Math Analysis 11

Vertex Vision

Mapping Domain, Range, and Extrema through Quadratic Graphs

Today's Goal

Students will define domain, range, and extrema for quadratic functions using graphs.

Vocabulary Check

• Vertex Form: \( y = a(x-h)^2 + k \)
• AOS: Axis of Symmetry (\( x = h \))
• Extrema: Max or Min values

Warm-up

Identify the Domain and Range of \( y = x^2 \) using interval notation.

Domain (-\infty, \infty)

Range [0, \infty)

Guided Exploration

Fill out your T-Chart

Watch the property lists

Discussion Point 1 (0:18)

The Sign Swap

"Change this sign, but don't change this one."

When looking at \( (x - 2)^2 - 3 \), why is the \( x \)-coordinate of the vertex positive 2?

Think: What value of \( x \) makes the expression inside the parentheses equal to zero?

Discussion Point 2 (3:15)

The Live Pivot

The narrator changed the coefficient from negative to positive during the second example.

Critical Question:

If he kept the -2 coefficient, where would the graph have gone? How would the range have changed?

The Multiplier Logic

"As you travel one unit to the right... go up two units."

Parent \( x^2 \)

1 → 1
2 → 4

Stretched \( 2x^2 \)

1 → 2
2 → 8

Visual Analysis (7:08)

Stretching the Limits

"As we increase the 'a' value, the graph becomes more narrow. It stretches vertically."

Domain

(-\infty, \infty)

AOS

x = h

Extrema

y = k

Property Hunt

You have 6 graphs defined by their Vertex and one Point.

01

Find the 'a' coefficient

02

Write Vertex Form

03

List all 4 Properties

Final Insight

"How does the sign of 'a' immediately tell you the range limit?"

Complete your journal prompt focusing on the connection between concavity and the k-value.

Vertex Vision Slides

Vertex Vision Slides

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