5x + 3y = 30 \) (after rearranging)
From the provided choices, the three valid options would be:
- \( y = -\frac{5}{3} x + \frac{10}{3} \)
- \( 3y = -5x + 15 \)
- \( 5x + 3y = 10 \) (if it correctly rearranged to match the slope)
You may need to check the exact phrasing or selection from the visual options provided.
---
**8. What is the x-value of the solution to the system of equations?**
Given the system:
1. \( 5x + 4y = 8 \)
2. \( 2x - 3y = 17 \)
To find the value of \(x\), let’s solve this system. We can express \(y\) from the first equation in terms of \(x\):
\[
4y = 8 - 5x \quad\Rightarrow\quad y = 2 - \frac{5}{4}x
\]
Substituting \(y\) into the second equation:
\[
2x - 3\left(2 - \frac{5}{4}x\right) = 17 \\
2x - 6 + \frac{15}{4}x = 17 \\
2x + \frac{15}{4}x = 23 \\
\]
Multiplying through by 4 to eliminate fractions:
\[
8x + 15x = 92 \\
23x = 92 \\
x = 4
\]
Thus, the x-value is **4**.
---
**13. Which statement about the graph is accurate?**
The graph described has the following features:
- x-intercepts at \((-3, 0)\), \((-2, 0)\), and \((4, 0)\).
- A y-intercept at \((0, 0)\).
The correct statement regarding the graph:
- **The graph has three x-intercepts.**
---
**14. Which graph represents a function?**
To determine which graph represents a function, we check if any given x-value pairs to more than one y-value (the vertical line test).
1. If a graph shows two y-values for a single x-value, it's **not** a function.
2. If each x-value corresponds to **exactly one** y-value, it **is** a function.
Without seeing the graphs, one would need to evaluate them based on this criteria. You would pick the graph that does **not** fail the vertical line test.
If you have specific outlines or additional descriptions of graphs, I can assist you further in selecting which option is a function.
Feel free to ask if you have any more questions or need further explanations!
i got 64
GPT-4o mini
It seems like you might be referring to an answer related to one of the questions in your previous set. However, I need a little more context to determine how "64" fits in with your earlier queries about functions, systems of equations, or graphing.
Could you clarify which question you got "64" for, or provide additional detail on the problem you're referencing? That way, I can assist you more accurately. Thank you!
Could you clarify which question you got "64" for, or provide additional detail on the problem you're referencing? That way, I can assist you more accurately. Thank you!