Understand Graphs as Complete Solution Sets

Lesson 1 of 1

In this lesson:

• A graph is every ordered pair that makes an equation true
• Verify whether a point is on a graph by substitution
• Build graphs from tables and read solutions from graphs

Learning Objectives for This Lesson

1. Explain that a graph shows ALL solution pairs
2. Verify whether a point is on a graph by substitution
3. Build graphs from a table of values
4. Distinguish one-variable from two-variable equation graphs
5. Recognize graphs as lines, parabolas, circles, or curves

What Is a Graph, Really?

Solving gives one answer: — mark it on a number line.

For :

• Every gives a different — infinitely many solutions
• The coordinate plane shows them all at once

The Graph Is the Entire Solution Set

— points on the graph satisfy it:

Graph Shows All Solutions at Once

Every point on the line satisfies — and NO point off the line does

"On the Graph" = "Is a Solution"

These two statements mean exactly the same thing:

• "Point is on the graph"
• " is a solution to the equation"

A point is on the graph if and only if it satisfies the equation.

Arrows on the graph indicate it continues infinitely beyond the paper.

Quick Check: On the Graph?

For :

Is on the graph? Try it.

Is on the graph? Try it.

Substitute and check before advancing to the answer.

Answers: On the Graph Check

: ✓ — YES

: — NO (graph requires at )

Verifying Points: The Substitution Test

To check whether is on the graph:

1. Substitute and
2. Simplify both sides
3. Equal → point IS on the graph
4. Unequal → point is NOT on the graph

Works for any equation — linear, quadratic, circle, or other.

Example: Testing Points on a Circle

For :

Example: Testing Points on a Quadratic

:

and are the x-intercepts

Your Turn: Point Verification Practice

Test each point using substitution:

: and

: and

Pause and test each point before the next slide

Answers to Point Verification Practice Problems

: ✓ ON — ✓ ON

: ✓ ON — : NOT ON

is inside the circle, not on it

Building a Graph: Choose, Compute, Plot, Connect

To graph any equation in two variables:

1. Choose several -values (spread across the domain)
2. Substitute each and solve for
3. Plot each pair
4. Connect with a line (linear) or smooth curve (non-linear)
5. Add arrows to show the graph continues

Graphing a Quadratic Equation: Worked Example

• at and — these are the x-intercepts
• Verify: ✓ and ✓

Connecting Points: Line vs. Smooth Curve

Between plotted points, there are infinitely more solutions

Worked Example: Graph

• for all — the graph never touches the x-axis
• As , but never reaches

Quick Check: Reading an Exponential Asymptote

For :

Does the graph ever touch the x-axis?

What would that require? Can for any real ?

Guided Practice: Graph

Complete the table, then plot and connect:

What shape will the graph be? Where does it cross each axis?

Graphs of Equations Take Many Shapes

All four are solution sets — same definition, different shapes

Graphs: Lines, Parabolas, Circles, and More

A "curve" includes lines — lines are a special case

Reading Solutions from a Graph

For , read answers visually:

• Zeros: graph crosses -axis →
• Output: at , read height →
• Input: for , intersect line →

Worked Example: Reading

Zeros: → crosses -axis at

For : horizontal line intersects at

Verify: ✓

Practice: Reading and Interpreting Graphs

Use the graph to answer each question:

1. Where does the parabola cross the -axis?
2. What is the -value when ?
3. For what values of is ?

Write your answers before the next slide

Answers to Graph-Reading Practice Problems

For :

1. -intercepts: and
2. -intercept:
3. : intersects at and

Key Takeaways from This Lesson

✓ Graph = complete solution set of the equation

✓ On the graph ⟺ is a solution

✓ Test: substitute → equal sides → on the graph

Connect smoothly — infinitely many solutions between plotted points

Graph includes ALL pairs, not just intercepts

What Comes Next: Key Features of Graphs

Next lesson — HSA.REI.D.11:

If and , where do they have the same output?

→ Graph both equations → find their intersection point(s)

The intersection is where two solution sets share a point — a solution to both equations simultaneously.