Quadratic Functions: Factoring and the Square

Lesson 1 of 2

In this lesson:

• Factored form reveals zeros (x-intercepts)
• Completing the square reveals the vertex (extreme value)

What You Will Learn Today

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

1. Factor a quadratic to find its zeros
2. Complete the square to find the vertex
3. Identify the axis of symmetry from vertex form
4. Determine maximum vs. minimum from the sign of

Each Quadratic Form Tells a Different Story

Factored Form Reveals the Function's Zeros

Setting :

Zeros and are the x-intercepts — where the parabola crosses the x-axis.

Factoring to Find Zeros: Example

Factor:

Zeros: and

The parabola crosses the x-axis at and .

Zeros Connect to the Graph

Axis of symmetry: midpoint of zeros =

Zeros in Context: Break-Even Points

Zeros at and :

• Break even at 3 units and at 7 units
• Profit is positive between 3 and 7 units (: opens down)

Quick Check: Factor to Find Zeros

Factor and find the zeros.

Quick Check Verified: Zeros at −3 and 4

• Zero at : ✓
• Zero at : ✓
• Axis of symmetry:

What Factoring Doesn't Tell You

Factoring reveals:

• ✓ Zeros (x-intercepts)
• ✓ Axis of symmetry (midpoint of zeros)

Factoring does not directly reveal:

• ✗ The vertex's y-coordinate
• ✗ The maximum or minimum value of the function

Solution: Complete the square → vertex form

Vertex Form Reveals the Extreme Value

• : minimum; : maximum
• Axis: ; extreme value: (y-coordinate)

Completing the Square: Five-Step Method

Convert to vertex form:

1. Group:
2. Compute:
3. Add and subtract:
4. Factor:
5. Simplify

Completing the Square: Step-by-Step (a = 1)

Vertex: . Since , this is a minimum.

Completing the Square When a ≠ 1

Vertex: . Maximum value: at .

Maximum vs. Minimum: The One-Rule Check

Language: "The [maximum/minimum] value is , occurring at ."

Quick Check: Complete the Square

Complete the square and identify the vertex and extreme type.

Quick Check Verified: Vertex at (5, −7)

• Half of is ; squared:
• Vertex: . Minimum: at

Watch Out: The Balance Rule

Error: ✗

Adding changes the value — subtract to compensate.

Watch Out: Reading the Vertex Sign

Rule: when . Vertex x-coordinate is .

→ vertex at

Complete the Square: Second Practice

Factor out 2, complete the square, find the vertex.

Second Practice Verified: Vertex at (3, −4)

Vertex: . Minimum value: at .

Lesson 1 Summary: Two Algebraic Techniques

Factoring → zeros , ; axis at

Completing the square → vertex ; extreme value at

• : minimum; : maximum

Watch out: Vertex of is , not

Coming Up: Symmetry and Real-World Meaning

In Lesson 2, you will:

• Use symmetry to graph parabolas efficiently
• Connect zeros and vertex into a complete description
• Interpret quadratic features in context (projectile, profit)

Next: why knowing the vertex and zeros together gives the full picture