Rate of Change and Initial Value

Interpreting Rate of Change and Initial Value

Please take notes & concentrate on the questions

Bonnie has decided to join a gym for the next 3 months.

Both gyms Bonnie is researching charge a monthly fee but she noticed that there might be other fees she has to pay.

​How can she compare the two options to determine the better overall value?​

​Bonnie continues to research gym options and finds out that Fabulous Fitness has no sign-up fee, while Greg’s Gym charges $50 for the initial sign-up.

​Take a minute to examine the information before answering the next few questions.

​She makes the following table to help her compare the two options.

Fill in the Blanks

Fill in the Blanks

The initial value of any function represents the value of the function at the beginning.

Initial Value

Fabulous Fitness has an initial value of 0 because they do not charge a sign-up fee.

Greg's Gym has an initial value of 50 because they charge a sign- up fee of $50.

Fill in the Blanks

Fill in the Blanks

The rate of change of any function represents the constant value at which the function increases or decreases.

Rate of Change

Fabulous has a rate of change of 45 because they charge $45 per month

Greg's Gym has a a rate of change of 33 because when split evenly between 3 months, it costs $33 per month.

Multiple Choice

If the data for each gym was graphed on a coordinate plane, which gym would have a point on the origin (0,0)?

• Fabulous Fitness

• Greg's Gym

The owner of Fabulous Fitness wants to buy name tags for her staff.

She writes the following function to represent the cost, y, for a certain number of name tags, x.

y = $2.00x + $7.00

​

Multiple Choice

What is the rate of change (CROC) for the cost of each tag based on the equation?

y = $2.00x + $7.00

• $2.00 per tag

• $7.00 per tag

Multiple Choice

What do you think is the initial (starting) value for the cost of the name tags based on the equation?

y = $2.00x + $7.00

• $2.00

• $7.00

Multiple Choice

What is the rate of change in the equation?
y=8x+6

• 8
• 6

Multiple Choice

Find the rate of change

• 1
• 2
• 11
• Not a constant slope/Non-Linear

Multiple Choice

Find the rate of change.

• -2 minutes per gallon
• -20 minutes per gallon
• -2 gallons per minute
• -20 gallons per minute

Multiple Choice

Tim bought a new phone.  He paid $125 on the day of purchase, then $50 per month until it was paid off. What is the initial value in this situation?

• 50
• 125

Multiple Choice

Tim bought a new phone.  He paid $125 on the day of purchase, then $50 per month until it was paid off. What is the rate of change in this situation?

• 50
• 125

Multiple Choice

What is the initial value of this graph?

• 20000 kg

• 1 h

• 0 kg

• 0 h

Multiple Choice

Analyze the point (3, 27). What does the point represent? 

• The point (3,27) represents 27 shirts for $3. 
• The point (3,27) represents $27 for 3 shirts. 

Multiple Choice

Which of the following are ways to find the rate of change?

• change in y over change in x
• rise over run
• subtract y coordinates over subtract x coordinates
• all of the above

Multiple Choice

Which statement describes the rate of change of the line?

• $20 each week

• $30 each week

• $25 each week

• $70 each week

Multiple Choice

What is the meaning of the rate of change?

• The cost for the service without any hours of calling is $5

• The cost of the service without any hours of calling is $6

• The cost per minute is $0.05

• The cost per minute is $0.50

Multiple Choice

What is the meaning of the initial value?

• The cost for the service without any hours of calling is $5

• The cost of the service without any hours of calling is $6

• The cost per minute is $0.05

• The cost per minute is $0.50

Multiple Choice

What is the meaning of the rate of change?

• Colby's initial deposit was $100.

• Colby's initial deposit was $150

• Colby is adding $5 per year

• Colby is adding $10 per year

Multiple Choice

The amount Mike gets paid weekly can be represented by the equation p(α)=2.75a + 200, where a is the number of fashion accessories he sells that week.

Which of the following statements is true?

I. 2.75 is the amount he gets paid per cellphone accessory

II. a is the number of accessories he sells that week

III. p(a) is the total amount he gets paid

IV. 200 is the amount he is guaranteed to be paid each week

• Statement I only

• Statements I and II

• Statements I, II, and IV only

• All statements I, II, III and IV, are true

Multiple Choice

An employer uses the equation

y = 9x + 100 to determine how much to pay his employees for the holidays including a bonus. In the equation, y is the total money paid and x is the number of hours worked. What is the meaning of the initial value?

• the amount of hours need to earn overtime

• money earned each hour

• the bonus for employees

• the pay for a really good work

​Cost of Name Tags

The graph shows how the cost of the name tags for the equation

y = 2.00x + 7.00


x represents the number of tags purchased

y represents the total cost

Fill in the Blanks

As a reminder, when an equation is written slope-intercept form, the "m" represents the slope/CROC

Now we know that the "b" represents the y-intercept or the "initial value" of the function.

Slope-Intercept Form

OR

INITIAL VALUE!

OR

rate of change!

Multiple Choice

The graph represents the money in a savings account.  

What is the y-intercept and what does it represent in this graph?

• The initial deposit was $30

• $30 is added to the account each week

• It took 30 weeks to same enough money to open an account

• not enough information

Multiple Choice

The graph represents the height of a burning candle.  

What is the y-intercept of the graph and what does it represent?

• the time it takes to burn the entire candle

• the rate at which the candle is losing height

• the change in the height of the candle each hour

• the original height of the candle

Multiple Choice

An air conditioning salesman earns a base salary plus a commission for each air conditioner he sells.

The equation y = 125x + 375 represents his pay based on the number of air conditioners he sells each week.

What is the slope and what does it represent?

• The slope is 375 and represents the amount he earns for each air conditioner.

• The slope is 125 and represents the amount he earns for each air conditioner he sells.

• The slope is 375 and represents the base salary.

• The slope is 125 and represents the base salary.

Multiple Select

Janice is saving money for retirement. The equation y = 500x + 12,000 models her retirement savings where y is the total she has saved in dollars, and x is the number of months.

Which two statements are correct based on the given equation?

• Janice started with $500 in her account

• Janice started with $12,000 in her account

• Each month, Janice saves an additional $500

• Each month, Janice saves an additional $12,000