Welcome to Parent Functions Part II

Today, we are taking a closer look at graphs to gain a better understanding of the transformations occurring within function families. Just as family members share common traits, mathematical functions belong to families based on their characteristics. The Parent Function is the simplest function in that family, and every other function is just a transformation of that parent.

Key Concepts: Identifying the Parent

To know what a graph looks like, you first need to identify which family it belongs to. We do this by looking at the power of $x$ (the degree) or the root involved. As detailed in our class notes, here are the four main parent functions we are focusing on:

  • Linear: $$f(x) = x$$ (A straight line passing through the origin)
  • Quadratic: $$f(x) = x^2$$ (A U-shaped parabola)
  • Cubic: $$f(x) = x^3$$ (An S-shaped curve)
  • Square Root: $$f(x) = \sqrt{x}$$ (Starts at 0 and curves to the right)

Understanding Transformations

Once you know the parent, you can see how the "child" function has moved. We specifically looked at Vertical Translations today. By adding or subtracting a constant to the function rule, we slide the graph up or down without changing its shape.

For example, let's look at the quadratic family:

  • The parent is $f(x) = x^2$, which has a vertex at $(0,0)$.
  • If we have $g(x) = x^2 + 5$, the graph moves up 5 units.
  • If we have $g(x) = x^2 - 3$, the graph moves down 3 units.

We also touched on Reflections. For instance, $g(x) = (-x)^2$ represents a reflection across the y-axis (though for a parabola centered at the y-axis, it looks the same!).

Class Resources

To help you practice identifying these families and describing their movements, please utilize the following materials used in class today:

  • Class Notes: Review the slides on identifying objectives and vocabulary.
  • Reading Strategies Worksheet: Use the models provided to visualize the differences between families.
  • In-Class Assignment: Complete the 'Ready to Go On? Skills Intervention' page.
  • Homework: Finish the Practice Worksheet, paying close attention to describing the translations in words.

Discussion Question of the Day

Let's get creative with our understanding of function families!

The Task: Using an online graphing calculator (like Desmos or GeoGebra) or a drawing tool, create a "Function Tree." Start with the four parent functions at the top. Then, create at least four children (transformations) under each parent. For example, under $x^2$, you might have $x^2+2$, $x^2-5$, etc.

Place the link to your online creation in the comments of this post!