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Equivalent Forms of Expressions | Lesson 1 of 1

Choose Equivalent Forms to Reveal Properties

Lesson 1 of 1

In this lesson:

  • Three forms of a quadratic reveal different features of its graph
  • Factor to find zeros; complete the square to find the vertex
  • Transform exponential expressions to reveal rates in different units
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Learning Objectives for This Lesson

  1. Factor a quadratic to reveal its zeros in context
  2. Complete the square to find the vertex form
  3. Convert exponential expressions between time scales
  4. Choose the right form based on the property needed
  5. Explain what each equivalent form reveals
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

The Same Expression, Three Faces

Form Expression Reveals
Standard -int
Factored zeros
Vertex vertex

Same parabola — three lenses

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Which Form for Which Question?

Visual showing a parabola with three labeled features: y-intercept labeled "Standard form reads this", two x-intercepts labeled "Factored form reads these", and vertex labeled "Vertex form reads this"

  • Need zeros? Use factored form:
  • Need max/min? Use vertex form:
  • Need -intercept? Use standard form: read off
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: All Three Forms

Standard: → -intercept

Factored: → zeros at and

Vertex: → vertex , minimum

Verify: ✓ and ✓

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Quick Check: Choose the Form

For :

  1. You want the -intercepts. Which form do you use?

  2. You want the lowest point of the parabola. Which form do you use?

  3. You want . Which form, and what is the answer?

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Guided Practice: Three Forms of

Write in all three forms:

  1. Standard: already given — what is the -intercept?
  2. Factored: factor
  3. Vertex: complete the square

State what each form reveals

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

How Factored Form Reveals the Zeros

Set :

Zeros = -intercepts. In context: break-even points, ground-level times, or inputs producing zero output.

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Factoring a Quadratic

Zeros: and

Check: ✓ and ✓

-intercepts at and

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Leading Coefficient

Zeros: and

Check: ✓

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Factoring in Profit Context

Context: Weekly profit (thousands) at units (hundreds):

  • Break-even: (100 units) and (500 units)
  • Profitable when : production between 100 and 500 units
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Practice Problems: Factoring Quadratic Expressions

Factor each expression and identify the zeros:

  3. — interpret the zeros as revenue break-even points
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Answers to Factoring Practice Problems

  1. → zeros

  2. → zeros

  3. → zeros

Break-even: ; profitable for

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Completing the Square: Why It Works

We want to rewrite as a perfect square trinomial:

To maintain equality: add and subtract

Algebra tile diagram showing x² + bx as an L-shaped region and the missing square piece (b/2)² that completes it into a full square

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Complete the Square ()

Step 1: Group:

Step 2: Completing term ; add and subtract:

Vertex: . Minimum:

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Complete the Square ()

Step 1: Factor out 3:

Step 2: ; adding 4 inside adds total — subtract 12:

Vertex: . Minimum:

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Finding Maximum Height

(height in feet)

Step 1:

Step 2: ; adds — compensate:

Maximum: 256 feet at seconds

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Quick Check: Reading Vertex Form

  1. What is the vertex?
  2. Is it a maximum or minimum? Why?
  3. What is the minimum/maximum value?

Be careful with the sign of the -coordinate.

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Guided Practice: Complete the Square

Step 1:

Step 2: Half of is ; squared is .

Step 3: Add and subtract to complete; factor the perfect square.

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Practice Problems: Completing the Square

Find the vertex form and identify the vertex:

  3. — find the maximum height in context
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Answers to Vertex Form Practice Problems

  1. → vertex , minimum

  2. → vertex , minimum

  3. → vertex , maximum height at

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Exponential Forms and Time Scales

This shows annual growth of 15%. What is the equivalent monthly rate?

Key identity:

Compute:

Monthly rate: per month

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Why Not Just Divide by 12?

Simple division gives: per month

But

Correct monthly rate: , and ✓

Compounding makes the correct monthly rate slightly less than

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Worked Example: Hourly to Daily Decay

Rewrite to show the daily decay rate (24 hours per day):

Compute:

Each day, about of the substance decays

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Guided Practice: Finding the Quarterly Rate

An investment grows at per year:

Find the equivalent quarterly growth factor.

(There are 4 quarters per year. Use the identity .)

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Practice Problems: Rewriting Exponential Form

Rewrite each expression to reveal the rate in the new time unit:

  1. (annual) → find the monthly rate
  2. (doubling every 5 years) → find the annual growth rate
  3. (per day) → find the weekly decay factor
Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Answers to Exponential Form Practice

  1. Monthly rate: → about per month

  2. Annual rate: → about per year

  3. Weekly decay factor: → about decays each week

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

Key Takeaways from This Lesson

✓ Standard → read intercept from constant

✓ Factored → read zeros from factors

✓ Vertex → read and max/min

✓ Exponential → reveals rate in any time unit

⚠️ Vertex sign trap: , not

⚠️ Monthly rate annual — use

Grade 9 Algebra | HSA.SSE.B.3
Equivalent Forms of Expressions | Lesson 1 of 1

What Comes Next: Creating Equations

  • Zeros → HSA.REI.B.4: quadratic formula derived by completing the square
  • Vertex form → function transformations: shifts and
  • Exponential rewriting → logarithms: solving requires logs

These forms are vocabulary for all future function analysis.

Grade 9 Algebra | HSA.SSE.B.3

Click to begin the narrated lesson

Choose equivalent forms to reveal properties