Create and Graph Two-Variable Equations

Lesson 1 of 1

In this lesson:

• Write equations in two variables from real-world contexts
• Graph equations with labeled axes, correct scale, and point interpretation
• Model linear, quadratic, and exponential relationships

Learning Objectives for This Lesson

1. Write a two-variable equation to model a relationship
2. Identify independent and dependent variables for each axis
3. Create a graph with appropriate labels and scales
4. Interpret graph points as solution pairs in context
5. Write equations for linear, quadratic, and exponential relationships

One Variable vs. Two Variables

One variable: Buy 10 items at $5 each → (one answer)

Two variables: Cost for any quantity → (a relationship)

Solution pairs: infinitely many

Two Variables Describe a Relationship

— = items, = cost in dollars

• Every gives a different — infinitely many solution pairs
• Each pair represents a real (items, cost) combination
• The graph shows all solution pairs at once

What Points on the Graph Mean

• : at the start, 500 liters remain
• : after 10 minutes, 300 liters remain

Every Point Has Meaning in Context

What does the point mean?

After 15 minutes, there are 200 liters remaining in the tank.

Check: ✓

Every point on the graph is a valid (time, volume) combination.

Quick Check: Interpreting a Point

For :

Explain the meaning of the point in context.

What happens at ?

Answers to Interpreting the Intercept

means: after 25 minutes, the tank holds 0 liters — it is empty.

✓

The -intercept of the graph tells us when the tank empties.

Choosing Variables, Axes, and Scale

When setting up a two-variable graph:

• Independent variable (freely chosen input) → -axis
• Dependent variable (output that depends on input) → -axis
• Scale: cover the full relevant domain and range
• Labels: variable name AND units — e.g., "Time (minutes)" not just ""

Well-Labeled Graph:

• Axes labeled with variable names and units
• Scale: from 0 to 25; from 0 to 500
• Graph ends at — the domain is

Worked Example: Building a Fuel Equation

Car: 12 gallons, 30 miles per gallon →

• = miles driven (independent → -axis)
• Domain:
• -intercept: 12 (full tank); -intercept: 360 (empty)

Worked Example: Budget Equation (Standard Form)

$12 budget: apples $1.50, oranges $2.00 →

• -intercept: — spend all on apples
• -intercept: — spend all on oranges
• Only non-negative integers make sense in context

Guided Practice: Taxi Cost Equation

Context: A taxi charges $3.50 base fee plus $1.75 per mile.

Set up the graph:

• Which variable is independent? Which is dependent?
• What are appropriate labels and scale for 0 to 20 miles?
• What does the point mean in context?

Practice Problems: Graphing Linear Equations

Graph each with labeled axes and 3+ labeled points:

1. ( = hours, = products remaining)

2. (spending equation)

Find both intercepts and explain their meaning

Answers to Linear Graphing Practice Problems

: starts at 200, hits 0 at ; domain

:

• -intercept: ; -intercept:
• Slope : as increases, decreases

Poor vs. Good Graph Comparison

• Left: unlabeled axes, arbitrary scale — uninterpretable
• Right: labeled with units, domain-appropriate scale — clear and complete

Quick Check: Axis Assignment Error

A student graphs and labels:

• -axis: "Cost ()"
• -axis: "Number of items ()"

What error did the student make? Why does it matter?

Overview of Non-Linear Two-Variable Equations

Identify the type before graphing:

Worked Example: Modeling Projectile Height

Ball thrown at 48 ft/s: ( = seconds)

• Parabola opening downward
• -intercepts: (launch), (landing)
• Vertex at : maximum height = 36 ft

Worked Example: Exponential Population Growth

Colony: 500 bacteria, doubles hourly →

• J-shaped curve (exponential growth)
• : ; :
• Use large -axis scale — values grow quickly

Worked Example: Area and Perimeter Quadratic

Rectangle, perimeter 20 cm, side :

• Domain:
• Maximum area at (square): cm²

Guided Practice: Setting Up a Quadratic Equation

— profit (thousands), = hundreds of units

1. What domain makes sense?
2. What does the vertex represent?
3. Where does ?

Practice Problems: Modeling Non-Linear Equations

For each situation, set up the two-variable equation, identify the type, and sketch the general shape:

1. A ball falls from rest: ( in meters, in seconds)

2. An investment triples every decade: ( in decades, in dollars)

Answers to Non-Linear Practice Problems

: upward parabola; at ; domain

: exponential J-curve; at ; at

Key Takeaways from This Lesson

✓ Two-variable equations model relationships, not single values

✓ Independent → -axis; dependent → -axis

✓ Label axes with units; choose appropriate scale

✓ Every graph point has a real-world meaning

Don't graph beyond the valid domain

What Comes Next: Systems of Equations

HSA.REI.D.10: Every graph point is precisely a solution — formalizing today's work.

HSA.CED.A.3 & REI.C.6: Inequalities and systems — multiple constraints combined.

Today's graphs are the tools for all upcoming lessons.