Quadratic Core Study Guide
Algebra 2 & Honors Reference Guide
Characteristics of Quadratic Functions
Reference Sheet Page 1 • Foundations & Forms
PART 1
The Three Core Algebraic Forms
Standard Form General
\( f(x) = ax^2 + bx + c \)
- AOS: \( x = -\frac{b}{2a} \)
- Vertex: \( (h, k) = \left(-\frac{b}{2a},\, f\left(-\frac{b}{2a}\right)\right) \)
- \(y\)-Intercept: \( (0, c) \)
- Best For: Quadratic formula & \(y\)-int
\(a > 0\): opens up (min) • \(a < 0\): opens down (max)
Vertex Form Shifted
\( f(x) = a(x - h)^2 + k \)
- Vertex: \( (h, k) \) (flip \(h\) sign!)
- AOS: \( x = h \)
- Extremum: \( k \) is min/max value
- Best For: Reading vertex & shifts
Formed by completing the square on standard form
Intercept Form Factored
\( f(x) = a(x - p)(x - q) \)
- \(x\)-Intercepts: \( (p, 0) \) and \( (q, 0) \)
- AOS: \( x = \frac{p + q}{2} \) (midpoint of zeros)
- Vertex: Evaluate \( f\left(\frac{p+q}{2}\right) \)
- Best For: Real zeros & trajectory modeling
Parabola has bilateral symmetry across midpoint
PART 2
Visual Anatomy & Critical Attributes
Parabola Anatomy Map \(a > 0\)
x y AOS: x = h Vertex (h, k) Root (p,0) Root (q,0) y-int (0,c)
Vertex: Extremum \( (h, k) \) • Min if \(a > 0\), Max if \(a < 0\)
AOS: Vertical fold-line \(x = h\) dividing curve into mirror halves
| Attribute | Opens Up (\(a > 0\)) | Opens Down (\(a < 0\)) |
|---|
| Extremum | Minimum at \(y = k\) | Maximum at \(y = k\) |
| Domain | \((-\infty, \infty)\) or all real numbers \(\mathbb{R}\) (always quadratic domain) | |
| Range | \([k, \infty)\) | \(y \ge k\) |
| Increasing | \((h, \infty)\) (right of AOS) | \((-\infty, h)\) (left of AOS) |
| Decreasing | \((-\infty, h)\) (left of AOS) | \((h, \infty)\) (right of AOS) |
| End Behavior | As \(x \to \pm\infty, y \to +\infty\) | As \(x \to \pm\infty, y \to -\infty\) |
Honors Rule: Intervals of increase/decrease use \(x\)-coordinates only with strictly open parentheses \( (a, b) \). Range includes the vertex with bracket \([k\) or \(k]\).
PART 3
The Discriminant & Nature of Solutions (\(\Delta = b^2 - 4ac\))
\( \Delta > 0 \) (Positive) 2 Real Roots
• Crosses \(x\)-axis at two points.
• If \(\Delta\) is a perfect square: 2 rational roots.
• If non-square: 2 irrational conjugate roots.
\( \Delta = 0 \) (Zero) 1 Real Root
• Tangent to \(x\)-axis (bounces).
• Double root with multiplicity 2.
• Vertex lies exactly on axis: \( (h, 0) \).
\( \Delta < 0 \) (Negative) 2 Complex Roots
• Parabola never touches \(x\)-axis.
• 0 Real roots, 2 complex conjugate roots.
• Roots: \( x = -\frac{b}{2a} \pm \frac{\sqrt{|\Delta|}}{2a}i \).
Parent Function & Transformations: \( g(x) = a(x - h)^2 + k \) from \( f(x) = x^2 \)
Dilation (\(a\)): \( |a| > 1 \): Vertical stretch
\( 0 < |a| < 1 \): Vertical compress
Reflection (\(\pm a\)): \( a < 0 \): Reflect over \(x\)-axis
\( (-x)^2 = x^2 \): Even symmetry
Horizontal Shift (\(h\)): \( (x - h) \): Shift Right \( h \)
\( (x + h) \): Shift Left \( h \)
Vertical Shift (\(k\)): \( +k \): Shift Up \( k \)
\( -k \): Shift Down \( k \)
Advanced Concepts & Problem Solving
Honors Topics, Master Problem & Exam Traps
Mastery Suite Page 2 • Honors & Review
HONORS
Geometric Conic Definition & Average Rate of Change
Focus, Directrix & Focal Length (\(p\)) Conics
A parabola is equidistant from Focus point and Directrix line.
