The Watchman's Workbook
Edition 13, Lesson 13.1: Finding the Turning Point
x = -b / 2a
Part I: The Basic Pivot
For each equation, find the 'x' coordinate of the vertex using the Pivot Formula. Then, find the 'y' coordinate by substituting x back into the equation.
1.
The Humble Valley: y = x² - 4x + 7
a = ____ b = ____ c = ____
Vertex (x, y): (____, ____)
2.
The Rising Mountain: y = -x² + 6x - 2
a = ____ b = ____ c = ____
Vertex (x, y): (____, ____)
3.
The Narrow Descent: y = 2x² + 8x + 1
Vertex (x, y): (____, ____)
The Watchman's Check:
Did you check the "Spirit" of the arc? If 'a' is positive, your vertex should be the lowest point. If 'a' is negative, it should be the highest point. Does your math match the spirit?
Part II: Identifying the Spirit
State whether the vertex is a **Maximum** or a **Minimum** and find its coordinates.
4.
y = -3x² - 12x + 5
Spirit: ________________ (Max or Min?)
Vertex: (____, ____)
5.
y = x² + 10x
Spirit: ________________ (Max or Min?)
Vertex: (____, ____)
Part III: The Double Turn (Comparing Arcs)
Calculate the vertices for both equations and explain which one represents a "Deeper Valley."
6.
Equation A: y = x² - 2x + 5
Equation B: y = x² - 2x + 1
Vertex A: (____, ____) | Vertex B: (____, ____)
Which valley is lower? ________________
Part IV: The Zero Check (Connecting Edition 12 and 13)
The Vertex must always sit exactly between the two Zeros (Roots) of the parabola.
7.
The Symmetrical Proof: y = x² - 6x + 8
1. Factor the trinomial to find the Zeros (x₁ and x₂).
Zeros: x = ____ and x = ____
2. Find the midpoint of the Zeros: (x₁ + x₂) / 2.
Midpoint: x = ____
3. Use the Pivot Formula (x = -b/2a) to find the vertex x.
Vertex x: x = ____
Do they agree? ____
Part V: Kingdom Modeling
8.
The Flight of the Stone:
A stone is thrown into the air. Its height (h) at time (t) is modeled by:
h = -t² + 10t + 5.
Task: Find the Vertex. What is the maximum height of the stone, and at what time does it reach that height?
Time (t): ____ Height (h): ____
9.
The Prodigal's Return:
A person's "Level of Estrangement" from the Father is modeled by
E = x² - 12x + 40.
Find the Vertex. At what time (x) does the person reach their closest point to home (the minimum estrangement)?
Time of the Turn: ____ Min Estrangement: ____
Part VI: The Echad Extension (Transmission)
10.
The Mentoring Challenge:
Use a ball or beanbag to demonstrate a parabola to a younger sibling or friend. Ask them to point to the exact moment the ball "turns." Explain to them that in math, we can calculate that moment using the numbers 'a' and 'b'. Write a summary of how you explained the "Turning Point" as a moment of peace.
Part VII: Logic and Reflection
11.
The Axis of Symmetry:
For the equation y = x² - 8x + 15, find the vertex. Then, write the equation for the vertical line that passes through the vertex (the Axis of Symmetry).
Vertex: (____, ____) | Axis: x = ____
12.
The Theology of the Pivot:
Why is the formula for the turn (x = -b/2a) called the "Formula of the Pivot"? How does finding the vertex help you understand the *limits* of a difficult situation? How does the "Sign Flip" of the 'b' term reflect the change of heart required for repentance?
Part VIII: Graphical Witness (Vision and Logic)
The Vertex is the "Center of Vision." For the following equation, calculate the vertex and then briefly sketch the curve on the grid provided.
13.
y = x² - 2x - 3
1. Calculate Vertex: x = ____ y = ____
2. Vertex Coordinate: (____, ____)
Theological Note: Why is the Vertex the most important point to plot first when drawing an Arc?
"I vow to show every step of my calculation. I will not stop at the 'when', but will persevere until the 'where' of the vertex is established. I will honor the turning point as the holy ground of restoration. My math shall be a mirror of my metanoia, clear and true in the sight of the Father."
[VOLUME 2 WORKBOOK SPEC: 13.1]
This workbook focuses on the properties of the parabola, specifically the vertex. It applies the Standard Form coefficients to the vertex formula and requires substitution to complete the coordinate.
Total Practice Items: 9
Theological Anchor: Metanoia/Turning Points