Vertex Form, Factored Form, and Max/Min

Deck 2 of 2: Reading Forms and Real Applications

In this deck:

• Read the vertex directly from vertex form
• Read roots directly from factored form
• Apply max/min to real-world problems

What You Will Learn in Deck Two

In this deck you will:

3. Graph quadratics from vertex form — vertex is immediate
4. Graph quadratics from factored form — roots are immediate
5. Determine whether a quadratic has a maximum or minimum
6. Label all key features on a complete quadratic graph

Vertex Form: Reading the Vertex Directly

• Vertex: — no computation needed
• Axis: ; Direction: up, down

Sign trap: gives , not .

Graphing Vertex Form: f(x) = −2(x−3)² + 8

• Vertex: ; axis: ; opens down ()
• y-int: ; x-ints:

Factored Form: Reading Roots Directly

• Roots: and — read immediately; set each factor to zero
• Axis: (midpoint); Vertex: substitute in

Sign trap: gives , not .

Graphing Factored Form: g(x) = 2(x+1)(x−5)

• Roots: and ; axis:
• Vertex: →
• y-intercept: →

Three-Form Comparison: What Each Reveals

Choose your starting strategy based on the form you're given.

Check: Read Vertex Without Calculating

For :

1. State the vertex.
2. Is this a maximum or minimum?
3. What is the axis of symmetry?

Answer all three before the next slide — no calculation needed.

Answer: Vertex of 3(x − 4)² + 1

• Vertex:
• → opens upward → minimum at
• Axis of symmetry:

The minimum VALUE is 1. It OCCURS AT x = 4.

The Vertex Answers Real-World Questions

The vertex answers every optimization question:

• Maximum height → vertex of height function
• Maximum revenue → vertex of revenue function
• Minimum cost → vertex of cost function

Find the vertex; report value (y) and location (x) separately.

Finding Maximum Height for a Projectile

• → vertex is a maximum
• Axis: s; height: ft

Maximum height: 69 ft at t = 2 s.

Finding the Price That Maximizes Revenue

• → vertex is a maximum
• Axis: ; max revenue:

Maximum revenue of 450 dollars at a price of 30 dollars.

Max/Min Value vs. Where It Occurs

Both answers are required — they are different numbers.

Check: Max or Min and Location

For :

1. Does this function have a maximum or minimum?
2. At what time does it occur?
3. What is that maximum or minimum value?

Answer all three before the next slide.

Answer: Maximum Height at t = 2 Seconds

• → opens down → maximum
• Axis: seconds
• Max value:

Maximum height of 23 feet occurs at t = 2 seconds.

Key Takeaways from Deck Two

✓ Vertex form: vertex immediate — sign of flips

✓ Factored form: roots immediate — sign of each flips

✓ Axis = midpoint of roots; substitute to find vertex

✓ VALUE = y-coord; OCCURS AT = x-coord

Coming Up: Graphing Root and Piecewise Functions

This standard is complete. Next: graphing non-polynomial families.

HSF.IF.C.7b topics:

• Square root and cube root parent functions
• Transformed root functions and restricted domains
• Piecewise-defined, step, and absolute value functions