Geogebra: Polynomials and Rational Functions

Patterns in Graphs

Display three graphs:

• A straight line
• A parabola
• A curve with multiple turns

Ask:

• Which graph looks most complex?
• Where does each graph cross the x-axis?
• What do all the graphs have in common?

Introduce the idea that x-intercepts are often connected to factors.

Co-Use Activity

Partners examine the graphs and identify:

• Number of x-intercepts
• Whether the graph crosses or touches the x-axis

Factors and Zeros

Review:

• Polynomial
• Factor
• Zero
• X-intercept
• Degree

Example:

[(x-2)(x+1)]

Ask:

What values make the expression equal zero?

Discuss:

• x = 2
• x = -1

These values are called zeros.

Co-Use Activity

Student A identifies a factor.

Student B identifies the corresponding zero.

Switch roles.

Exploring the GeoGebra Interactive

Students access the GeoGebra activity.

Teacher models:

• Changing factors
• Observing graph behavior
• Identifying x-intercepts
• Examining turning points

Ask:

• What happens when a new factor is added?
• How does the graph change?
• Where are the zeros located?

Co-Use Activity

Partners share a device.

• Student A = Navigator
• Student B = Recorder

Students switch roles halfway through the activity.

Polynomial Investigation Teams

Students work in pairs.

Activity 1: Finding Zeros

Use the interactive to investigate:

[(x-1)(x+2)]

Record:

• Zeros
• Number of x-intercepts

 
Activity 2: Multiplicity Exploration

Investigate:

[(x-2)^2]   or [(x-2)(x-2)]

and

[(x-2)^3] or [(x-2)(x-2)(x-2)]

Questions:

• Does the graph cross or touch the x-axis?
• What changes when multiplicity increases?

 
Activity 3: Degree Investigation

Compare:

• Quadratic
• Cubic
• Quartic

Students record:

Degree.                       Number of Turning Points              End Behavior
 
Activity 4: Predict and Test

Students predict graph behavior before manipulating the interactive.

Co-Use Activity

One student predicts.

One student tests using GeoGebra.

Partners explain whether the prediction was correct.

Exit Ticket

Given the polynomial:

[(x+3)(x-2)^2] or  [(x+3)(x-2)(x-2)]

Answer:

1. What are the zeros?
2. Which zero has multiplicity 2?
3. Will the graph cross or touch the x-axis at x = 2?
4. Explain your reasoning.

Success Criteria

Students can:

✓ Identify factors and zeros.

✓ Locate x-intercepts.

✓ Explain multiplicity.

✓ Interpret polynomial graphs.

Co-Use Activity

Partners compare explanations before submitting.

Real-World Modeling

Discuss how polynomial functions are used to model:

• Business profit and loss
• Population growth
• Engineering design
• Projectile motion

Co-Use Activity
Partners choose one application and discuss why understanding graph behavior would be useful.