Parabola Roots Slides
Algebra II • Quadratic Functions
Parabola Roots
Mastering Factored Form, the Zero-Product Property, & Intercept Connections
\( f(x) = a(x - r_1)(x - r_2) \)
Spot Zeros Instantly
Zero-Product Logic
Graph with Precision
Core Foundations
Anatomy of Factored Form
01 / Concept
\( f(x) = a(x - r_1)(x - r_2) \)
\(a\)
Vertical stretch / direction: Opens upward if \(a > 0\), opens downward if \(a < 0\).
\(r_1\)
First zero: Value where the first linear factor equals zero.
\(r_2\)
Second zero: Value where the second linear factor equals zero.
Direct Visibility
Standard form hides where the graph hits the ground. Factored form hands you the roots on a silver platter.
Symmetry Clue
The vertex and axis of symmetry are always perfectly centered halfway between \(r_1\) and \(r_2\).
Key Rule: Notice the subtraction sign • \( (x - r) \) means the root has the opposite sign of the written constant. Sign Reversal Alert
The Mathematical Engine
The Zero-Product Property
02 / Principle
Fundamental Algebraic Axiom
If \( A \cdot B = 0 \), then \( A = 0 \) or \( B = 0 \)
Two quantities can only multiply to zero if at least one of those quantities is itself zero.
1
Set to Zero
Ensure the entire quadratic equation equals zero:
\( (x - 3)(x + 5) = 0 \)
2
Split Factors
Create two simple, independent linear equations:
\( x - 3 = 0 \quad\text{or}\quad x + 5 = 0 \)
3
Solve for \(x\)
Isolate variable in both branches to find the solutions:
\( x = 3 \quad\text{or}\quad x = -5 \)
A quadratic degree 2 polynomial yields at most two real solutions.
Vocabulary & Geometry
Factors, Zeros, & \(x\)-Intercepts
03 / Bridge
1. The Factor
\( (x - r) \)
The algebraic expression inside the parentheses multiplying the rest of the function.
2. The Zero (Root)
\( x = r \)
The numerical input value that causes the function output \(f(x)\) to become exactly zero.
3. The \(x\)-Intercept
\( (r, 0) \)
The physical coordinate point on the Cartesian graph where the parabola crosses the horizontal axis.
Factor: \((x - 7)\)
Zero: \(x = 7\)
Coordinate: \((7, 0)\)
Factor: \((x + 4)\)
Zero: \(x = -4\)
Coordinate: \((-4, 0)\)
Common test mistake: Writing a zero as a point or writing an intercept as just a number. Keep them distinct!
Step-by-Step Modeling
Worked Example 1: Direct from Factors
Example 1
Given the quadratic function: \( f(x) = (x - 2)(x + 6) \) Find the zeros & \(x\)-intercepts.
Step 1: Set \(f(x) = 0\)
Replace function output with zero to find horizontal ground hits:
\( (x - 2)(x + 6) = 0 \)
Step 2: Apply Property
Break down into two linear equations:
\( x - 2 = 0 \implies \mathbf{x = 2} \)
\( x + 6 = 0 \implies \mathbf{x = -6} \)
Step 3: State Results
Express as zeros and coordinate points:
Zeros: \( x \in \{-6, 2\} \)
Intercepts: \( (-6, 0), (2, 0) \)
Midpoint bonus: Axis of symmetry is \( x = \frac{-6 + 2}{2} = -2 \)
Vertex \(x\)-coordinate is always exactly centered!
Step-by-Step Modeling
Worked Example 2: Factoring First
Example 2
Solve the equation by factoring: \( x^2 - 5x - 14 = 0 \) Find the solutions for \(x\).
Phase 1: Find the Factors
We need two integers \(m\) and \(n\) such that:
- Product: \( m \cdot n = -14 \)
- Sum: \( m + n = -5 \)
Numbers: -7 and +2
\((-7) \times 2 = -14\), and \((-7) + 2 = -5\).
Phase 2: Factored Form & Solve
Write factored expression equal to zero:
\( (x - 7)(x + 2) = 0 \)
Apply Zero-Product Property:
\( x = 7 \quad\text{or}\quad x = -2 \)
Graphical meaning: The parabola crosses the \(x\)-axis at \((-2, 0)\) and \((7, 0)\). Solutions Verified
Special Cases
Example 3: GCF & What \(a \neq 1\) Means
Example 3
Find the zeros of the function: \( g(x) = 2x^2 - 8x \) Factoring with a GCF
1. Factor Out GCF
Both terms share \(2x\). Pull it out front:
\( 2x(x - 4) = 0 \)
2. Zero-Product Logic
The constant 2 is never zero. Focus on variable factors:
\( 2x = 0 \implies x = 0 \)
\( x - 4 = 0 \implies x = 4 \)
3. Geometric Zeros
Roots at origin and positive 4:
Zeros: \( x = 0, x = 4 \)
Intercepts: \( (0, 0), (4, 0) \)
Does the leading coefficient \(a = 2\) change the zeros?
No! The multiplier \(a\) stretches the parabola vertically, making it steeper, but the curve still crosses the \(x\)-axis at exactly the same points.
Active Engagement
Your Turn: Practice Problems
Student Challenge
Problem A Factored Form Given
\( y = (x + 3)(x - 8) \)
1 Identify the two zeros of the function.
2 Write both \(x\)-intercepts as coordinate points.
Work time: 2 minutes
Problem B Factor & Solve
\( x^2 + 4x - 12 = 0 \)
1 Factor the quadratic into two binomials.
2 Apply the Zero-Product Property to solve for \(x\).
Work time: 3 minutes
Tip: Check your signs carefully when converting from factor \((x + c)\) to zero! Answers on next slide →
Answer Key & Verification
Practice Solutions Breakdown
Solutions
Solution A: \( y = (x + 3)(x - 8) \)
1. Set factors equal to zero:
\( x + 3 = 0 \implies \mathbf{x = -3} \)
\( x - 8 = 0 \implies \mathbf{x = 8} \)
2. \(x\)-Intercept Coordinates:
\( (-3, 0) \quad\text{and}\quad (8, 0) \)
Solution B: \( x^2 + 4x - 12 = 0 \)
1. Factoring (product \(-12\), sum \(+4\)):
Numbers are \(+6\) and \(-2\):
\( (x + 6)(x - 2) = 0 \)
2. Zero-Product Property Solutions:
\( x = -6 \quad\text{or}\quad x = 2 \)
Sign Check: If the factor is \((x + 6)\), the zero is \(x = -6\). Never forget to flip the sign!
Summary
Recap: The Factored Form Toolkit
Key Takeaways
1. Instant Zeros
In \( y = a(x - r_1)(x - r_2) \), the zeros \(r_1\) and \(r_2\) are directly visible. Always watch the sign flip.
2. Zero-Product Power
If \(A \cdot B = 0\), split the quadratic into two simple linear equations. This is the fastest way to solve.
3. Graph Anchors
Every zero \(r\) anchors an \(x\)-intercept at \((r, 0)\). The parabola's axis of symmetry sits exactly between them.
Ready for graphing, finding vertices, and modeling real-world parabolas!
Q.E.D.
