Which table represents viable solutions for y = 5x, where x is the number of tickets sold for the school play and y is the amount of money collected for the tickets?A 2-column table with 4 rows titled School Play Tickets. The first column is labeled tickets (x) with entries negative 100, negative 25, 40, 600. The second column is labeled money collected (y) with entries negative 500, negative 125, 250, 3000.A 2-column table with 4 rows titled School Play Tickets. The first column is labeled tickets (x) with entries negative 20, 20, 100, 109. The second column is labeled money collected (y) with entries negative 100, 100, 500, 545.A 2-column table with 4 rows titled School Play Tickets. The first column is labeled tickets (x) with entries 0, 10, 51, 400. The second column is labeled money collected (y) with entries 0, 50, 255, 2000.A 2-column table with 4 rows titled School Play Tickets. The first column is labeled tickets (x) with entries 5, 65, 80, 200. The second column is labeled money collected (y) with entries 25, 350, 400, 1000.

Question

2. Which table represents a direct variation function?A table with 6 columns and 2 rows. The first row, x, has the entries, negative 3, negative 1, 2, 5, 10. The second row, y, has the entries, negative 4.5, negative 3.0, negative 1.5, 0.0, 1.5.A table with 6 columns and 2 rows. The first row, x, has the entries, negative 5.5, negative 4.5, negative 3.5, negative 2.5, negative 1.5. The second row, y, has the entries, 10, 8, 6, 4, 2.A table with 6 columns and 2 rows. The first row, x, has the entries, negative 5.5, negative 5.5, negative 5.5, negative 5.5, negative 5.5. The second row, y, has the entries, negative 3, negative 1, 2, 5, 10.A table with 6 columns and 2 rows. The first row, x, has the entries, negative 3, negative 1, 2, 5, 10. The second row, y, has the entries, negative 7.5, negative 2.5, 5.0, 12.5, 25.0.

3. How many intersections are there of the graphs of the equations below?

One-halfx + 5y = 6

3x + 30y = 36

none
one
two
infinitely many

4. Line JK passes through points J(–3, 11) and K(1, –3). What is the equation of line JK in standard form?

7x + 2y = –1
7x + 2y = 1
14x + 4y = –1
14x + 4y = 1

5. The function graphed is reflected across the x-axis to create a new function.Which is true about the domain and range of each function?
Both the domain and range change.
Both the range and domain stay the same.
The domain stays the same, but the range changes.
The range stays the same, but the domain changes.

6. Ray was in charge of filling the soccer team’s water jug for practice. At the beginning of practice he filled the jug. Halfway through practice the team took a water break. After the break Ray refilled the jug so the players could have water after practice. At the end of practice Ray drained the jug. Which graph could represent the level of the water in the jug during and after practice? 1. A graph titled Water Break. The horizontal axis shows Practice time (minutes), numbered 6 to 60, and the vertical axis shows Water (quarts), numbered 2 to 20. The line starts at 15 quarts, is constant to 25 minutes, decreases to 5 quarts at 35 minutes, increases to 14 quarts at 37 minutes, decreases. 2. A graph titled Water Break. The horizontal axis shows Practice time (minutes), numbered 6 to 60, and the vertical axis shows Water (quarts), numbered 2 to 20. The line starts at 0 quarts, increases to 15 quarts at 5 minutes, constant to 25 minutes, decreases to 5 quarts at 35 minutes, decreases. 3. A graph titled Water Break. The horizontal axis shows Practice time (minutes), numbered 6 to 60, and the vertical axis shows Water (quarts), numbered 2 to 20. The line starts at 15 quarts, constant to 25 minutes, decreases to 5 quarts at 35 minutes, constant to 55 minutes, decreases. 4. A graph titled Water Break. The horizontal axis shows Practice time (minutes), numbered 6 to 60, and the vertical axis shows Water (quarts), numbered 2 to 20. The line starts at 0 quarts, increases to 15 quarts at 5 minutes, constant to 25 minutes, decreases to 5 quarts at 35 minutes, increases to 14 quarts at 37 minutes, constant to 55 minutes, decreases.

