Welcome back to math class! Today, we're diving into the fascinating world of parent functions. Understanding these fundamental functions is crucial because every other function you'll encounter can be seen as a transformation of one of these parent functions. Just like in geometry where all shapes can be traced back to a few basic shapes, all functions are based on parent functions.

What is a Parent Function?

A parent function is the simplest function within a family of functions that share common characteristics. Think of it as the "original" function before any transformations are applied. Functions within the same family are transformations of their parent function.

Key Parent Functions

Let's explore some of the most common parent functions:

  • Constant Function: $f(x) = c$, where c is a constant. This is a horizontal line.
  • Linear Function: $f(x) = x$. This is a straight line passing through the origin with a slope of 1.
  • Quadratic Function: $f(x) = x^2$. This is a parabola with its vertex at the origin.
  • Cubic Function: $f(x) = x^3$. This function has a characteristic "S" shape.
  • Square Root Function: $f(x) = \sqrt{x}$. This function starts at the origin and increases gradually. The domain is $x \geq 0$.

Identifying Transformations

Once you know your parent functions, you can start identifying transformations. Let's look at some examples:

  • Example 1: $g(x) = x - 3$. This is a transformation of the linear parent function $f(x) = x$. Specifically, it's a vertical translation 3 units down. Notice that 'x' still has a power of 1, which indicates it is still a linear function.
  • Example 2: $g(x) = x^2 + 5$. This is a transformation of the quadratic parent function $f(x) = x^2$. It's a vertical translation 5 units up. The power of 2 on the 'x' indicates that this function is quadratic.
  • Example 3: $g(x) = x^3 + 2$. This is a transformation of the cubic parent function $f(x) = x^3$. It represents a vertical translation 2 units up. The power of 3 on the 'x' indicates that this function is cubic.
  • Example 4: $g(x) = (-x)^2$. This is a transformation of the quadratic parent function $f(x) = x^2$. It's a reflection across the y-axis.

Why are Parent Functions Important?

  • Understanding Transformations: Parent functions provide a base for understanding how transformations (shifts, stretches, reflections) affect the graph and equation of a function.
  • Graphing: Knowing the parent function helps you quickly sketch a graph by applying the appropriate transformations.
  • Problem Solving: Recognizing parent functions makes it easier to solve problems involving functions and their properties.

Resources

Here are the resources we used in class today:

Homework

For tonight, please complete problems pg. 18-21 #11-13, 17-22, 47-49. Don't hesitate to ask questions in class tomorrow!

Discussion Question

If the above are parent functions and every other function is just a transformation of them then think back to geometry and tell me the parent shapes?