Parabola Anatomy Guided Notes Algebra 2 Focus & Directrix Introduction
Parabola Anatomy: Focus & Directrix
Discovering the geometric locus definition of a quadratic curve
Name: _________________
Date: ___________
Period: _______________
Teacher: __________
1. What Exactly is a Parabola?
In Algebra 1, you graphed parabolas as quadratic curves. Geometrically, a parabola is the set of all points that are equidistant (the exact same distance) from:
A fixed point called the FOCUS (inside curve).
A fixed line called the DIRECTRIX (outside curve).
2. Visual Anatomy & Key Terms
Equidistant Property: \(d_1 = d_2\)
x y Directrix: y = -p Vertex (0,0) Focus (0, p) p p P(x,y) d₁ d₂
By Definition: Distance d₁ = Distance d₂
The Value of \(p\): The distance from the vertex to the focus, and also from the vertex to the directrix.
Vertex: The turning point of the parabola. It lies exactly halfway between the focus and the directrix. (For today: vertex at \((0,0)\)).
Focus vs. Directrix:
Focus: Always a point inside the curve.Directrix: Always a line outside the curve.
3. The Core Equation: Connecting \(a\) and \(p\)
Core Concept
You already know standard quadratic form: \(y = ax^2\) . The stretch factor \(a\) is connected to focal distance \(p\) by one simple relation:
To Find \(a\) from \(p\):
\[a = \frac{1}{4p}\]
To Find \(p\) from \(a\):
\[p = \frac{1}{4a}\]
Opens UPWARD (\(a > 0\))
Focus is above vertex: \((0, p)\)below vertex: \(y = -p\)
Opens DOWNWARD (\(a < 0\))
Focus is below vertex: \((0, -p)\)above vertex: \(y = p\)
Parabola Anatomy Guided Notes • Focus & Directrix Page 1 of 2
Page 2
Guided Examples & Step-by-Step Practice
Core Equation: \(a = \frac{1}{4p}\)
Example 1: Given Focus → Write Equation Vertex at \((0,0)\)
A parabola has its vertex at \((0,0)\) and focus at \((0, 2)\). Find the directrix and write the equation of the parabola.
Step 1: Identify \(p\)
Distance from \((0,0)\) to \((0,2)\):
Step 2: Calculate \(a\)
\(a = \frac{1}{4(p)} = \frac{1}{4(2)}\)
Step 3: Directrix & Equation
Directrix: \(y = -2\)
Example 2: Given Equation → Find Features (Guided) Vertex at \((0,0)\)
Given the equation \(y = \frac{1}{12}x^2\), find the value of \(p\), the focus coordinates, and the directrix equation.
1. Identify \(a\) & direction
\(a = \underline{\hspace{26px}}\)
2. Solve for \(p\) using \(\frac{1}{4a}\)
\(4p = \underline{\hspace{26px}}\)
3. Write Focus & Directrix
Focus: \((0, \underline{\hspace{18px}})\)
Example 3: Opening Downward (\(a < 0\)) Watch the Negative!
Given the parabola equation \(y = -x^2\):
1. Stretch factor & direction: \(|a| = 1\), opens DOWN
2. Solve for distance \(p\): \(1 = \frac{1}{4p} \implies 4p = 1 \implies p =\) \(\frac{1}{4}\)
3. Focus & Directrix: Focus: \((0, -\frac{1}{4})\) | Directrix: \(y = \frac{1}{4}\)
Key Insight: Distance \(p\) is always positive. The minus sign in front of \(x^2\) tells you the curve faces down !
5. Check Your Understanding: Complete the Table
Parabola Equation Value of \(p\) Opens Focus Point Directrix Line \(y = \frac{1}{16}x^2\) \(p = \underline{\hspace{20px}}\) _________ \((0, \underline{\hspace{18px}})\) \(y = \underline{\hspace{20px}}\) \(y = \underline{\hspace{35px}}x^2\) \(p = 5\) Up \((0, \underline{\hspace{18px}})\) \(y = \underline{\hspace{20px}}\) \(y = \frac{1}{4}x^2\) \(p = \underline{\hspace{20px}}\) _________ \((0, \underline{\hspace{18px}})\) \(y = \underline{\hspace{20px}}\) \(y = \underline{\hspace{35px}}x^2\) \(p = 2\) Down \((0, \underline{\hspace{18px}})\) \(y = \underline{\hspace{20px}}\)
TIP
Three-Way Check: Vertex is always in the exact middle. Focus is inside the curve. Directrix never touches the parabola!
