Welcome to our new 6-day cycle focused on Quadratics! In this unit, we are going to look beyond just graphing; we will master how to identify the most critical features of a parabola—the intercepts, the vertex, and the axis of symmetry—simply by looking at the equation.

To get started, please download the comprehensive workbook we will be using for the next week:

Download: Quadratics Packet & Notes

The Three Forms of a Quadratic Equation

As outlined in the class notes, there are three primary formats you need to know. Understanding the strengths of each will help you solve problems much faster.

1. Standard Form

$$f(x) = ax^2 + bx + c$$

  • Direction: If $a$ is positive, the graph opens up (happy). If $a$ is negative, it opens down (sad).
  • Axis of Symmetry (A.O.S.): Calculate this using the formula $x = \frac{-b}{2a}$.
  • Vertex: Plug your A.O.S. value back into the original equation to find the y-coordinate. Write your answer as a point $(x, y)$.
  • Y-Intercept: This is the easiest to find in standard form! It is simply $(0, c)$.

2. Vertex Form

$$f(x) = a(x-h)^2 + k$$

  • The Vertex: As the name implies, this form gives you the vertex $(h, k)$ immediately.
  • Crucial Hint: Remember to change the sign of $h$ from the equation, but keep $k$ the same. For example, if you see $(x-3)^2$, the x-coordinate is positive 3.
  • Axis of Symmetry: This is simply $x = h$.

3. Intercept Form

$$f(x) = a(x-p)(x-q)$$

  • The Intercepts: This form reveals the roots (or zeros) of the function. The x-intercepts are located at $(p, 0)$ and $(q, 0)$.
  • Hint: Just like with vertex form, change the signs of $p$ and $q$ when extracting them from the equation.
  • Axis of Symmetry: The A.O.S. is exactly halfway between the intercepts. Calculate it using $x = \frac{p+q}{2}$.

Connecting the Forms

You will often need to switch between forms. For example, converting from Vertex Form to Standard Form involves expanding the squared binomial and distributing the coefficient $a$.

Example from notes:
$$f(x) = 2(x-3)^2 + 4$$
Step 1: Expand $(x-3)(x-3)$ to get $x^2 - 6x + 9$.
Step 2: Distribute the 2 to get $2x^2 - 12x + 18$.
Step 3: Add the constant 4.
Result: $f(x) = 2x^2 - 12x + 22$

Factoring with the GCF

Later in the packet, we will review factoring techniques, starting with the Greatest Common Factor (GCF). Always look for a GCF first! For example, in $8x^4 - 12x^3 - 16x^2$, the GCF is $4x^2$. Factoring it out leaves $4x^2(2x^2 - 3x - 4)$.

Video Resources

Watch these videos to reinforce the concepts of graphing and finding intercepts for each form: