Session 3 Lecture Notes Sketching Quadratics - Factorization

Published

September 15, 2026

1. Review of Linear Graphs (y = mx + c)

  • General Strategy: To plot a linear graph, determine two key points on the axes:
    • y-intercept: Set x = 0 \implies y = c. The graph passes through (0, c).
    • x-intercept: Set y = 0 \implies mx + c = 0 \implies x = -\frac{c}{m}. The graph passes through \left(-\frac{c}{m}, 0\right).
  • Why two points work: Because y = mx + c is a straight line, finding two points is sufficient to draw the complete line.

2. Understanding Functions (y = f(x))

  • Intuitive Meaning: y changes simultaneously as x changes (y depends on x).
  • Mathematical Meaning: y can be expressed as an algebraic combination/expression involving x.
  • Universal Plotting Strategy for any Curve y = f(x):
    1. Find the y-intercept: Substitute x = 0 into y = f(x) to evaluate f(0).
    2. Find the x-intercept(s): Set y = 0 and solve the equation f(x) = 0.
    3. Determine Curve Geometry: Identify additional points, symmetry, or vertices to complete the sketch.

3. Example: Plotting a Non-Linear Curve (y^2 = 4 - x^2)

  1. y-intercepts: Set x = 0 \implies y^2 = 4 \implies y = \pm 2. Points: (0, 2) and (0, -2).
  2. x-intercepts: Set y = 0 \implies 0 = 4 - x^2 \implies x^2 = 4 \implies x = \pm 2. Points: (2, 0) and (-2, 0).
  3. Shape: The given equation is the equation of a circle. Connecting these four points smoothly forms a circle centered at the origin with radius r = 2 (x^2 + y^2 = 2^2).

4. Basic Parabolas (y = ax^2)

  • Standard Form (y = x^2 or y = ax^2):
    • Setting x = 0 \implies y = 0 and setting y = 0 \implies x = 0. The x- and y-intercepts meet at the origin (0,0).
    • The curve is an upward-turned parabola symmetrical around the y-axis.
    • Fixing the scale: Substitute one additional x-value (e.g., x = 2 \implies y = 4a) to fix the curve, then draw symmetrically.
  • Negative Coefficient (y = -2x^2):
    • Forms a downward-turned parabola passing through (0,0) (e.g., at x = 2, y = -8).

5. Quadratic Curves of the Form y = ax^2 + bx

Step-by-Step Derivation & Formulas

  1. y-intercept: Set x = 0 \implies y = 0. Point: (0, 0).
  2. x-intercepts: Set y = 0 \implies ax^2 + bx = 0 \implies ax\left(x + \frac{b}{a}\right) = 0 \implies x = 0 \text{ or } x = -\frac{b}{a}.
  3. Axis of Symmetry & Vertex:
    • Parabolas are symmetrical around their vertex line.
    • Vertex x-coordinate (x_v): Midpoint of the two x-intercepts: x_v = \frac{0 + \left(-\frac{b}{a}\right)}{2} = -\frac{b}{2a}.
    • Vertex y-coordinate (y_v): Substitute x_v = -\frac{b}{2a} back into y = ax^2 + bx: y_v = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) = \frac{b^2}{4a} - \frac{b^2}{2a} = -\frac{b^2}{4a}.
    • Vertex Coordinates: \left(-\frac{b}{2a}, -\frac{b^2}{4a}\right).

Worked Example: y = 2x^2 + 6x

  • y-intercept: x = 0 \implies y = 0 \implies \mathbf{(0, 0)}.
  • x-intercepts: Solve 2x^2 + 6x = 0 \implies 2x(x + 3) = 0 \implies x = 0 \text{ or } x = -3 \implies \mathbf{(0,0) \text{ and } (-3,0)}.
  • Vertex x-value: x_v = \frac{0 + (-3)}{2} = \mathbf{-1.5}.
  • Vertex y-value: y_v = 2(-1.5)^2 + 6(-1.5) = 4.5 - 9 = \mathbf{-4.5}.
  • Vertex Coordinates: (-1.5, -4.5).
Figure 1: The graph of y=2x^2+6x, showing its intercepts, vertex, and axis of symmetry.

6. General Quadratic Curves y = ax^2 + bx + c

  1. y-intercept: Set x = 0 \implies y = c \implies \mathbf{(0, c)}.
  2. x-intercepts: Solve ax^2 + bx + c = 0 using factorization or quadratic methods.
  3. Symmetry Shortcut: y = c occurs at both x = 0 and x = -\frac{b}{a}. The midpoint between them gives the axis of symmetry x = -\frac{b}{2a}.

Factorizable Example: y = x^2 + x - 12 = (x - 3)(x + 4)

  • y-intercept: Set x = 0 \implies y = (0-3)(0+4) = -12 \implies \mathbf{(0, -12)}.
  • x-intercepts: Set y = 0 \implies (x - 3)(x + 4) = 0 \implies x = 3 \text{ or } x = -4 \implies \mathbf{(3, 0) \text{ and } (-4, 0)}.
  • Vertex x-value: x_v = \frac{3 + (-4)}{2} = \mathbf{-0.5}.
  • Vertex y-value: y_v = (-0.5 - 3)(-0.5 + 4) = (-3.5)(3.5) = \mathbf{-12.25}.
  • Vertex Coordinates: (-0.5, -12.25).
Figure 2: The graph of y=x^2+x-12, showing its intercepts, vertex, and axis of symmetry.

7. Homework & Pre-Work Assignment

  1. Task 1: Find x- and y-intercepts for various given functions y = f(x).
  2. Task 2: Sketch curves of the form y = ax^2 + bx.
  3. Task 3: Sketch factorizable quadratic curves y = ax^2 + bx + c.
  4. Mandatory Formatting: Write mathematical steps on the left side of the page and the step-by-step narrative reasoning on the right side.
  5. Class Preparation: Practice explaining each step out loud to a classmate prior to the next session.