7. Graph the function y = 8 –StartRoot 2 x + 6 EndRoot. Which is the best approximation of a point on the function?

(4.4, 3.421)
(4.6, 4.232)
(5.2, 4.025)
(5.9, 3.781)

8. Elias and Niko are polishing the silver at the heritage museum. Elias could polish all the silver himself in 40 minutes. That task would take Niko 50 minutes to complete alone. Which table is filled in correctly and could be used to determine how long it would take if Elias and Niko polished the silver together? 1. A table showing Rate in part per minute, Time in minutes, and Part of Silver Polished. The First row shows Elias, and has StartFraction 1 Over 40 EndFraction, t, and StartFraction 1 Over 40 EndFraction t. The second row shows Niko and has, StartFraction 1 Over 50 EndFraction, t, and StartFraction 1 Over 50 EndFraction t. 2. A table showing Rate in part per minute, Time in minutes, and Part of Silver Polished. The First row shows Elias, and has r, StartFraction 1 Over 40 EndFraction, and StartFraction 1 Over 40 EndFraction r. The second row shows Niko and has, r, StartFraction 1 Over 50 EndFraction, and StartFraction 1 Over 50 EndFraction r. 3. A table showing Rate in part per minute, Time in minutes, and Part of Silver Polished. The First row shows Elias, and has 40, t, and 40 t. The second row shows Niko and has, 50, t, and 50 t . 4. A table showing Rate in part per minute, Time in minutes, and Part of Silver Polished. The First row shows Elias, and has r, 40, and 40 r. The second row shows Niko and has, r 50, and 50 r .

9. What is the solution to 4|0.5x – 2.5| = 0?

x = 1.25
x = 5
x = –1.25 or x = 1.25
x = –5 or x = 5

10. The graph represents the function where electricity usage in kilowatts per hour of a clock radio varies directly with the number of days it is plugged into the wall current.A coordinate grid showing Clock Radio Electricity Usage, with Days Connected to Current on the x-axis and Electricity Usage in kilowatt hours on the y-axis with a line starting at (0, 0) and passing through (2, 0.5) and (6, 1.5).Which is a reasonable estimate of the constant of variation?
0.25 kWh per day
0.50 kWh per day
2.40 kWh per day
4.00 kWh per day

11. What is the solution to the linear equation?

–12 + 3b – 1 = –5 – b

b = –2
b = –1.5
b = 1.5
b = 2

12. The crew of a garbage truck takes 5 hours to empty all the bins in one neighborhood on pickup day. The city assigned a second garbage truck to this neighborhood. During their training, it took the crew from the second truck 8 hours to empty all the bins.A table showing Rate in part per hour, Time in hours, and Part of Bins Emptied. The first row shows First Truck and has, StartFraction 1 Over 5 EndFraction, t, and StartFraction 1 Over 5 EndFraction times t. The second row shows Second Truck and has, StartFraction 1 Over 8 EndFraction, t, and StartFraction 1 Over 8 EndFraction times t. When the two crews start working together, what part of all the garbage bins will the first garbage truck empty?
0.6
0.5
0.7
0.4

13. Becky created a graph to represent her distance away from home one afternoon. She left home and ran to the park, met some friends and stayed at the park, and then walked back home using the same route. Which graph could Becky have created? 1. A graph titled Becky's distance from home. The horizontal axis shows time (minutes) and the vertical axis shows distance (miles). Both axes are unnumbered. The line shows an increase, constant, and decrease in distance. 2. A graph titled Becky's distance from home. The horizontal axis shows time (minutes) and the vertical axis shows distance (miles). Both axes are unnumbered. The line shows an decrease, constant, and decrease in distance. 3. A graph titled Becky's distance from home. The horizontal axis shows time (minutes) and the vertical axis shows distance (miles). Both axes are unnumbered. The line shows an increase and decrease in distance. 4. A graph titled Becky's distance from home. The horizontal axis shows time (minutes) and the vertical axis shows distance (miles). Both axes are unnumbered. The line shows a constant, decrease, and increase in distance.

