The Mirror’s Edge
Edition 13, Lesson 13.2: The Axis of Symmetry
Part I: Identifying the Mirror
Use the formula x = -b/2a to find the equation for the Axis of Symmetry. Write your answer as an equation.
1.
The Centered Curve: y = x² - 6x + 5
a = ____ b = ____
Axis of Symmetry: x = ________
2.
The Leftward Shift: y = x² + 10x + 21
a = ____ b = ____
Axis of Symmetry: x = ________
3.
The Narrow Mountain: y = -2x² + 8x - 3
Axis of Symmetry: x = ________
The Weaver's Check:
Did you write your answer as "x = ___"? A mirror is not just a point; it is an infinite vertical line. Honor the geometry of the spine.
Part II: The Dance of the Twins (Reflecting Points)
For each given point and axis, find the Reflected Point on the other side of the parabola.
4.
Axis: x = 2
Source Point: (0, 1)
Distance from Axis: ________ units
Reflected Point: (____, ____)
5.
Axis: x = -3
Source Point: (-1, 10)
Distance from Axis: ________ units
Reflected Point: (____, ____)
Part III: The Inverse Mirror (Finding the Axis from Points)
Use the Midpoint Rule to find the Axis of Symmetry without using the formula -b/2a.
6.
The Twin Witnesses:
A parabola passes through (2, 10) and (8, 10).
1. Calculate the midpoint of the x-values: ________________
2. Write the Axis of Symmetry: x = ________
3. Where must the Vertex be located? (____, ____) *Use the x you found.*
7.
The Grounded Anchors:
The Zeros of a function are at x = -5 and x = 1.
1. Calculate the midpoint: ________________
2. Write the Axis of Symmetry: x = ________
Part IV: Symmetrical Proofs (Substitution Check)
8.
The Integrity Test:
Given y = x² - 4x + 7.
1. Find the Axis of Symmetry: x = ____
2. Calculate the height at x = 1: y = ____
3. Find the reflected point of x=1: x = ____
4. Prove the height is the same by plugging your reflected x into the equation.
Part V: Kingdom Modeling
9.
The Bridge of Unity:
A bridge over a holy river is modeled by
y = -x² + 12x.
1. Find the Axis of Symmetry (the center pillar).
2. If one side of the bridge is anchored at (0, 0), where must the other side be anchored? Use symmetry to find the second zero.
Center Pillar (Axis): x = ____ Second Zero: x = ____
10.
The Vineyard's Balance:
A vineyard produces its maximum fruit at the Vertex (5, 50). The production follows a quadratic arc. If the harvest is 30 units at x = 2, at what other location (x) will the harvest also be 30 units?
Mirror Point: x = ____
Part VI: The Echad Extension (Transmission)
11.
The Mirror Challenge:
Sit with a younger sibling or friend. Show them how to draw a vertical line (the Axis). Draw a simple shape on the left side and have them try to draw the perfect mirror image on the right. Explain to them that in the Kingdom, we want our "Right hand" to know what our "Left hand" is doing—we want a life of symmetry. Write down one thing they learned about balance.
Part VII: Logic and Reflection
12.
The Broken Mirror:
If a curve has points at (1, 5) and (5, 5), but its vertex is at (2, -1), is it a symmetrical parabola? Why or why not? Use the Midpoint Rule to explain the "Rupture."
13.
The Mirror of the Word:
How does the Axis of Symmetry reflect the character of God's consistency? If God's love is the Axis of our lives, how should our "left side" (private life) and our "right side" (public life) relate to one another? How does the Axis protect the Arc from collapsing?
14.
The Sower's Symmetry:
A sower is planting two seeds that must be symmetrical around the center of the field ($x=0$). If the first seed is at $x=5$, where is the second seed? What is the Axis of Symmetry for this field?
"I vow to be consistent. I will draw my Axis with precision and find the partner for every point. I will not settle for an unbalanced story, but will walk in the perfect symmetry of the Father's truth. I will honor the Mirror of the Word in all my ways."
[VOLUME 2 WORKBOOK SPEC: 13.2]
This workbook reinforces the relationship between the Vertex, the Axis, and the reflecting points of a parabola. It emphasizes the midpoint property of symmetry.
Total Practice Items: 9
Theological Anchor: Consistency/Integrity