Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The...

14
Families of Functions Name: Notes Date: Sarah is making an assignment for her math class. She wants to create one question that has as many different answers as possible related to a single family of functions with transformations. She has come up with three questions and wants to know all the possible transformations that could create the transformed function from the parent function. Sarah must decide which is the hardest one to put on the assignment. Write the definition of the term and include an image or example that represents it. Term Definition Example Parent Function Quadratic Cubic Power Function Absolute Value Function Linear Function Exponential Function

Transcript of Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The...

Page 1: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

 

 

Families of Functions  Name:   

Notes  Date:   

 

 

Sarah is making an assignment for her math class. She wants to                       

create one question that has as many different answers as possible                     

related to a single family of functions with transformations. She                   

has come up with three questions and wants to know all the                       

possible transformations that could create the transformed             

function from the parent function. Sarah must decide which is the                     

hardest one to put on the assignment. 

 

 

Write the definition of the term and include an image or example that represents it. 

Term  Definition  Example 

Parent 

Function  

 

Quadratic   

 

Cubic   

 

Power 

Function  

 

Absolute 

Value Function  

 

Linear 

Function  

 

Exponential 

Function  

 

Page 2: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

Term  Definition  Example 

Logarithmic 

Function  

 

Vertical Shift   

Horizontal 

Shift  

Narrowing/ 

Widening  

Reflections   

 

 

 

What are Functions?  The Conceptualizer! 

A function is a relation between a set of                 

inputs and a set of permissible outputs with               

the property that each input is related to               

exactly one output. 

 

Typically, we use function notation      (x)f =      

in talking about the function, and            y =notation when talking about graphing the           

function. 

 

When graphing, you might have .          (x)y = f  

The output value (from the “function box”)             

is the dependent variable, y. Graph the             

points , the same as .x, (x))( f x, )( y  

 

 

 

© Clark Creative Education 

Page 3: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

Families of Functions & Parent Functions  The Conceptualizer! 

The different operations (square roots,         

exponents, etc) that are possible on a             

variable are different functions.  

 

They are different families of functions and             

each have different graphs and features. 

 

Each family will have consistent Domain           

and Range features and may have unique             

characteristics.  

 

We refer to the most basic as parent               

functions.   

All Domain and Range information and facts             

will be specific to the parent, but by               

understanding transformations you will see         

relationships here too.  

 

 

 

Constant Function  The Conceptualizer! 

The constant function is a horizontal line. 

 

(x)f = 0  

 

The most basic is , on the x-axis.y = 0  

 

Domain: − , )( ∞ ∞  

Range: [0] 

 

 

Linear Functions  The Conceptualizer! 

The linear function is a straight line. 

 

(x)f = x  

 

Domain: − , )( ∞ ∞  

Range: − , )( ∞ ∞  

 

 

 

© Clark Creative Education 

Page 4: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

Quadratic Function  The Conceptualizer! 

The quadratic function is a second-degree           

polynomial that makes a parabola. 

 

(x)f = x2  

 

Domain: − , )( ∞ ∞  

Range: 0, )( ∞  

 

The vertex is the point where the function               

reaches its minimum or maximum value. 

 

 

Cubic  The Conceptualizer! 

The cubic function makes a stretched “S”             

shape. 

 

(x)f = x3  

 

Domain: − , )( ∞ ∞  

Range: − , )( ∞ ∞  

 

 

 

Power Up 

The functions discussed so far are polynomials -- x raised to some                       

power. There is no value of x that is not valid.   

Therefore, the domain of these functions is “all real numbers”, but                     

the ranges are different. So all x values have corresponding y values.  

 

Absolute Value Functions  The Conceptualizer! 

The absolute value function makes a sharp             

“V” shape made out of two linear functions.  

(x) x|f = |  

 

Domain: − , )( ∞ ∞  

Range: 0, )( ∞  

 

The vertex is where the two lines intersect. 

 

 

© Clark Creative Education 

Page 5: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Rational Functions  The Conceptualizer! 

Rational functions create hyperbolas and         

have horizontal and vertical asymptotes.  

