Welcome back to class! In our previous lessons, we mastered the art of taking a function rule, such as $f(x) = (x-2)^2$, and graphing it using transformations. Today, we are flipping the script in a lesson I like to call Parent Functions III: Revenge of the Tables.

From Data to Functions

Instead of starting with an equation, we are starting with a table of points. This is a critical skill for mathematical modeling because, in the real world, we often get raw data first and have to find the equation that fits it. To succeed in this, follow these three steps discussed in our Class Notes:

  1. Plot the Points: sketch the coordinates on a graph to visualize the shape.
  2. Identify the Parent Function: Does it look like a U-shape ($x^2$)? A snake ($x^3$)? A shooting star ($\sqrt{x}$)?
  3. Determine the Transformation: Compare the "vertex" or "center" of your data to the origin $(0,0)$ to find the vertical and horizontal shifts.

Key Examples

Let's look at an example from today's lecture notes. Consider the following table:

  • $(-2, 6)$
  • $(-1, 3)$
  • $(0, 2)$
  • $(1, 3)$
  • $(2, 6)$

When we plot these, we see the symmetric U-shape of a Quadratic Function. However, the lowest point (the vertex) isn't at $(0,0)$; it is at $(0,2)$. This indicates a vertical translation upwards by 2 units. Therefore, our function is $$f(x) = x^2 + 2$$

We also looked at a table that formed a cubic shape "snaking" through the point $(1,0)$. Because the inflection point moved to the right by 1 unit, the function is $f(x) = (x-1)^3$.

Class Resources

Please make sure to review the attached PDF notes, as they contain several more examples, including square root functions and linear simplifications.

  • Class Notes: Detailed graphs and handwritten examples.
  • Worksheets: Vocabulary and example worksheet, Core Math Worksheet.
  • Homework: Practice Worksheet 1-2 B.

Discussion Question of the Day

Let's practice creating our own data sets. Give me a table of points that you created.

The rules are simple: it must be based on one of our parent functions but must include at least one transformation (shift, stretch, or reflection). I will reply to your comment identifying the function and the transformation. If you stump me, but you can provide the correct answer, you earn two extra points!