17. The tables below show the number of jumping jacks completed after a given period of time in seconds. Kimberly: Time (seconds) Jumping Jacks 3 17 8 37 12 53 16 69 Katrina: Time (seconds) Jumping Jacks 2 10 5 25 12 60 20 100 (3 points) a. Which person is doing more jumping jacks per second? b. Which person had done more jumping jacks initially before the timer started? c. Which person shows a proportional relationship?

Answer 1

Let's analyze the data presented for both Kimberly and Katrina to answer the questions.

a. Jumping Jacks Per Second

To find out who is doing more jumping jacks per second, we can calculate the rate of jumping jacks per second.

Kimberly:

• 3 seconds: 17 jumping jacks → 17/3 = 5.67 jacks/second
• 8 seconds: 37 jumping jacks → 37/8 = 4.625 jacks/second
• 12 seconds: 53 jumping jacks → 53/12 = 4.42 jacks/second
• 16 seconds: 69 jumping jacks → 69/16 = 4.3125 jacks/second

Average rate for Kimberly: (5.67 + 4.625 + 4.42 + 4.3125) / 4 ≈ 4.77 jacks/second

Katrina:

• 2 seconds: 10 jumping jacks → 10/2 = 5 jacks/second
• 5 seconds: 25 jumping jacks → 25/5 = 5 jacks/second
• 12 seconds: 60 jumping jacks → 60/12 = 5 jacks/second
• 20 seconds: 100 jumping jacks → 100/20 = 5 jacks/second

Average rate for Katrina: (5 + 5 + 5 + 5) / 4 = 5 jacks/second

Conclusion for a:
Katrina is doing more jumping jacks per second, averaging 5 jacks/second compared to Kimberly's average of approximately 4.77 jacks/second.

b. Jumping Jacks Initially

To find out which person had done more jumping jacks before the timer started, we can refer to the number of jumping jacks each person had completed at their initial time point.

Kimberly:

• At 0 seconds: 0 jumping jacks (not given but implied)

Katrina:

• At 0 seconds: 0 jumping jacks (not given but implied)

Both Kimberly and Katrina started at 0 jumping jacks.

Conclusion for b:
Neither person had done any jumping jacks initially before the timer started; both had completed 0 jumping jacks.

c. Proportional Relationship

To determine which person shows a proportional relationship, we need to see if the ratios of jumping jacks to time remain constant.

Kimberly's Results:

• The rates decrease as time increases; therefore, it does not show a proportional relationship.

Katrina's Results:

• Each rate is consistent at 5 jacks/second, meaning it is directly proportional.

Conclusion for c:
Katrina shows a proportional relationship in the number of jumping jacks to time, as the ratio remains constant across the intervals measured.

Answer 2

18. Function A: on a graph we have (-4, 0) (-2, 1) (0, 2) (2, 3) Function B: X axis has 0 4 8 Y axis has 3 -9 -21 Function C: y=x+5 (4 points) a. Which function has the greatest rate of change (slope)? b. Which function has the smallest y-intercept? c. Which function would you want to represent your money earned per hour? d. Which function would give you the highest initial amount of money?

Answer 3

To analyze the functions and answer the questions, let's break down the information for each function.

Function A: Points Given

The points for Function A are:

• (-4, 0)
• (-2, 1)
• (0, 2)
• (2, 3)

To find the slope (rate of change), we can choose two points from the set and use the slope formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's calculate the slope using two points (for example, (0, 2) and (2, 3)):

• \( \text{slope} = \frac{3 - 2}{2 - 0} = \frac{1}{2} = 0.5 \)

You can check other pairs to confirm the slope remains constant:

• Using (-2, 1) and (0, 2):
  • \( \text{slope} = \frac{2 - 1}{0 - (-2)} = \frac{1}{2} = 0.5 \)

Function A has a slope of \( 0.5 \).

Function B: Points Identified

The X-axis has points at \(0, 4, 8\) and the Y-axis has points at \(3, -9, -21\). We can check the slope between two points (let's take (0, 3) and (4, -9)):

• \( \text{slope} = \frac{-9 - 3}{4 - 0} = \frac{-12}{4} = -3 \)

Next, we check the slope between (4, -9) and (8, -21):

• \( \text{slope} = \frac{-21 - (-9)}{8 - 4} = \frac{-12}{4} = -3 \)

Function B has a slope of \( -3 \).

Function C: \( y = x + 5 \)

The slope for this linear function is \( 1 \) (the coefficient of \( x \)), and the y-intercept is \( 5 \) (the constant term).

Summary of Functions' Slopes:

• Function A: Slope = \( 0.5 \)
• Function B: Slope = \( -3 \)
• Function C: Slope = \( 1 \)

Analyze the Questions:

a. Which function has the greatest rate of change (slope)?
Answer: Function C, with a slope of \(1\).

b. Which function has the smallest y-intercept?
The y-intercepts are:

• Function A: y-intercept at \(2\) (when \(x = 0\)).
• Function B: y-intercept at \(3\) (when \(x = 0\)).
• Function C: y-intercept at \(5\) (when \(x = 0\)).

Answer: Function A has the smallest y-intercept, which is \(2\).

c. Which function would you want to represent your money earned per hour?
To represent money earned per hour, we usually look for a function with a positive slope.

Answer: Function C (slope of \( 1 \)) would be ideal for representing money earned per hour as it indicates consistent earning.

d. Which function would give you the highest initial amount of money?
The initial amount of money is represented by the y-intercept:

Answer: Function C (y-intercept \(5\)) gives you the highest initial amount of money.

Final Summary of Answers:

a. Function C has the greatest rate of change.
b. Function A has the smallest y-intercept.
c. Function C would represent money earned per hour.
d. Function C gives you the highest initial amount of money.