(x)f = x1

 

Domain: − , ) (0, )( ∞ 0 ∪

∞  

Range: − , ) (0, )( ∞ 0 ∪

∞  

The asymptotes of the parent function are             

at and .x = 0 y = 0   

 

 

Square-Root Function  The Conceptualizer! 

Square root functions are curves that are             

limited in that negative values are           

undefined in the square root.   

(x)f = √x   

Domain: 0, )( ∞  

Range: 0, )( ∞   

The square-root function is also      y = √x      

. It is also a power function. They = x 21

               

graph “starts” at .x = 0  

 

 

Exponential Functions  The Conceptualizer! 

Exponential function can change quickly.   

 

(x)f = 2x  

 

Domain: − , )( ∞ ∞  

Range: 0, )( ∞  

 

The graph changes curvature with different           

bases (instead of 2), but will still have               

on its graph unless there are other0, )( 1              

transformations. 

 

For base values larger than 1 it shows               

exponential growth. For values between 0           

to 1, it will show exponential decay. 

 

 

 

© Clark Creative Education 

Page 6: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Logarithmic Functions  The Conceptualizer! 

Logarithmic functions are inverses of         

exponential functions.  

 

(x) og(x)f = l  

 

Domain: 0, )( ∞  

Range: − , )( ∞ ∞  

 

The graph changes curvature with different           

bases (instead of base-10), but will still             

have on its graph.1, )( 0  

 

The graph is the reflection across of            y = x  

the exponential function with the same           

base. 

 

 

 

Write examples functions from each family in the correct box. 

Quadratic  Cubic  Absolute Value 

     

Rational  Square Root  Exponential 

     

 

 

 

 

 

 

© Clark Creative Education 

Page 7: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Graphing Functions  Notes 

Graph . What is its domain and range?y = x4    

 

 

Functions from Parent Functions  The Conceptualizer! 

As totally awesome as is, you’d        y = x2      

almost certainly want (or need) to graph             

different parabolas.  

Fortunately, you can express all other           

parabolas as changes to this basic parent             

function.  

How does the graph compare to the graph               

of the parent function?  

You express the other parabolas as           

transformations of the parent parabola.  

#mamaParabola 

 

 

© Clark Creative Education 

Page 8: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

What are the Transformations?  The Conceptualizer! 

When transforming a function, there are only             

a few possibilities: translations, reflections         

and dilations (and combinations of these).  

 

This is a general function of all             

transformations:  

[b(x )]a · f + c + d  

 

 

Applying Transformations  The Conceptualizer! 

Isn’t it a lot to keep track of all these                   

different kinds of functions and how they             

can be transformed? 

 

Not at all! 

 

It turns out that the transformations that             

turn into are exactly  y = x2   y = (x )− 3 2 + 5    

the transformations that turn into        x|y = |    

.x |y = | − 3 + 5   

 

As we tour these transformations, we’ll use             

a parabola as the sample for how             

transformations affect the graph -- but they             

are true for other function types.  

 

 

 

Vertical Translations  The Conceptualizer! 

Compared to , what is the graph of    y = x2            

?y = x2 + 1  

 

Following the order of operations, you           

would square the x value -- then 1 is added                   

“at the end”. 

 

The numerical effect is to raise the final y                 

value by 1. 

 

The visual effect is to move the graph up by                   

one unit. 

 

 

 

© Clark Creative Education 

Page 9: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Vertical Translations  Notes 

Graph x|y = | − 2  

 

 

 

 

Graphing  Notes 

Identify the function from the graph: 

 

 

 

 

Horizontal Translations  The Conceptualizer! 

Think about the vertex of the parabola. If               

the graph is shifted 3 units to the right, the                   

vertex is now at .3, )( 0    

The y value there is 0, BUT that’s the value                   

for when .y = x2 x = 0   

So for this function, you need to subtract 3                 

from the x value. That gets back to            x = 3    

, so that the function gives the same yx = 0                  

value. Thus the function is .y = (x )− 3 2 

 

The operation is applied only to the x,               

inside the function, before other operations           

are applied. 