14. Percy works two part-time jobs to help pay for college classes. On Monday, he works 3 hours at the library and 2 hours at the coffee cart and earns $36.50. On Tuesday, he works 2 hours at the library and 5 hours at the coffee cart and earns $50. His hourly wage at the library, x, and hourly wage at the coffee cart, y, can be determined using the system of equations below.

3x + 2y = 36.50
2x + 5y = 50.00
At which job does Percy earn the greater hourly wage? How much does Percy earn each hour at this job?

Percy earns a greater hourly wage of $7.00 at the library.
Percy earns a greater hourly wage of $7.00 at the coffee cart.
Percy earns a greater hourly wage of $7.50 at the library.
Percy earns a greater hourly wage of $7.50 at the coffee cart.

15. The table represents a function.A 2-column table with 4 rows. The first column is labeled x with entries negative 5, negative 1, 6, 9. The second column is labeled f of x with entries 4, 0, negative 1, negative 3.What is the value of f(–1)?
f(-1)=3
f(-1)=-1
f(-1)=0
f(-1)=6

16. Yesterday, Selma read 75 pages of her book. If she reads at a pace of 2 pages per minute today, which table shows only viable solutions for the total number of pages she has read, y, after x minutes have elapsed? A 2-column table with 4 rows titled Selma's Reading. The first column is labeled minutes of reading (x) with entries 2, 14, 39, 55. The second column is labeled total pages read (y) with entries 79, 101, 153, 185. A 2-column table with 4 rows titled Selma's Reading. The first column is labeled minutes of reading (x) with entries negative 16, 6, 27, 52. The second column is labeled total pages read (y) with entries 43, 87, 129, 179.A 2-column table with 4 rows titled Selma's Reading. The first column is labeled minutes of reading (x) with entries 0, 19, 32, 47. The second column is labeled total pages read (y) with entries 0, 113, 139, 169. A 2-column table with 4 rows titled Selma's Reading. The first column is labeled minutes of reading (x) with entries 0, 11, 20.25, 58. The second column is labeled total pages read (y) with entries 75, 97, 115.5, 191.

17. A sequence is defined by the recursive function f(n + 1) = one-halff(n). If f(3) = 9 , what is f(1) ?

1
3
27
81

18. Which is the graph of y – 3 = -2/3(x + 6)? 1. A coordinate grid with a line passing through the points at (negative 3, 1), (0, negative 1), and (3, negative 3).  2. A coordinate grid with a line passing through the points at (negative 2, 1), (0, 0), and (4, negative 3). 3. A coordinate grid with a line passing through the points at (negative 3, 0), (0, negative 2), and (3, negative 4).  4. A coordinate grid with a line passing through the points at (negative 3, 3), (0, 1), and (3, negative 1).

19. What is the solution (a, c) to this system of linear equations?

2a – 3c = –6

a + 2c = 11

(negative StartFraction 76 over 7 EndFraction, StartFraction 17 over 7 EndFraction)
(3, –4)
(3, 4)
(StartFraction 87 over 7 EndFraction, negative StartFraction 5 over 7 EndFraction)

20. A function g(x) is defined as shown.

g(x) = StartLayout enlarged left-brace 1st Row 1st column 2 + 3 x, 2nd column 0 less-than-or-equal-to x less-than 4 2nd Row 1st column 0.5 x + 10, 2nd column 4 less-than-or-equal-to x less-than 8 3rd Row 1st column 16, 2nd column x greater-than-or-equal-to 8 EndLayout

What is the value of g(4)?

10
12
14
16

GPT-4o mini · Answer

Let's analyze each question one by one.

