Welcome back to class! In this lesson, we are diving deep into the visual language of algebra: Graphing and Functions. Tonight's topics bridge the gap between simple equations and analyzing complex behaviors of graphs, such as parabolas and piecewise-defined functions. Whether you are finding the vertex of a quadratic or shifting a graph horizontally, understanding these visual cues is essential for success in Pre-Calculus.

1. Identifying and Analyzing Functions

Before we can graph complex equations, we must define what a function actually is. We start with the Vertical Line Test, a visual method to determine if a relation is a function. Remember, a graph represents a function only if no vertical line intersects the graph at more than one point.

Once identified, we look at the behavior of the graph:

  • Intervals: We will determine where a function is increasing (going up from left to right), decreasing (going down), or constant using interval notation (e.g., $(-\infty, 3] \cup [5, \infty)$).
  • Parabolas: For quadratic functions, we identify the vertex $(h, k)$, the intercepts, and the axis of symmetry. We also look at how the leading coefficient affects the shape—specifically, how a larger coefficient can "stretch" the graph vertically, making it narrower.

2. Piecewise and Square Root Functions

One of the more challenging topics tonight is the Piecewise-Defined Function. These are functions defined by different formulas for different parts of their domain. We will tackle three levels of difficulty (Problem types 1, 2, and 3), learning how to graph "pieces" of lines or curves that stop and start at specific $x$-values.

We will also explore Square Root Functions, typically in the form $f(x) = \sqrt{x-h} + k$. The key here is identifying the starting point of the "arc" and determining the domain restrictions, as we cannot take the square root of a negative number in the real number system.

3. Transformations and Parent Graphs

Finally, we will look at the "family tree" of math: Parent Graphs. By memorizing the basic shapes (like $y=x^2$, $y=|x|$, and $y=\sqrt{x}$), we can apply transformations to graph any variation quickly.

Key transformations we will practice include:

  • Reflections: Flipping a graph over the $x$-axis or $y$-axis.
  • Translations: Moving a graph up/down (vertical) or left/right (horizontal). For example, to write an equation after a translation, remember that $f(x-c)$ shifts the graph to the right by $c$ units, while $f(x) + c$ shifts it up.
  • Leading Coefficients: Understanding how negative coefficients flip the graph and how the absolute value affects the "V" shape of functions like $f(x) = a|x|$.

Take your time matching the parent graphs to their equations and pay close attention to the signs when performing translations. Let's get graphing!