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.

___
linear
___
exponential
___
quadratic
___
polynomial
___
rational
___
absolute value
___
logarithmic
___
trigonometric
___
radical

$y = | x |$
$y = \sin (x)$ or $y = \cos (x)$
$y = x$
$y = \log_{b} (x)$
$y = x^{2}$
$y = \frac{1}{x}$
$y = b^{x}$
$y = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_{0}$
$y = \sqrt[n]{x}$

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.

___
linear
___
exponential
___
quadratic
___
polynomial
___
rational
___
absolute value
___
logarithmic
___
trigonometric
___
radical

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	$(x, y)$	$(x, y) \to (x, y + 2)$
function notation	$f (x) = x^{2}$	$f_{1} (x) = x^{2} + 2$
relationship to parent function	$f (x)$	$f_{1} (x) = f (x) + 2$
selected points that fit this image	$(- 2, 4)$	$(- 2, 6)$	$(- 2, 8)$	$(- 1, 4)$
	$(- 1, 1)$	$(- 1, 3)$	$(- 1, 2)$	$(- \frac{1}{2}, 1)$
	$(0, 0)$	$(0, 2)$	$(0, 0)$	$(0, 0)$
	$(1, 1)$	$(1, 3)$	$(1, 2)$	$(\frac{1}{2}, 1)$
	$(2, 4)$	$(2, 6)$	$(2, 8)$	$(1, 4)$

4.

Complete the Table:

	pre-image (parent graph)	image 1	image 2	image 3
geometric notation	$(x, y)$
function notation	$f (x) = 2^{x}$
relationship to parent function	$f (x)$	$f_{1} (x) = 4 f (x)$
selected points that fit this image	$(- 2, \frac{1}{4})$	$(- 2, 1)$	$(- 2, - \frac{1}{4})$	$(- 3, \frac{1}{4})$
	$(- 1, \frac{1}{2})$	$(- 1, 2)$	$(- 1, - \frac{1}{2})$	$(- 2, \frac{1}{2})$
	$(0, 1)$	$(0, 4)$	$(0, - 1)$	$(- 1, 1)$
	$(1, 2)$	$(1, 8)$	$(1, - 2)$	$(0, 2)$
	$(2, 4)$	$(2, 16)$	$(2, - 4)$	$(1, 4)$

Pause and Reflect

5.

Complete the table:

	pre-image (parent graph)	image 1	image 2	image 3
geometric notation	$(x, y)$
function notation	$f (x) = \| x \|$
relationship to parent function	$f (x)$
selected points that fit this image	$(- 2, 2)$	$(- 2, - 4)$	$(- 1, 2)$	$(- 5, 2)$
	$(- 1, 1)$	$(- 1, - 2)$	$(- \frac{1}{2}, 1)$	$(- 4, 1)$
	$(0, 0)$	$(0, 0)$	$(0, 0)$	$(- 3, 0)$
	$(1, 1)$	$(1, - 2)$	$(\frac{1}{2}, 1)$	$(- 2, 1)$
	$(2, 2)$	$(2, - 4)$	$(1, 2)$	$(- 1, 2)$

6.

Complete the table:

	pre-image (parent graph)	image 1	image 2	image 3
geometric notation	$(x, y)$
function notation	$f (x) = \sin (x)$
relationship to parent function	$f (x)$
selected points that fit this image	$(0, 0)$	$(0, 2)$	$(0, 0)$	$(0, 0)$
	$(\frac{π}{2}, 1)$	$(\frac{π}{2}, 3)$	$(\frac{π}{4}, 1)$	$(\frac{π}{2}, - 2)$
	$(π, 0)$	$(π, 2)$	$(\frac{π}{2}, 0)$	$(π, 0)$
	$(\frac{3 π}{2}, - 1)$	$(\frac{3 π}{2}, 1)$	$(\frac{3 π}{4}, - 1)$	$(\frac{3 π}{2}, 2)$
	$(2 π, 0)$	$(2 π, 2)$	$(π, 0)$	$(2 π, 0)$

7.

Complete the table:

	pre-image (parent graph)	image 1	image 2	image 3
geometric notation	$(x, y)$
function notation	$f (x) = \sqrt{x}$
relationship to parent function	$f (x)$
selected points that fit this image	$(0, 0)$	$(0, 0)$	$(0, 0)$	$(3, 0)$
	$(1, 1)$	$(1, \frac{1}{2})$	$(2, 1)$	$(4, 1)$
	$(4, 2)$	$(4, 1)$	$(8, 2)$	$(7, 2)$
	$(9, 3)$	$(9, \frac{3}{2})$	$(18, 3)$	$(12, 3)$
	$(16, 4)$	$(16, 2)$	$(32, 4)$	$(19, 4)$

Ready for More?

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

You can start with this transformation: $(x, y) \to (2 x + 3, 2 y - 1)$

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 .

$f (x) + k$ behaves like .
$k \cdot f (x)$ behaves like .

An operation performed inside of the function .

$f (x + k)$ behaves like .
$f (k x)$ behaves like .

Vocabulary

parent function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we explored the transformations of functions by comparing tables of data of parent functions and transformations of those parent functions. We found that all functions are transformed in the same ways, and we looked for explanations for why horizontal transformations—such as shifting left or right, or stretching or shrinking horizontally—behave the way they do.

Retrieval

1.

Graph the following linear functions on the grid. The function $f (x) = x$ has been graphed for you. For each new function, explain what the number $2$ does to the graph of $f (x) = x .$ Pay attention to the $y$ -intercept, the $x$ -intercept, and the slope. Identify what changes in the graph and what stays the same.

$f_{1} (x) = x + 2$
$f_{2} (x) = x - 2$
$f_{3} (x) = 2 x$

2.

Calculate $f (x)$ .

a.

$f (125) = \log_{5} x$

b.

$f (\frac{1}{125}) = \log_{5} x$

c.

$f (1) = \log_{5} x$