Viable solutions for y = 5x:
- We calculate $ y $ from the possible $ x $ values:
  - For $ x = -100 $, $ y = 5(-100) = -500 $
  - For $ x = -25 $, $ y = 5(-25) = -125 $
  - For $ x = 40 $, $ y = 5(40) = 200 $
  - For $ x = 600 $, $ y = 5(600) = 3000 $
- The only table that includes these calculated values with valid $ x $ is:
  - Table 1 (ticks: -100, -25, 40, 600; collects: -500, -125, 250, 3000)
Direct variation function:
- A direct variation function has the form $ y = kx $ where $ k $ is a constant.
- From the options, the only case where $ y/x $ is constant is Table 2 (entrees: -5.5 to -2.5).
Intersection of the equations (One-halfx + 5y = 6; 3x + 30y = 36):
- The second equation can be simplified to one-half of the first equation.
- Therefore, these two lines are the same, so they intersect infinitely many times.
Equation of line JK in standard form:
- The slope between points J(-3, 11) and K(1, -3) is calculated as:
  - \[ m = \frac{-3 - 11}{1 + 3} = -\frac{14}{4} = -\frac{7}{2} \]
- The equation in point-slope form is:
  - \[ y - 11 = -\frac{7}{2}(x + 3) \]
- Standard form gives:
  - 14x + 4y = 1 after rearranging.
Reflection across x-axis:
- Reflecting a function across the x-axis changes the range but keeps the domain the same.
- Thus, The domain stays the same, but the range changes.
Graph representing Ray refilling the jug:
- The graph that reflects the actions: starting full, decreasing for usage, a refill, and draining would be:
- Option 4 (starts at 0, increases to 15 quarts, constant, decreases lower)
Approximation of the function $ y = 8 - \sqrt{2x + 6} $:
- Substituting $ x $ into the function allows checking; calculations indicate:
- The closest answer approximates at (5.9, 3.781).
Table determining time taken together:
- For rates and time, correct entries should effectively yield the work done by Elias and Niko.
- Thus, Table 1 reflects $ e.g., \frac{1}{40} + \frac{1}{50} $.
Solution to $ 4|0.5x - 2.5| = 0 $:
- This leads to $ |0.5x - 2.5| = 0$, yielding $ x = 5$.
- So, the solution is x = 5.
Estimate of constant of variation:
- Checking points [(2,0.5) to (6,1.5)], the slope changes effectively reflect consistent daily usage.
- Thus, viable estimate: 0.50 kWh per day.
Solution to linear equation:
- Simplifying gives $ b = -1.5 $.
- Hence, the answer is b = -1.5.
Part of all bins emptied by the first garbage truck:
- Adjusting rates for teamwork gives a combined effect.
- Thus, effective work fraction for the first truck can be calculated to yield 0.6.
Graph of Becky’s distance from home:
- The logical graph reflects a scenario of increase to the park, staying constant, and returning home.
- Hence, Option 1 is the best representation.
Percy’s hourly wage comparison:
- Solving the system leads to hourly wage comparison, yielding $7.50 at the library.
Value of f(–1):
- The function lookup yields $ f(-1) = 0 $.
- Thus, the answer is f(-1)=0.
Viable solutions for Selma’s reading:
- After valid conversions, the strong candidate with no negative or illogical entries is Table 4 (reflecting adequate minutes).
Value finding from recursive function:
- Revising the sequence, $ f(n) = 9 $ collapses to prior workings, producing $ f(1) = 36 $.
Graph examining equation:
- Translating $ y - 3 = -\frac{2}{3}(x + 6) $ represents reflected values such that:
- Best candidate graph is Option 4.
Solution to the system of equations:
- Proper solving for values leads to $(3, 4)$.
- Hence, the answer is (3, 4).
Value of g(4):
- Defined conditions lead directly to $ g(4) = 12$.
- Thus, the outcome is 12.

This covers all respective queries based on the descriptions and scenarios provided.