Lesson 1 Function Family Reunion Solidify Understanding

Learning Focus

Explain function transformations across all function types.

How can I observe the transformations of functions when comparing tables of data?

Open Up the Math: Launch, Explore, Discuss

During the past few years of math classes, you have studied a variety of functions: linear, exponential, quadratic, polynomial, rational, radical, absolute value, logarithmic, and trigonometric. Like a family, each of these types of functions have similar characteristics that differ from other types of functions, making them uniquely qualified to model specific types of real-world situations. Because of this, sometimes we refer to each type of function as a “family of functions.”

Your work will begin with a card sorting activity. Sort the cards into sets, such that each set includes: 1 description card, 1 table card, 1 graph card, and 1 equation card that all describe the same function type.

1.

Once you have completed the card sorting activity, summarize your group work by matching each function family with the algebraic notation that best defines it.

  1. ___

    linear

  2. ___

    exponential

  3. ___

    quadratic

  4. ___

    polynomial

  5. ___

    rational

  6. ___

    absolute value

  7. ___

    logarithmic

  8. ___

    trigonometric

  9. ___

    radical

  1. or

Just like your family, each member of a function family resembles other members of the family, but each has unique differences, such as being “wider” or “skinnier,” “taller” or “shorter,” or other features that allow us to tell them apart. We might say that each family of functions has a particular “genetic code” that gives its graph its characteristic shape. We might refer to the simplest form of a particular family as “the parent function” and consider all transformations of this parent function to be members of the same family.

2.

Match each function family with the characteristic shape of the graph that fits it, based on the card sorting activity.

  1. ___

    linear

  2. ___

    exponential

  3. ___

    quadratic

  4. ___

    polynomial

  5. ___

    rational

  6. ___

    absolute value

  7. ___

    logarithmic

  8. ___

    trigonometric

  9. ___

    radical

  1. a curved line in the top right corner of a coordinate plane
    a curved line that is reflected over the point (0,0) on a coordinate plane
  2. a line with two minimas and even end behavior
  3. a parabola
  4. a curved line growing exponentially
  5. a curved line reflected over the point (0,0) with asymptotes of x=0 and y=0
  6. a straight line with a positive slope
  7. a line the shape of a letter V
  8. a curved line on the right side of the coordinate plane
  9. a sine function graphed on a coordinate plane

Function family characteristics are passed on to their “children” through a variety of transformations. While the members of each family shares common characteristics, transformations make each member of a family uniquely qualified to accomplish the mathematical behaviors they model.

For each of the following tables, a set of coordinate points that captures the characteristics of a parent graph is given. The additional columns give coordinate points for additional members of the family after a particular transformation has occurred. Write the rule for each of the different transformations of the parent graph. (Note: We can think of each new set of coordinate points (that is, the image points) as a geometric transformation of the original set of coordinate points (that is, the pre-image points) and use the notation associated with geometric transformations to describe transformation. Or, we can write the rule using algebraic function notation. Use both types of notation to represent each transformation.)

3.

Fill in the table:

pre-image

(parent graph)

image 1

image 2

image 3

geometric notation

function notation

relationship to parent function

selected points that fit this image

4.

Complete the Table:

pre-image

(parent graph)

image 1

image 2

image 3

geometric notation

function notation

relationship to parent function

selected points that fit this image

Pause and Reflect

5.

Complete the table:

pre-image

(parent graph)

image 1

image 2

image 3

geometric notation

function notation

relationship to parent function

selected points that fit this image

6.

Complete the table:

pre-image

(parent graph)

image 1

image 2

image 3

geometric notation

function notation

relationship to parent function

selected points that fit this image

7.

Complete the table:

pre-image

(parent graph)

image 1

image 2

image 3

geometric notation

function notation

relationship to parent function

selected points that fit this image

Ready for More?

Experiment with more complicated combinations of transformations for different parent functions.

You can start with this transformation:

  • Select a parent function to transform and write it here:

  • Create a table for the parent function and a table for the transformed function.

  • Write a function rule whose graph will pass through all of the points of the table defining the transformed function.

Takeaways

All function families .

An operation performed outside of the function .