1.1: basic functions and...
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Pre-calculus 12 Chapter 1: Function Transformations
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1.1: Basic Functions and Translations
Here are the Basic Functions (and their coordinates!) you need to get familiar with.
1. Quadratic functions (a.k.a. parabolas)
2y x
Ex. 2( 2) 1y x
2. Radical functions (a.k.a. square root function)
y x
Ex. 3 4y x
3. Absolute-value functions
| |y x
Ex. | 1| 2y x
4. Reciprocal functions
1
yx
Ex. 1
12
yx
Note: You are expected to remember the shape of the above (left) functions!
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Regardless of the type of function ( )y f x , the transformed function
y f (xh) k tells us:
Ex. 1: Find the equations for the base functions and their transformed graphs.
a)
Base function: Transformed function:
b)
Base function: Transformed function:
c)
Base function: Transformed function:
“h” = ________________________________
“k”= ________________________________
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Ex. 2: For the function
y4 f (x2) state the value of h and k that represent the horizontal and vertical
translations applied to
y f (x)
Ex. 3: Determine the new function when ( 6) 1y f x is translated 4 units to the left and 2 units
downward.
Ex. 4: Transform the following graph. Describe the transformations in words.
Given: ( )y f x
Graph: ( 2) 3y f x
Transformation: __________________________________________________________
**To translate, choose key points on the graph and then translate each one to graph its corresponding image
point on the transformed graph.
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1.2 (part 1): Vertical and Horizontal Reflections
A reflection can be identified with a “negative sign.” A reflection is a mirror image of a given function.
Using a graphing calculator, let’s explore the effect of having a “negative sign” at different locations of a
function.
Ex. 1: Graph 2y x and 2y x on the same grid.
Ex. 2: Graph | |y x and | |y x on the same grid.
Ex. 3: Graph y x and y x on the same grid.
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Observations:
y f (x) ______________________________________________________ Mapping: ( , )
y f (x) ______________________________________________________ Mapping: ( , )
Ex. 4: Without using a graphing calculator,
a) graph ( ) 3 2f x x with a solid line.
____ b) graph ( )y f x with a dotted line.
. . . c) graph ( )y f x with a broken line.
_ _ _
Ex. 5: Given 3 24 2 1f x x x , write a new function after applying the following reflection:
a) over the x-axis b) over the y-axis
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Ex. 6: Given
f (x), graph the indicated relation. State the domain and
range for each of them. Determine if it is a function.
a) Graph y f x
Domain: Range: Function?
b) Graph y f x
Domain: Range: Function?
Homework: p. 28 – 29: # 1, 3, 4, 5c , 5d, 7b, 7d
* use graph paper when drawing graphs (available from Ms. Dobson)
* “mapping notation”: (x,y) ( , )
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1.2 (part 2): Expansions and Compressions
Ex. 1: Consider the following function:
f x x
Graph the indicated function using the table of values on the graphing calculator:
a) 2y f x
b) 1
2y f x
c) 2y f x
d) 1
2y f x
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Observations:
y af (x) ______________________________________________________ Mapping: ( , )
y f (bx) ______________________________________________________ Mapping: ( , )
Ex. 2: Using )(xfy as a base function, describe the transformation when x is replaced by 2x and y is
replaced by y3
1 in words and in mapping notation.
Ex. 3: Given the graph of
f (x), perform each of the following transformations:
a) a vertical expansion by a factor of 2 b) a horizontal compression by a factor of
1
2
Homework: p. 28 – 31: # 2, 5a, 5b, 6, 7a, 7c, 8, 9, 14
* use graph paper when drawing graphs (available from Ms. Dobson)
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1.3 (part 1): Combining Transformations
Does order matter? Let’s explore.
Ex. 1: Graph y x
Vertical Expansion first vs. Vertical Translation first: (Reverse Order)
1) VE by factor of 2 1) VT by 1 unit down
2) VT by 1 unit down 2) VE by a factor of 2
Did the order effect the outcome?
Horizontal Expansion first vs. Horizontal Translation first: (Reverse Order)
1) HE by factor of 2 1) HT by 1 unit right
2) HT by 1 unit right 2) HE by a factor of 2
Did the order effect the outcome?
x y
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“Build-it-up” Method: Replace x with “new x” – Put in “new y” setting.
Ex. 2: Given the description, write the following transformations in function notation.
a) VE of 2
VT down 3
HT right 1
HC of 1/3
b) HE of 2
VC of 1/6
VT up 1
HT left 6
Combinations of Transformations:
We will perform transformations in the order S (Stretches), R (Reflections), T (Transformations)
First, re-write the function as
y af (b(xh)) k to be able to read all of the transformations directly.
Notice: there is no coefficient on x it must be factored out!
We will perform the transformations in order SRT for both vertical and horizontal:
1) A Vertical Stretch by a factor of a; a Horizontal Stretch by a factor of
1
b
2) A Vertical Reflection if a < 0 in the x-axis; a Horizontal Reflection if b < 0 in the y-axis
3) A Vertical Translation by a factor of k; a Horizontal Translation by a factor of –h
Ex. 3: Describe the order of transformations that occur for the following functions.
a) 3 2 4y f x b) 2 1y f x
c) 2 1y f x d) 1
2 1 53
y f x
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Ex. 4: Given ( )y f x :
a) Describe the transformation 2 1 2 3y f x
b) Graph the indicated functions on the grid provided.
Ex. 5: Write the equation for both the base function and the
transformed function.
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1.4: Inverse Functions The inverse of a relation is found by interchanging the x and y coordinates of the ordered pairs of that relation. Mapping: (x,y) ( , ) Graphically, this is the same as a reflection across the line y = x
The notation for an inverse is )(1 xf
Ex. 1: Given the function 32)( xxf :
a) Determine )(1 xf
b) Graph )(xf and )(1 xf on the same grid
c) Invariant point(s):
Ex. 2: Consider the function 4)( 2 xxf
a) Graph f(x) on the grid provided and state the
domain and range.
b) Find the inverse of the function 4)( 2 xxf .
c) Graph the inverse and state the domain and range.
d) Is the inverse of f a function? If not, how could the domain or range of f(x) be restricted so
that the inverse of f is a function?
Recall: an invariant point is any point that is unchanged
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Ex. 3: For each of the following functions
Find the inverse function using the notation )(1 xf , where appropriate
State the domain and range of the given function and its inverse.
a) 2)( xxf b) 2)3()( xxf c) 7
( )3
xf x
x
Ex. 4: a) Accurately graph the inverses of the following functions on the same grid.
b) Is the inverse a function? Justify your answer.
Homework: p. 51 – 53: # 2, 4ac, 5ace, 7a, 9ce, 11, 13 (use graphing calculator), 15
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-10 -5 5 10
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-10 -5 5 10