Quadratic Functions: Symmetry and Interpretation

Lesson 2 of 2

In this lesson:

• Symmetry: every point on a parabola has a mirror image
• Connecting zeros, vertex, and symmetry into a complete picture
• Interpreting quadratic features in real-world contexts

What You Will Learn Today

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

1. Use symmetry to graph parabolas more efficiently
2. Explain why the axis of symmetry bisects the zeros
3. Interpret zeros and vertex in real-world contexts
4. Write complete interpretive statements with units

The Vertex Reveals a Symmetry Property

For , axis of symmetry is .

Points equidistant from give equal output values. The graph mirrors across .

Verifying Symmetry with Numerical Examples

and ✓

and ✓

Graphing Shortcut: Reflect for Free

Vertex + one point → reflected point is free.

(vertex), :

• is 2 units right of
• is 2 units left — also on the graph

Zeros and Vertex: The Midpoint Connection

Zeros at and → axis at → vertex on the axis.

Both zeros are equidistant from the axis; their average gives the axis x-value.

Zeros at and : axis at , vertex at .

Complete Sketch from Zeros and Axis

Zeros at and , :

• Axis:
• Vertex: → , minimum

Three points: , , → complete sketch.

Quick Check: Axis and Vertex from Zeros

The zeros of are and .

Find the axis of symmetry and the vertex.

Quick Check Verified: Axis at x = 2

Zeros at and :

Vertex: . Minimum value at .

Connecting Algebra to Real-World Problems

What does each feature mean in the specific situation?

Applied Example: Projectile Height (Part 1)

Set → → s (ground)

Discard (negative time is not physical).

Applied Example: Projectile Height (Part 2)

What is the maximum height? Complete the square:

Vertex: . Maximum height: 42 feet at seconds.

Complete Projectile Interpretation: All Three Features

• Starts at feet (platform height)
• Reaches maximum 42 feet at seconds
• Hits ground at seconds

Maximum vs. Minimum Value: One More Time

• Maximum value: feet (y-coordinate)
• Occurs at seconds (x-coordinate)

Correct: "Maximum height is 42 feet at seconds."

is WHEN, not the maximum height.

Guided Practice: Profit Function Analysis

Analyze this profit function:

1. Factor to find break-even points (zeros)
2. Complete the square to find maximum profit (vertex)
3. Write interpretive statements for each

Work through all three steps before advancing.

Guided Practice: Profit Analysis Answer

• Zeros: , — break-even at 1 and 7 units
• Vertex: — max profit at 4 units

Independent Practice: Two Quadratic Problems

1. : find zeros, vertex, and sketch

2. : interpret as area (in square feet) where is width in feet — find maximum area and zeros

For each: factor, complete the square, interpret.

Practice Answers: Zeros and Vertex Found

1. : zeros , ; vertex ; minimum

2. : zeros , ; vertex

"Positive area between 3 and 7 ft. Maximum area: 8 sq ft at width 5 ft."

Complete Toolkit Summary: Quadratic Analysis

Factoring → zeros: x-intercepts, break-even, ground-level
Completing the square → vertex: max/min value and where it occurs
Symmetry → axis bisects zeros; reflect one side for the other

Extreme value = (y-coordinate). It occurs at .

Next: exponential forms and comparing function types

Coming Up: Exponential Function Forms

In the next standard, you will:

• Interpret growth and decay rates from exponential expressions
• Convert between time scales using exponent rules
• Apply the same "form reveals meaning" principle to exponential functions

The same question: what does each algebraic form tell you?