Welcome to Week 10! Despite the disruptions caused by Hurricane Sandy, we are pressing forward with one of the most important topics in algebra: Quadratic Functions. This week, we aren't just moving $x$'s and $y$'s around; we are looking at how these functions are used in the real world to find maximums, minimums, and optimal solutions.

1. The Three Forms of a Quadratic Graph

As we discussed in the lecture notes, quadratic functions can be written in three distinct formats. Being able to recognize these will help you graph parabolas quickly and efficiently:

  • Standard Form: $$y = ax^2 + bx + c$$
    Useful for: Using the Quadratic Formula and finding the $y$-intercept $(0, c)$. The axis of symmetry is found using $x = \frac{-b}{2a}$.
  • Vertex Form: $$y = a(x-h)^2 + k$$
    Useful for: Identifying the vertex $(h, k)$ instantly. This is the turning point of the graph.
  • Intercept Form: $$y = a(x-p)(x-q)$$
    Useful for: Finding the $x$-intercepts (roots) at $(p, 0)$ and $(q, 0)$.

Key Concept: The leading coefficient $a$ tells you the direction. If $a$ is positive, the graph opens up (has a minimum). If $a$ is negative, it opens down (has a maximum).

2. Quadratics in Disguise (U-Substitution)

Sometimes you will encounter equations that don't look like quadratics at first glance but follow the same structure. We call these "Quadratics that don't look like Quadratics." We solve these using U-Substitution.

For example, looking at the equation $$x^4 - 9x^2 + 8 = 0$$

This looks like a 4th-degree polynomial, but notice the powers are double each other ($x^4$ and $x^2$). We can let $u = x^2$, which means $u^2 = x^4$. The equation transforms into a simple quadratic:

$$u^2 - 9u + 8 = 0$$

After factoring to find $u$, don't forget to substitute back to solve for $x$!

3. Applications: Optimization in the Real World

Why do we care about the vertex? Because in real-world models, the vertex represents the maximum or minimum value.

  • Precipitation Models: In our class notes, we looked at Sonoma, CA rainfall modeled by $P(x) = 0.2x^2 - 2.8x + 9.8$. By finding the vertex (specifically the minimum), we determined that July ($x=7$) has the least precipitation.
  • Area Optimization: We also solved the "Patio Design" problem. Given a limited amount of stone for a perimeter, we used a quadratic equation to calculate the dimensions that would yield the maximum possible area ($450$ sq ft).

4. Class Resources

Due to the storm, our schedule shifted slightly. Below you will find the notes covering the "Sandy Stole My Homework" session and the subsequent lectures, as well as the answer key for the recent Chapter 9 and 10 test.

Keep practicing those graphs and remember: check your signs when using the quadratic formula!