1.  Pre-Calculus 30

2.  PC30.7 PC30.7  Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.  PC30.8 PC30.8  Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

3.  Transformations  Mapping  Translations  Image Point  Reflection  Invariant Point  Stretch  Inverse of a Function  Horizontal Line Test

5.  PC30.7 PC30.7  Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.

6.  First off, a transformation is when a functions equation is altered resulting in any combination of location, shape and/or orientation changes of the graph  Every point on the original graph corresponds to a point on the transformed graph  The relationship between the points is called mapping

7.  Mapping Notation is a way to show the relation between the original function and the transformed function. original (x,y) translation (x,y+3)  Mapping Notation: (x,y)  (x,y+3)

8.  Translation is a type of transformation  A translation can move a graph left, right, up and down.  In a translation the location of the graph changes but not the shape or orientation.

9.  Lets look at a quick example to see how a translation works and what it looks like in an equation  Graph: y=x 2, y-2=x 2, y=(x-5) 2

11.  Now before we graph the following 3 functions let’s predict what we think will happen?  Graph: y=x 2, y+1=x 2, y=(x+3) 2

13.  So with vertical and horizontal translations we shift the graph of a function vertically and/or horizontally by applying one or both of the changes to the equation  Vertical Shift: y-k=f(x)  Horizontal Shift: y=f(x-h)  Both: y-k=f(x-h)

17.  Ex. 1.1 (p.12) #1-14 #1-13 odds, 17-19

18.  PC30.7 PC30.7  Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.  PC30.8 PC30.8  Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

19.  A Reflections of a functions graph is the mirror image in a line called the Line of Reflection  Reflections do not change the shape of the graph but does change the orientation of the graph

20.  When output of a function is multiplied by -1 the result is y=-f(x)  Vertical Reflection (reflect in x-axis)  (x,y)  (x,-y)  Line of reflection=x-axis

21.  When input of a function is multiplied by -1 the result is y=f(-x)  Horizontal Reflection (reflect in y-axis)  (x,y)  (-x,y)  Line of reflection=y-axis

23.  A Stretch changes the shape of a graph but not its location  A vertical stretch can make the function shorter or taller bc the stretch multiplies or divides the y-values by a constant while the x is unchanged

25.  A Horizontal Stretch can make the function narrower or wider because the stretch multiplies or divides the x-values by a constant while the y-values are unchanged

27.  If the a or b values are negative there would also be a reflection.

33.  Ex. 1.2 (p.28) #1-12 #1-6, 7-9 odds in each, 10-12, 15, 16

34.  PC30.7 PC30.7  Extend understanding of transformations to include functions (given in equation or graph form) in general, including horizontal and vertical translations, and horizontal and vertical stretches.  PC30.8 PC30.8  Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

36.  Multiple transformations can be applied to a function using the General Transformation Model y-k=af(b(x-h)) or y=af(b(x-h)) +k  The same order of operations are used as when you are working with numbers (BEDMAS)  So multiplying and dividing (stretches, reflections) are done first then add and subtract (translations)

37.  Steps to graph combinations: 1. Horizontal stretch and reflect in the y-axis (if b<0) 2. Vertical stretch and reflect in the x-axis (if a<0) 3. Horizontal and/or vertical Translations (h and k)

38.  Lets look at the transformations in mapping notation for y=af(b(x-h)) +k

44.  Ex. 1.3 (p.38) #1-12 odds in each with multiple parts #3-16 odds in each with multiple parts

45.  PC30.8 PC30.8  Demonstrate understanding of functions, relations, inverses and their related equations resulting from reflections through the: x-axis, y-axis, line y=x

47.  The Inverse of a Function y=f(x) is denoted y=f -1 (x) if the inverse is a function.  The -1 is not an exponent because f represents a function, not a variable. (just like in sin -1 (x))

48.  The inverse of a function reverses the processes represented by that function.  For example, the process of squaring a number is reversed by taking the square root. Taking the reciprocal of a number is reversed by taking the reciprocal again.

49.  For example, for f(x)=2x+1 we are multiplying by 2 and adding 1.  What would the inverse be?

50.  To determine the inverse of a function, interchange the x and y coordinates Function  Inverse (x,y)  (y,x) y=f(x)  x=f(y) reflect in the line y=x

51.  When working with an equation of a function y=f(x), interchange the x for y.  Then solve for y to get the equation for the inverse, if the inverse is a function, then y=f -1 (x)

53.  Restricting the domain is necessary for any function that changes direction (increasing to decreasing, or vise versa) at some point in the domain of the function Unrestricted domainRestricted domain x≤0

58.  Ex. 1.4 (p.51) #1-16 odds in questions with multiple parts #4-20 odds in questions with multiple parts