Beauty Salon Slopes: Finding Rate of Change and Initial Value

Materials Needed

• Paper or notebook
• Pencil and colored pencils or highlighters
• Graph paper or a digital graphing tool
• Calculator, optional
• Practice worksheet and answer key below

Lesson Information

Standard: MA.912.AR.2.4 — Given a table, equation, or written description of a linear function, graph the function and determine and interpret its key features.

Topic: Beauty Salon

Age: 16

Duration: 15 minutes

Student-Friendly Goal: I can determine the rate of change and initial value of a linear function from an input-output table or graph, and explain what they mean in a beauty salon situation.

Learning Objectives

By the end of the lesson, the learner will be able to:

1. Identify the rate of change, or slope, from a table or graph.
2. Identify the initial value, or y-intercept, from a table, graph, or equation.
3. Write a linear equation in the form y = mx + b.
4. Interpret the slope and initial value in a real-world beauty salon context.
5. Graph a linear function using its initial value and rate of change.

Success Criteria

I am successful if I can:

• Correctly calculate slope using change in output ÷ change in input.
• Locate or calculate the initial value.
• Use the equation y = mx + b.
• Plot at least two correct points and draw a straight line.
• Explain what the slope and initial value mean in the salon situation.

Introduction: Hook and Objectives — 2 Minutes

Hook: Imagine you own a beauty salon. You charge a one-time booking fee plus a price for every service. How could you predict the cost of a customer’s appointment before they arrive?

Explain that a linear function can model this situation. The initial value is the starting fee, and the rate of change is the amount added for each service or unit.

Say: “Today we will learn how to find these two numbers from tables and graphs, then use them to graph and explain salon prices.”

Body: Instruction and Practice

I Do: Teacher or Parent Models — 4 Minutes

Use this beauty salon example:

1. Find the rate of change: The cost increases by $20 whenever the number of services increases by 1. Therefore, the rate of change is $20 per service.
2. Find the initial value: When x = 0, the cost is $25. Therefore, the initial value is $25.
3. Write the equation: Using y = mx + b, substitute m = 20 and b = 25:
   y = 20x + 25
4. Interpret the equation: The salon charges a $25 booking fee plus $20 for each service.
5. Graph the function: Plot the initial point (0, 25). Move up $20 and right 1 to plot (1, 45). Connect the points with a straight line.

Quick Check: Ask: “What does the 20 represent? What does the 25 represent?”

Expected response: 20 is the cost per service, and 25 is the starting or booking fee.

We Do: Guided Practice — 3 Minutes

Work through this example together:

Ask the learner:

• How much does the price increase each time?
• What is the initial value?
• What is the equation?
• What would the price be for 4 appointments?

Answers: The rate of change is $20 per appointment. The initial value is $15. The equation is y = 20x + 15. Four appointments cost $95.

You Do: Independent Practice — 4 Minutes

Have the learner complete the worksheet below. Encourage the learner to:

1. Look for how much the output changes.
2. Find the output when the input is 0.
3. Use y = mx + b.
4. Label graph axes and units.
5. Explain answers using salon vocabulary.

Beauty Salon Practice Worksheet

Name: ____________________________    Date: __________________

Directions: Show your work. Use y = mx + b when appropriate. For graphing questions, label both axes.

1. A salon charges a $30 styling fee plus $15 for each treatment.
   a. Identify the rate of change: ____________________
   b. Identify the initial value: ____________________
   c. Write an equation for the total cost: ____________________
2. Use the table to find the rate of change.

   x	0	1	2	3
   y	40	55	70	85

   
   Rate of change: ____________________
3. Using the table in Question 2, find the initial value and write the equation.
   Initial value: ____________________
   Equation: ____________________
4. A nail salon’s equation is y = 12x + 18.
   a. What is the rate of change? ____________________
   b. What is the initial value? ____________________
   c. What is the cost of 5 services? ____________________
5. A salon’s cost increases from $50 for 2 services to $80 for 5 services. Find the rate of change.
   Rate of change: ____________________
6. A makeup artist charges a $20 travel fee and $35 per client.
   a. Write an equation for the total cost, y, for x clients: ____________________
   b. Find the cost for 4 clients: ____________________
7. Real-World Problem: A hairstylist tracks the following prices:

   Number of Hair Treatments, x	0	1	2	3
   Total Price, y	$25	$50	$75	$100

   
   a. Find the rate of change: ____________________
   b. Find the initial value: ____________________
   c. Write the equation: ____________________
   d. Interpret both values in this situation.
8. Graph y = 10x + 20 for x-values 0, 1, 2, and 3. Label the initial value and rate of change.
   
   Points: ______________________________________________________
9. A salon graph crosses the y-axis at 45 and rises 8 units for every 1 unit it moves right.
   a. What is the initial value? ____________________
   b. What is the rate of change? ____________________
   c. Write the equation: ____________________
10. Create your own beauty salon linear function. Include:
    • An initial fee
    • A price per service
    • An equation
    • The total cost for 3 services
    • A sentence explaining what the slope means
    
    My scenario: __________________________________________________
    Equation: ___________________________________________________
    Total cost for 3 services: ___________________________________
    Meaning of the slope: ________________________________________

Conclusion: Closure and Recap — 2 Minutes

Ask the learner to complete these sentence starters aloud or in writing:

• “The rate of change tells me ______________________________.”
• “The initial value tells me ________________________________.”
• “In the equation y = mx + b, m represents __________ and b represents __________.”
• “In a salon situation, the slope could represent __________.”

Exit Ticket: A salon charges $10 per manicure and a $25 booking fee. Write the equation and explain the meaning of the slope and initial value.

Expected answer: y = 10x + 25. The slope, 10, is the $10 cost per manicure. The initial value, 25, is the booking fee.

Assessment Plan

Formative Assessment

• Listen to responses during the “I Do” and “We Do” examples.
• Check whether the learner identifies the change in y-values correctly.
• Ask the learner to point to the initial value on a table or graph.
• Review the first three worksheet questions before independent work continues.

Summative Assessment

Use the worksheet and exit ticket. The learner demonstrates mastery by correctly identifying rate of change and initial value, writing an equation, graphing a linear function, and interpreting both values in context.

Differentiation and Flexibility

• Support: Provide the formula m = change in y ÷ change in x, highlight the row where x = 0, and use a table with equal intervals.
• Additional support: Allow the learner to use a calculator or digital graphing tool and complete the worksheet orally.
• Visual support: Use one color for the initial value and another color for the repeated change.
• Extension: Ask the learner to compare two salons and determine which is less expensive for 2, 5, and 10 services.
• Choice: The learner may complete the final problem as a written scenario, a table, or a labeled graph.

Answer Key

1. a. $15 per treatment; b. $30; c. y = 15x + 30
2. $15 per appointment
3. Initial value: $40; Equation: y = 15x + 40
4. a. 12; b. 18; c. $78
5. $10 per service, because (80 − 50) ÷ (5 − 2) = 30 ÷ 3 = 10
6. a. y = 35x + 20; b. $160
7. a. $25 per treatment; b. $25; c. y = 25x + 25; d. The slope means each treatment adds $25. The initial value is the $25 starting fee.
8. Points: (0, 20), (1, 30), (2, 40), (3, 50). Initial value: 20. Rate of change: 10.
9. a. 45; b. 8; c. y = 8x + 45
10. Answers will vary. The equation should match the learner’s initial fee and price per service.