Chapter 4: Quadratic Equations & Graphs

1. Standard Properties & Nature of Roots

Standard Equation: ax² + bx + c = 0 (a ≠ 0)

Roots Formula: x = [ -b ± √(D) ] / 2a (where Discriminant D = b² - 4ac)

The value of D establishes whether your roots cross the x-axis twice (D > 0), touch it at one spot (D = 0), or float completely above/below it without touching (D < 0).

2. Mechanics of the Parabolic Graph

Every quadratic function y = ax² + bx + c produces a symmetric U-shaped smooth curve known as a Parabola.

Key Parameters to Find Before Plotting:

Direction of Opening: If a > 0, the curve opens Upward (holds water). If a < 0, the curve opens Downward (sheds water).
The Vertex Coordinates (h, k): The absolute turning point of the curve.
h = -b / 2a
k = f(h) = a(h)² + b(h) + c
Axis of Symmetry: The central vertical line that cuts the parabola into two matching mirror halves: x = -b / 2a.

3. Transformations of Quadratic Functions

By converting standard equations into Vertex Form, you can quickly analyze geometric shifts away from the base parent curve y = x²:

Vertex Form: y = a(x - h)² + k

Parameter Shift	Mathematical Action	Geometric Effect on Graph
+k or -k	Vertical Shift Outer Value	Shifts the whole curve Up (+k) or Down (-k)
(x - h) or (x + h)	Horizontal Shift Inner Value	Shifts the curve Right (+h) or Left (-h) (Sign is opposite!)
\|a\| > 1	Vertical Stretch Factor	Stretches the curve vertically, making the parabola look Narrower
0 < \|a\| < 1	Vertical Compression Factor	Flattens the curve vertically, making the parabola look Wider
Negative (-a)	Vertical Reflection Axis	Flips / Reflects the entire parabola upside down across the x-axis

4. Solving Quadratic Equations Graphically

In the SEE exam, you are often asked to solve a quadratic equation like ax² + bx + c = 0 by breaking it into a system containing a standard base curve and a straight line.

Operational Step-by-Step Method:

Isolate the Base Quadratic Term: Rearrange your target equation so the base curve expression sits on one side. For example, change x² - 2x - 3 = 0 into x² = 2x + 3.
Split Into Two Separate Equations:
• Equation 1 (Base Parabola): y = x²
• Equation 2 (Straight Line Intercept): y = 2x + 3
Build Coordinate Value Tables:
• For y = x², calculate points around the origin: (-2,4), (-1,1), (0,0), (1,1), (2,4).
• For y = 2x + 3, find any 3 linear coordinate points: (-1,1), (0,3), (3,9).
Plot and Intersect: Draw both lines cleanly on the same graph sheet.
The final answers to your equation are the exact x-coordinates where the straight line crosses through the parabola.

Visual Concept Matrix:

Quadratic Equations: Algebraic & Graphical Mastery