Math Lesson 15.7.1 - Recalling the Concept of Parent Function

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Welcome to our Math lesson on Recalling the Concept of Parent Function, this is the first lesson of our suite of math lessons covering the topic of Transforming Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Recalling the Concept of Parent Function

In tutorial 15.5, we briefly explained the concept of parent function as the simplest from a list of functions of the same category. For example, the parent function of all linear functions is y = x, the graph of which passes through the origin and divides the first and third quadrants into two halves. All the other linear graphs such as y = ax, y = x + b, y = ax + b are obtained by transforming the parent linear graph y = x.

Likewise, the quadratic function y = x² is the parent graph of all the other quadratic functions of the type y = ax², y = x² + bx, y = x² + c, y = ax² + c, y = ax² + bx + c, etc., because all of them are all obtained by various transformations of the quadratic parent function y = x².

Some other parent functions we know so far are the parent exponential function y = ex, the parent cubic function y = x³, the parent logarithmic function y = log x, the parent reciprocal function y = 1/x and so on.

Example 1

Identify the parent function of the following functions.

1. y = 3x² - 5x + 6
2. y = 2 - 5e^-3x
3. y = (5x - 4) / (2x + 1)
4. y = 1 - 4x

Solution 1

The task in this exercise is not to solve the equation or plot the graph, etc., but simply to determine which kind of functions they are so that we can identify the corresponding parent function, which - as stated above in theory - represents the simplest function of a given type. Hence, we make the following reasoning.

1. This is an example of a quadratic function, as the right side represents a second-degree polynomial. Therefore, the parent function of y = 3x² - 5x + 6 is y = x², as this is the simplest quadratic function.
2. This is an example of an exponential function, as the right side contains an exponential term. Therefore, the parent function of y = 2 - 5e^-3 is y = e^x, as this is the simplest exponential function.
3. This time we have to do some transformations in the original function to understand what kind of function it is. We have
   y = 5x - 4/2x - 1
   = 4x - 2/2x - 1 + x - 2/2x - 1
   = 2(2x - 1)/2x - 1 + x - 1/2 - 3/2/2(x - 1/2)
   = 2 + x -1/2/2(x - 1/2) - 3/2/2(x - 1/2)
   = 2 + 1/2 - 3/2/2(x - 1/2)
   = 5/2 - 3/2/x - 1
   Now, it became evident that this is a reciprocal function of the general form
   y = a/x - h + k
   as discussed in tutorial 15.4, where a = 3/2, h = 1 and k = 5/2. The parent function of all reciprocal functions is y = 1/x. therefore, the parent function of y = 5x - 4/2x + 1 is y = 1/x.
4. This is an example of a linear function as the independent variable x is at the first power. Hence, the parent function of y = 1 - 4x is y = x.

Having covered a refresher lesson on the concept of parent function, we will now explain how to transform a given parent function graph to obtain all the corresponding graphs of the same family. There are several methods we can use to transform a parent graph, to obtain the graph of a new function of the same family. We will these methods in the following lessons.

You have reached the end of Math lesson 15.7.1 Recalling the Concept of Parent Function. There are 3 lessons in this physics tutorial covering Transforming Graphs, you can access all the lessons from this tutorial below.

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