Warm up Determine whether the graph of each equation is symmetric wrt the x-axis, the y-axis, the line y = x the line y = -x or none. 1. 2. x-axis, y-axis, origin none of these

Lesson 3-2 Families of Graphs Objective: Identify transformations of simple graphs and sketch graphs of related functions.

Family of graphs A family of graphs is a group of graphs that displays 1 or more similar characteristics. Parent graph – the anchor graph from which the other graphs in the family are derived.

Identity Functions f(x) =x y always = whatever x is

Constant Function f(x) = c In this graph the domain is all real numbers but the range is c. c

f(x) = x2 The graph is a parabola. Polynomial Functions f(x) = x2 The graph is a parabola.

Square Root Function f(x)=

Absolute Value Function f(x) =|x|

Greatest Integer Function (Step) y=[[x]]

Rational Function y=x-1 or 1/x

Reflections A reflection is a “flip” of the parent graph. If y = f(x) is the parent graph: y = -f(x) is a reflection over the x-axis y =f(-x) is a reflection over the y-axis

Reflections Parent Graph y =x3 y=f(-x) y=-f(x)

Translations y=f(x)+c moves the parent graph up c units y=f(x) - c moves the parent graph down c units

Vertical Translations Translations f(x) +c 2 4 6 8 x y = f(x) -2 -4 -6 Vertical Translations f(x) +2= x2 + 2 f(x) = x2 f(x) - 5 = x2 - 5

Translations y=f(x+c) moves the parent graph to the left c units y=f(x – c) moves the parent graph to the right c units

Translations y =f(x+c) 2 4 6 8 x y = f(x) -2 -4 -6 Horizontal Translations 2 5 f(x - 5) f(x) f(x + 2) In other words, ‘+’ inside the brackets means move to the LEFT

Translations y=c •f(x); c>1 expands the parent graph vertically (narrows) y=c •f(x); 0<c<1 compresses the parent graph vertically (widens)

Translations y=cf(x) 3f(x) 2f(x) f(x) 30 y = f(x) y co-ordinates tripled Stretches in the y direction 3f(x) 20 y co-ordinates doubled 2f(x) f(x) 10 Points located on the x axis remain fixed. x -6 -4 -2 2 4 6 8 -10 The graph of cf(x) gives a stretch of f(x) by scale factor c in the y direction. -20 -30

Translations y = cf(x);0<c<1 2 4 6 8 x y = f(x) -2 -4 -6 10 20 30 -10 -20 -30 The graph of cf(x) gives a stretch of f(x) by scale factor c in the y direction. f(x) ½f(x) 1/3f(x) y co-ordinates halved y co-ordinates scaled by 1/3

Translations y=f(cx); c>1 compresses the parent graph horizontally (narrows) y=f(cx); 0<c<1 expands the parent graph horizontally (widens)

Translations y=f(cx) f(3x) f(2x) f(x) 4 6 8 x y = f(x) -2 -4 -6 f(3x) f(2x) f(x) Stretches in x 1/3 the x co-ordinate ½ the x co-ordinate The graph of f(cx) gives a stretch of f(x) by scale factor 1/c in the x direction.

Translations y=f(cx) f(1/3x) f(1/2x) f(x) 4 6 8 x y = f(x) -2 -4 -6 Stretches in x f(1/3x) f(1/2x) f(x) The graph of f(cx) gives a stretch of f(x) by scale factor 1/c in the x direction. All x co-ordinates x 2 All x co-ordinates x 3

Even Functions Consider all functions with a domain and range in the element of reals. Some of these have an interesting property. Namely, they make no distinction between negative and positive numbers. For example, consider f(x) = x2. How does this function treat 3 vs. -3? We can prove it makes no distinction between positive and negative numbers. F(x) = (x)2 = x2 While F(-x) = (-x)2 = x2 Even Functions also have symmetry about the y-axis. To say that f(−x) = f(x) for all x in domain of f, is equivalent to saying that a point, (x,y), is on the graph of f if and only if (−x,y) is also on the graph, which is also equivalent to saying that the graph is symmetric about the y axis.

ODD Functions While even functions, by definition, map every x and −x to the same number, odd functions are defined to be those functions that map −x to the opposite of where x gets mapped to. That is, f(−x) = −f(x) for all x in the set of real numbers. Now consider how f treats positive and negative numbers. For example, how does f treat 5 as compared with −5? f(5) = (5)3 = 125 while f(−5) = (−5)3 = −125 Yet another way to put it is that one can always ’factor’ out the negative thru the function as in the above example, f(−5) = −f(5) Odd functions are symmetric in respect to the origin.

Algebraic Test - Evens We can determine if the graph is even without actually graphing the equation by Substituting (-x) in to the original equation. If we can manipulate it back to the Original expression, it is an even function. Example: |y| = 2 - |2x| |y| = 2 - |2(-x)| (substitute (-x) in for x.) |y| =2- |2x| since |-2x| = |2x| - This is an Even Function Example: xy = -2 (-x)y = -2 (substitute (-x) in for x.) -xy = -2 xy = 2 – THIS IS NOT AN EVEN FUNCTION.

Algebraic Test - Odds We can determine if the graph is even without actually graphing the equation by Substituting (-x) in to the original equation. If we can manipulate it back to the Original expression, it is an even function. Example: |y| = 2 - |2x| |-y| = 2 - |2x| (substitute (-y) in for y.) |y| =2- |2x| since |-y| = |y| This is also an odd function Example: xy = -2 x(-y) = -2 (substitute (-y) in for y.) -xy = -2 xy = 2 – THIS IS NOT AN ODD FUNCTION.