 

 

© Clark Creative Education 

Page 10: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Graphing  Notes 

Graph y = (x )− 2 2 

 

 

 

 

Combining Translations  The Conceptualizer! 

You can have both horizontal and vertical             

translations. 

 

By the normal order of operations, the             

horizontal shift “happens” first, as x is             

affected, with an addition of 2. Then, at               

the end, the vertical shift happens, as the y                 

value is increased by 1. 

 

 

 

Graphing  Notes 

Identify the function: 

 

 

 

 

© Clark Creative Education 

Page 11: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Reflection  The Conceptualizer! 

Where is compared to ? Across    −y = 5     y = 5    

the x-axis. Negating a calculated y value             

reflects it across the x-axis. 

 

The graph of is a reflection across the      (x)− f          

x-axis.   

 

And the graph of is a reflection across        (− )f x        

the y-axis. 

 

 

 

Combining Reflections & Translations  The Conceptualizer! 

What is the function that, compared to             

, has been shifted left 2 units,y = x2              

reflected across the x-axis, and then shifted             

up 3 units?  

You can write it, given that description:  

−y = (x )+ 2 2 + 3   

Once you’ve written it out, reread the             

description to see if it matches. 

 

 

 

Graphing  Notes 

Identify the function from the graph: 

 

 

 

 

 

© Clark Creative Education 

Page 12: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Dilations  The Conceptualizer! 

Compared to , what is the graph of    x|y = |            

? The y values are doubled.|x|y = 2   

 

On , there is a point . On  x|y = |           1, )( 1    

, there is a point . The|x|y = 2           1, )( 2    

transformation from the parent function         

maps to , expanding or  1, )( 1     1, )( 2      

stretching vertically. Therefore, the y         

values are getting big twice as fast. And in                 

this case, the slope of the lines is 2, not 1. 

 

 

Dilations  The Conceptualizer! 

Similarly, something like appears to      |x|y = 31

   

have widened, although it is actually a             

vertical compression. 

 

The terminology is awkward; there is a             

vertical change in y values by a factor of .                  31

 

However, some people would say that there             

is a horizontal stretch by a factor of 3.   

 

Be careful in your language. 

 

 

 

Vertical Stretch or Horizontal Compress? 

Our brains don’t easily see this as a vertical                 

stretching. Instead, our psychology is such           

that it is being compressed horizontally. This             

really doesn’t make a mathematical         

difference.  

 

Here’s what is useful:  

“Narrows” (stretched vertically, compressed       

horizontally) The scalar value is greater than 1               

inside or outside the function. 

“Widens” (stretched horizontally, compressed       

vertically) The scalar value is greater than 0, but                 

less than 1 inside or outside the function. 

2x|y = | |x|y = 2   x|y = | 21 |x|y = 2

 

 

 

© Clark Creative Education 

Page 13: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Combining All the Transformations  The Conceptualizer! 

Suppose is translated right 1 unit,  y = x2            

compressed horizontally by a factor of 2,             

and translated down by 3 units. 

 

What is this function? 

 

(x )y = 2 − 1 2 − 3  

 

 

 

Graphing  Notes 

Graph xy = 21 2 − 2

 

 

 

 

Compare functions that could be parent functions to those that have been transformed. 

Parent Functions  Transformations 

   

 

 

 

© Clark Creative Education 

Page 14: Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The different operations (square roots, exponents, etc) that are possible on a variable are

  

 

Create a function with the shifts as described to . Sketch your functions. y = x3  

Upward Translation  Left Translation  Downward Translation  Right Translation 

       

 

 

The graph of goes through . What is k?(x)f = √x − k 7, 2)(  

 

 

 

 

 

 

 

 

Wendy graphs the growth of an aggressive bacteria as a function of time. Sketch what the graph                                 

may look like and characterize it. 

 

 

 

 

 

 

 

 

Miguel observes the height of a homerun in the air as a function of time. Sketch what the graph                                     

may look like and characterize it. 

 

 

 

 

 

© Clark Creative Education