Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The...
Transcript of Families of Functions Name...Families of Functions & Parent Functions The Conceptualizer! The...
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
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
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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: − , )( ∞ ∞
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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.
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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.
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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
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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
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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.
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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.
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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:
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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:
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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
1
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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
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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.
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