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Functions

Functionsyouknow

Translations

Reflections

Inverses

Stretches

Combinations

Combining Functions

Review&Test

 

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 UNIT 2  : FUNCTIONS

 LESSON 9:  FUNCTIONS REVIEW & SUMMARY

 

1.  Functions: definition, notation, tests, domain and range - see lesson 1.

 

Def’n: A function is a relation such that for each value of x, there corresponds exactly one value of y.

 

Tests:

·        Vertical line test – if a vertical line cuts the graph more than once --  not a function

·        If no two ordered pairs have the same first element, then the relation is a function

·        Substitute in a value for x;  if you get more than one answer for y – not a function

Tips for finding domain:

·        Set of all first components of the ordered pairs.

·        Is there a largest value for x?  Is there a smallest value for x?

·        Are there any restrictions on x?

Tips for finding range: 

·        Set of all second components of the ordered pairs.

·        Is there a largest value for y?  Is there a smallest value for y?

 

      

 

 

 

 

 

                                                 

 

 

 

 

 

 

 

2.  Transformations:

 

 

                       

Text Box: In summary, to graph y = af [k(x – m)] + n from the graph of y = f(x), follow these ideas:

·	If a < 0, we have a reflection in the x-axis
·	If k < 0, we have a reflection in the y-axis
·	If –1 < a < 1, we have a vertical compression
·	If a > 1 or a < - 1, we have a vertical stretch
·	If –1 < k < 1, we have a horizontal stretch, factor 1/k
·	If k > 1 or k < - 1, we have a horizontal compression, factor 1/k
·	The value of m gives the horizontal translation (shift)
·	The value of n gives the vertical translation (shift)
 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example:  Describe  -2f [½ (x – 3)] – 1   relative a given function y = f(x)

·        Reflection in the x-axis

·        Vertical stretch factor 2

·        Horizontal stretch factor 2

·        Horizontal translation right 3

·        Vertical translation(shift) down 1

 

To graph, take points on original graph of y = f(x) and use the mapping form to determine image points  --  see Lesson 7.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.  Inverses:

 

 

           

 

 

 

 

 

 

 

           

 

 

 

 

In summary, if we restrict the domain of the given parabola to that half either to the right or left of the vertex, the inverse will be a function.

 
 

 

 

 

 

 

 


 

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