Quadratic Functions - Formulas & Properties
IB Mathematics Analysis & Approaches (SL & HL)
📐 Three Forms of Quadratic Functions
1. Standard (General) Form:
\[f(x) = ax^2 + bx + c\]
where \(a \neq 0\)
Shows: y-intercept at \((0, c)\)
Shape: Opens upward if \(a > 0\), downward if \(a < 0\)
2. Factored (Intercept) Form:
\[f(x) = a(x - p)(x - q)\]
Shows: x-intercepts (roots) at \(x = p\) and \(x = q\)
Axis of symmetry: \(x = \frac{p + q}{2}\)
3. Vertex (Completed Square) Form:
\[f(x) = a(x - h)^2 + k\]
Shows: Vertex at \((h, k)\)
Axis of symmetry: \(x = h\)
Note: \(h\) is subtracted inside the parentheses
🔑 Essential Formulas
Axis of Symmetry:
\[x = -\frac{b}{2a}\]
This formula is given in the IB formula booklet
Vertex Coordinates:
\[x = -\frac{b}{2a}\]
\[y = f\left(-\frac{b}{2a}\right)\]
Vertex = \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\)
y-Intercept:
\[y = c \quad \text{(point: } (0, c)\text{)}\]
Obtained by setting \(x = 0\) in \(f(x) = ax^2 + bx + c\)
✖️ Quadratic Formula
Solving \(ax^2 + bx + c = 0\):
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is given in the IB formula booklet
The Two Solutions:
\[x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\]
\[x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\]
🔍 The Discriminant
Definition:
\[\Delta = b^2 - 4ac\]
The discriminant determines the number and nature of roots
Case 1: Two Distinct Real Roots
\[\Delta > 0 \quad (b^2 - 4ac > 0)\]
Graph crosses the x-axis at two points
Case 2: One Repeated Real Root (Equal Roots)
\[\Delta = 0 \quad (b^2 - 4ac = 0)\]
Graph touches the x-axis at exactly one point (the vertex)
Case 3: No Real Roots (Two Complex Roots)
\[\Delta < 0 \quad (b^2 - 4ac < 0)\]
Graph does not cross the x-axis
🔧 Completing the Square
For \(x^2 + bx + c\) (when coefficient of \(x^2\) is 1):
Step 1: Take half of the coefficient of \(x\): \(\frac{b}{2}\)
Step 2: Square it: \(\left(\frac{b}{2}\right)^2\)
Step 3: Write as: \(\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c\)
General Formula:
\[x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c\]
For \(ax^2 + bx + c\) (when \(a \neq 1\)):
Step 1: Factor out \(a\): \(a\left(x^2 + \frac{b}{a}x + \frac{c}{a}\right)\)
Step 2: Complete the square inside the bracket
Step 3: Simplify and multiply through by \(a\)
➕ Sum and Product of Roots
For equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\):
\[\text{Sum of roots: } \alpha + \beta = -\frac{b}{a}\]
\[\text{Product of roots: } \alpha \cdot \beta = \frac{c}{a}\]
Forming a Quadratic from Roots:
\[x^2 - (\alpha + \beta)x + \alpha\beta = 0\]
Or equivalently: \(x^2 - (\text{sum})x + (\text{product}) = 0\)
🔄 Quadratic Transformations
From \(y = x^2\) to \(y = a(x - h)^2 + k\):
• \(a\): Vertical stretch by factor \(|a|\); reflection in x-axis if \(a < 0\)
• \(h\): Horizontal translation by \(h\) units (right if \(h > 0\), left if \(h < 0\))
• \(k\): Vertical translation by \(k\) units (up if \(k > 0\), down if \(k < 0\))
📊 Key Properties
Domain and Range:
Domain: \(x \in \mathbb{R}\) (all real numbers)
Range: If \(a > 0\): \(y \geq k\) (minimum at vertex)
If \(a < 0\): \(y \leq k\) (maximum at vertex)
Symmetry:
All quadratic functions are symmetric about their axis of symmetry \(x = -\frac{b}{2a}\)
Minimum/Maximum Value:
The minimum (if \(a > 0\)) or maximum (if \(a < 0\)) value occurs at the vertex and equals the y-coordinate of the vertex
🧮 Common Factoring Patterns
Difference of Squares:
\[a^2 - b^2 = (a - b)(a + b)\]
Perfect Square Trinomials:
\[a^2 + 2ab + b^2 = (a + b)^2\]
\[a^2 - 2ab + b^2 = (a - b)^2\]
Standard Factoring (when \(a = 1\)):
\[x^2 + (p + q)x + pq = (x + p)(x + q)\]
Find two numbers that add to \(b\) and multiply to \(c\)
🔀 Converting Between Forms
Standard → Vertex:
Use completing the square method or use vertex formulas \(h = -\frac{b}{2a}\) and \(k = f(h)\)
Standard → Factored:
Find roots using quadratic formula or factoring, then write as \(a(x - p)(x - q)\)
Vertex/Factored → Standard:
Expand the brackets and simplify to get \(ax^2 + bx + c\) form
💡 Exam Tip: The quadratic formula and axis of symmetry formula are given in the IB formula booklet. Always check which form is most useful for the question asked. Use your GDC to verify roots and vertex coordinates when solving problems.