MLF - Parent Functions (Lesson)

Parent Functions

So far in this module, we have mostly focused on linear functions. There are lots of different types of functions as well! In this lesson, we'll explore other function families. A parent function is the most basic function of the family. For example, the parent function of linear functions is y = x. Watch this video to explore a few other types of functions. 

Here is a chart that summarizes the different types of functions learned in the video.

Function Family	Parent Function	Graph of Parent Function
Linear Functions	$y=x$
Quadratic Functions	$y=x^2$
Absolute Value Functions	$y=\left|x\right|$
Square Root Functions	$y=\sqrt{x}$
Cube Root Functions	$y=\sqrt[3]{x}$
Exponential Functions	$y=2^x$

 

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Characteristics of Parent Functions

Each of these types of functions has different characteristics. The graphs of the parent functions differ greatly. These functions also have different values when it comes to their domain and range, x- and y- intercepts, end behavior, and intervals of increase and decrease. Let's take a closer look. 

Domain and Range

We've learned about domain and range in a previous lesson. Using the chart of the parent graphs above, can you determine the domain and range of each function? Try to figure it out and then come back here for the answers. Are there any similarities between functions?

The Linear Parent Function and the Cube Root Parent Function both have a domain and a range of all real numbers. The Quadratic Parent Function and the Absolute Value Parent Function both have a domain of all real numbers but a range of numbers greater than or equal to zero [0, ∞). The Square Root Parent Function only lives in quadrant I, which means it has a domain of [0,∞) and a range of [0, ∞). And lastly, the Exponential Parent Function has a domain of all real numbers and a range of (0, ∞). 

Intercepts

An intercept is the point at which the graph crosses either the x- or y-axis. There's something pretty interesting about our parent functions in the chart above. They all have similar intercepts except for one of them. Can you recognize the parent graph that differs from the rest in terms of intercepts? The Exponential Parent Function is the only function that has different intercepts. The y-intercept for this function is at (0, 1) and it does not have an x-intercept. For the rest of the parent functions, they all share the same value for their intercepts. They all have an x- and a y-intercept at (0,0) or the origin. 

End Behavior

The end behavior is the appearance of a graph as it is followed in the left direction and the right direction. To determine the end behavior, you would look at each end of the graph and determine where it is headed. For example, take a look at the left end of the quadratic parent function graph. Where is the graph headed at that point? If you said, to infinity, then you are right! The left end of the quadratic parent function graph is increasing to infinity and it's the same for the right end of the graph as well. Both ends are increasing to infinity. Can you determine the end behavior of the rest of the graphs?

Intervals of Increase and Decrease

On a graph, there can be different spots where the graph increases and where it decreases. These "spots" can be described using intervals. We can see where a graph is increasing and decreasing by looking at the y-values, but the interval is always described using the x-values. Take a look at this image:

This is the parent graph for quadratic functions. If you look at the graph from left to right, it decreases and then increases. It changes its direction at (0, 0). This graph has an interval of decrease from negative infinity, which is the left side of the graph 0 and an interval of increase from 0 to positive infinity, or the right side of the graph. We could write this as follows:

The interval of decrease is (-∞, 0) and the interval of increase is (0, ∞). Can you determine the intervals of increase and decrease for the rest of the parents graphs?

 

IMAGES CREATED BY GAVS