\( a = \frac{1}{4p} \iff p = \frac{1}{4a} \)
- • Standard Conic Form: \( (x - h)^2 = 4p(y - k) \)
- • Focus: \( (h,\, k + p) \) inside parabola curve
- • Directrix: Horizontal line \( y = k - p \)
- • Latus Rectum (focal width): Length \( = |4p| = \frac{1}{|a|} \)
Average Rate of Change & Finite Differences Calculus Prep
Unlike linear functions, quadratics have a variable rate of change:
\( \text{AROC}_{[a, b]} = \frac{f(b) - f(a)}{b - a} \) (secant slope)
- • Across symmetric interval \( [h - d, h + d] \), the \(\text{AROC} = 0\).
- • 2nd Finite Differences: For equal \(\Delta x\), second difference in \(y\) is constant: \(\Delta^2 y = 2a(\Delta x)^2\).
- • Confirms data set is exactly quadratic!
MASTER EXAMPLE
Full Comprehensive Analysis Walkthrough
Given: \( f(x) = -2x^2 + 8x + 10 \)
1. Orientation & Vertex Form
• \(a = -2, b = 8, c = 10\)
• Since \(a = -2 < 0\), opens down (maximum).
• AOS: \(x = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2\)
• Vertex \(y\): \(f(2) = -2(2)^2 + 8(2) + 10 = 18\)
• Vertex: \((2, 18)\)
• Vertex Form: \(f(x) = -2(x - 2)^2 + 18\)
2. Intercepts & Factored Form
• \(y\)-Intercept: \(f(0) = 10 \implies (0, 10)\)
• \(x\)-Intercepts: Set \(-2x^2 + 8x + 10 = 0\):
\(-2(x^2 - 4x - 5) = 0\)
\(-2(x - 5)(x + 1) = 0\)
\(\implies x = 5\) or \(x = -1\)
• Zeros: \((-1, 0)\) and \((5, 0)\)
• Factored Form: \(-2(x + 1)(x - 5)\)
3. Key Characteristics & Honors
• Domain: \((-\infty, \infty)\)
• Range: \((-\infty, 18]\) (\(y \le 18\))
• Increasing: \((-\infty, 2)\) • Decreasing: \((2, \infty)\)
• End Behavior: As \(x \to \pm\infty, y \to -\infty\)
• Honors Conics: \(p = \frac{1}{4(-2)} = -\frac{1}{8}\)
Focus: \((2, 17.875)\) • Directrix: \(y = 18.125\)
WARNING
Top 6 Exam Pitfalls & How to Avoid Them
1. The \( (x - h) \) Sign Flip In \( y = 3(x + 4)^2 - 5 \), vertex is \((-4, -5)\), NOT \((+4, -5)\). Formula has built-in minus sign.
2. AOS is an Equation! Writing "AOS \(= 3\)" loses credit. The Axis of Symmetry is a vertical line: must write \(x = 3\).
3. Range Bracket vs Parenthesis Vertex value is included, so use square bracket: \([k, \infty)\) or \((-\infty, k]\). Never open parenthesis.
4. Inc/Dec Uses \(x\)-Values State intervals along the \(x\)-axis only, never \(y\)-values. Always use open interval \((a, b)\) because slope at vertex is 0.
5. Sign of \(-b\) in AOS Formula Formula is \(x = \mathbf{-}\frac{b}{2a}\). If \(b = -6\), then \(-b = +6\). Double negatives are the #1 source of lost test points!
6. Factoring with \(a \neq 1\) Always factor out GCF or leading coefficient \(a\) first before factoring binomials, e.g. \(-2(x^2 - 4x - 5)\).
EXAM FORMULA VAULT:
AOS: \(x = -\frac{b}{2a}\) • Quad Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) • \(\Delta = b^2 - 4ac\) • Focus: \(p = \frac{1}{4a}\)