\(a = \frac{1}{4p}\)
Parabola Anatomy Guided Notes • Focus & Directrix Page 2 of 2
Parabola Anatomy Practice Worksheet Algebra 2 Practice Worksheet
Focus & Directrix of Parabolas
Using the focal relationship \(a = \frac{1}{4p}\) • All vertices at \((0,0)\)
Name: _________________
Date: ___________
Period: _______________
Score: _____ / 25
Key Formula: \(a = \frac{1}{4p}\) | \(p = \frac{1}{4a}\) | Opens Up: \(a > 0\) (Focus above) | Opens Down: \(a < 0\) (Focus below)
A
Part A: Anatomy & Vocabulary Check
D A B C
1. Letter A represents:
2. Letter B represents:
3. Letter C represents:
4. Letter D represents:
B
Part B: From Equation to Anatomy
Show work using \(a = \frac{1}{4p}\) or \(p = \frac{1}{4a}\)
5. \(y = \frac{1}{8}x^2\) Vertex: \((0,0)\)
Opens: ________________________
Show Work for \(p\):
Focus: \((0, \underline{\hspace{20px}})\)
Directrix: \(y = \underline{\hspace{20px}}\)
6. \(y = \frac{1}{20}x^2\) Vertex: \((0,0)\)
Opens: ________________________
Show Work for \(p\):
Focus: \((0, \underline{\hspace{20px}})\)
Directrix: \(y = \underline{\hspace{20px}}\)
7. \(y = -\frac{1}{16}x^2\) Negative!
Opens: ________________________
Show Work for \(p\):
Focus: \((0, \underline{\hspace{20px}})\)
Directrix: \(y = \underline{\hspace{20px}}\)
8. \(y = 2x^2\) Fractional \(p\)
Opens: ________________________
Show Work for \(p\):
Focus: \((0, \underline{\hspace{20px}})\)
Directrix: \(y = \underline{\hspace{20px}}\)
Parabola Anatomy Practice Worksheet • Focus & Directrix Page 1 of 2
Page 2
Writing Equations & Graphing Parabolas
Name:
C
Part C: From Features to Equation
9. Focus at \((0, 3)\), Directrix at \(y = -3\)
Vertex: \((0,0)\) • Distance \(p =\) ______ • Opens: ______
Calculate \(a = \frac{1}{4p}\):
Equation: \(y = \underline{\hspace{60px}}x^2\)
10. Focus at \((0, -2)\), Directrix at \(y = 2\)
Vertex: \((0,0)\) • Distance \(p =\) ______ • Opens: ______
Calculate \(a = -\frac{1}{4p}\):
Equation: \(y = \underline{\hspace{60px}}x^2\)
11. Focus at \((0, \frac{1}{2})\), Vertex at \((0,0)\)
Distance \(p =\) ______ • Directrix: \(y = \underline{\hspace{20px}}\)
Parabola Anatomy Teacher Key Teacher Guide & Key Parabola Anatomy Solutions
Guided Notes Answer Key & Teaching Guide
Introductory Focus & Directrix using \(a = \frac{1}{4p}\)
Recommended Time: 45–50 min
Target Level: Algebra 2 (Intro)
Pedagogical Tips & Common Student Traps
1. Distance is always positive: Students often plug in a negative \(p\) when a parabola opens downward. Remind them: \(p\) is physical geometric distance (\(p > 0\)); the negative sign belongs exclusively to the direction of opening in \(y = -ax^2\) and the focus coordinate \((0, -p)\).
2. Reciprocal Confusion: When solving \(\frac{1}{12} = \frac{1}{4p}\), students cross-multiply or simply equate denominators: \(4p = 12 \implies p = 3\). Reinforce equating numerators/denominators when the numerators are both 1.
1 Guided Notes Page 1 Solutions
Definition Blanks: A parabola is the set of all points that are equidistant from a fixed point called the FOCUS and a fixed line called the DIRECTRIX.
Core Formulas: \(a = \frac{1}{4p}\) | \(p = \frac{1}{4a}\) | Vertex is always halfway between focus and directrix.
2 Guided Notes Page 2: Model Examples & Table Solutions
Example 2 (Guided Try-It): \(y = \frac{1}{12}x^2\)
1. \(a = \frac{1}{12}\), opens UP (since \(a > 0\)).
2. \(4p = 12 \implies \mathbf{p = 3}\).
3. Focus: \(\mathbf{(0, 3)}\) • Directrix: \(\mathbf{y = -3}\).
Equation Value of \(p\) Opens Focus Directrix \(y = \frac{1}{16}x^2\) p = 4 Up (0, 4) y = -4 \(y = \frac{1}{20}x^2\) p = 5 Up (0, 5) y = -5 \(y = \frac{1}{4}x^2\) p = 1 Up (0, 1) y = -1 \(y = -\frac{1}{8}x^2\) p = 2 Down (0, -2) y = 2
Prompt for Deep Understanding: "If a parabola gets wider and flatter, does the focus move closer to or further away from the vertex?" further away from the vertex!
Parabola Anatomy Teacher Key • Reference & Solutions Page 1 of 2
Page 2 Key
