Graphing Linear Functions Worksheet for a 33-Year-Old — Math & Business

Instructions

This worksheet will guide you through graphing linear functions using a practical, real-world scenario. Complete each section in order. You will need a pen or pencil and a straight edge for drawing lines.

1. Warm-Up: Refresh your skills by graphing a basic linear equation.
2. Scenario: Apply your knowledge to a business scenario to see how linear functions model real-world earnings.
3. Analysis: Use your graph to interpret the data and make business decisions.
4. Challenge: Compare two different business models to find the most profitable option.

Part 1: The Basics - A Quick Refresher

Let's start with the fundamental equation: y = 2x + 1

1. Identify the Slope (m) and y-intercept (b):

   • Slope (rise/run):
   • y-intercept (where the line crosses the y-axis):

2. Complete the Table of Values: Calculate the y value for each x value given. One example is done for you.

3. Plot the Points: Use the coordinates from your table to plot the points on a graph and draw a straight line through them.

Part 2: Real-World Scenario - The Freelance Consultant

You are a freelance consultant who charges a $50 flat fee for any project plus $25 per hour of work.

This pricing model can be represented by the linear function: E = 25h + 50

• E represents your total Earnings in dollars.
• h represents the hours you work.

1. Complete the Earnings Table: Calculate your total earnings (E) for the different hours worked (h). An example is provided.

2. Graph Your Earnings: Plot the coordinate points from the table on the graph below. Label your axes clearly (h for the horizontal x-axis, E for the vertical y-axis). Draw a straight line connecting the points.

Suggestion: For the vertical (E) axis, let each grid line represent $50. For the horizontal (h) axis, let each grid line represent 1 hour.

(A blank 10x10 grid would be here on a printable sheet. For this format, visualize plotting on a graph with an x-axis from 0-10 and a y-axis from 0-500).

Part 3: Analysis and Interpretation

Use the graph you created in Part 2 to answer the following questions. You can also use the equation E = 25h + 50 to check your work.

1. If you work for 5 hours, what will your total earnings be? (Find h=5 on your graph and see where it intersects the line).

2. A client has a budget of $200. What is the maximum number of full hours you can work on their project?

3. What does the y-intercept (the point where h=0) represent in this business scenario? Why is it not $0?

Part 4: Challenge Question - Comparing Business Models

A competitor consultant doesn't charge a flat fee. Instead, they charge $35 per hour. Their earnings equation is: E = 35h.

1. On the same graph you used in Part 2, plot the line for your competitor's model. You can create a quick table of values for h=0, 2, 4, 6, 8 if you need it.

2. Analyze the two lines on your graph. At what number of hours do both you and your competitor earn the same amount of money? (This is the point where the lines intersect).

3. Based on your graph, for which projects is your pricing model (E = 25h + 50) a better deal for the client (i.e., cheaper)? For which projects is the competitor's model (E = 35h) a better deal?

Answer Key

Part 1: The Basics

1. Slope (m) = 2; y-intercept (b) = 1

2. Table of Values:	x	y	(x, y)
   -2	-3	(-2, -3)
   -1	-1	(-1, -1)
   0	1	(0, 1)
   1	3	(1, 3)
   2	5	(2, 5)

Part 2: Real-World Scenario

1. Earnings Table:	h	E	(h, E)
   0	$50	(0, 50)
   2	$100	(2, 100)
   4	$150	(4, 150)
   6	$200	(6, 200)
   8	$250	(8, 250)
   10	$300	(10, 300)

2. Graph should be a straight line starting at (0, 50) and passing through the points above.

Part 3: Analysis and Interpretation

1. For 5 hours, earnings are $175. (E = 25(5) + 50 = 125 + 50 = 175).
2. For a $200 budget, you can work a maximum of 6 hours. (200 = 25h + 50 -> 150 = 25h -> h = 6).
3. The y-intercept represents the $50 flat fee. This is the base charge for any project, even if it takes zero hours of work (e.g., for the initial consultation and project setup).

Part 4: Challenge Question

1. The competitor's line should start at (0, 0) and go through points like (2, 70), (4, 140), (6, 210), etc.
2. The lines intersect at 5 hours. At this point, both models result in earnings of $175. (25h + 50 = 35h -> 50 = 10h -> h = 5).
3. Your model is a better deal for clients with projects longer than 5 hours. For projects shorter than 5 hours, the competitor's model is cheaper.