Graph Intersections as Equation Solutions

Why the Crossing Point Gives the Answer

In this lesson:

• Explain why intersection x-coordinates solve
• Find approximate solutions graphically using technology

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Explain why intersection x-coordinates solve
2. Set up any equation as ; graph both sides
3. Use graphing technology to find approximate solutions
4. Count solutions from the graph; refine approximations
5. Distinguish approximate graphical solutions from exact algebraic ones

Recall: f(x) Notation and Graph Meaning

• : output of at input
• : the graph — on it iff
• Two functions: and on the same axes

If , what is ?

Why Does the Crossing Point Give ?

You graphed and and found they cross at . You said: "the solution is ."

But why? What does crossing at have to do with solving ?

From Graph to Intersection to Solution

REI.D.10: is on iff .

At an intersection, lies on both graphs:

• On :
• On :

Therefore:

The Forward Argument: Intersection → Solution

At the intersection: , so solves .

The Backward Argument: Solution → Intersection

If , let . Then is on both graphs — so it is an intersection. ✓

Solutions create intersections; intersections create solutions.

The connection is exact, not approximate.

Argument in Action: Linear Example

, . Cross at .

, . So . ✓ Solution: .

The -value 2 is their common output, not the answer.

Setup: Rewrite, Graph, Read

To solve any equation graphically:

1. Let $f(x) = $ (left side) and $g(x) = $ (right side)
2. Graph both on the same plane
3. Find -coordinates of all intersection points
4. Those -values are the solutions

Works for linear, polynomial, exponential, logarithmic, absolute value.

Worked Example 1: Verifiable Algebraically

Solve . Let , .

Graph: parabola and horizontal line cross at and .

Verify algebraically: ✓

Worked Example 2: Technology Required

Solve .

, . No standard algebraic method works.

Graph in Desmos: three intersections at , , .

Verify exact cases: ✓; ✓. Third: approximate only.

Common Error: Reading the Wrong Coordinate

At intersection :

• = the solution — the where and are equal
• = the shared output — both and equal

Wrong: "answer is "   Right: "solution is "

Technology: Enter, Graph, Find Intersection

TI-84: Enter , → GRAPH → CALC → 5:intersect → follow prompts

Desmos: Type and → click intersection dot → read

Take the -value as your solution. Write to signal approximation.

Technology Practice: Finding All Solutions

Graph both sides; find intersections; record ; verify.

1. (linear)
2. (polynomial — count first)
3. (absolute value)
4. (exponential)

Zoom out first; then zoom in.

Counting Solutions from the Graph

Count crossings before computing — it sets the expectation.

Approximate vs. Exact: Know the Difference

Equation:

Use for graphical results. Prefer algebraic when available — it's exact.

Your Turn: Write the Argument

In your own words, explain:

Why are the -coordinates of the intersections of and the solutions to ?

Your explanation must mention: what it means to be on a graph, and why the two equations are equal at the intersection.

Unscaffolded: Graph and Solve Completely

Solve by graphing. Then solve algebraically and compare.

1. Name and ; graph both
2. Count intersections — how many solutions do you expect?
3. Find approximate -coordinates using technology
4. Solve algebraically and verify both solutions exactly

What You Can Do Now

✓ Argue: at intersection → is a solution

✓ Set up and ; graph; read -coordinates

✓ Zoom out; use intersection tool; write

Read , not

Write — graphical results are approximate

Zoom out — see all intersections before zooming in

Coming Up: Inequalities and Graphical Regions

Next: HSA.REI.D.12 — graphing inequalities like .

At the intersection: boundary. Above or below: the solution region.

REI.D.11 found where the curves meet. REI.D.12 asks where one is above the other.