Transformations and Parent Graphs

Transformations Change a graph in some way Move it Reflect it Stretch it or compress it The shape of the graph does not change

The following table describes changes to the graph of f(x) How it transforms (x,y) f(x) + b f(x) – b f(x + b) f(x – b) -f(x) f(-x) bf(x) f(bx) Moves graph up (x, y+b) Moves graph down (x, y-b) Moves graph left (x-b, y) Moves graph right (x+b, y) Reflects over x-axis (x, -y) Reflects over y-axis (-x, y) Vertical stretch of b (x, by) Horizontal compression of 1/b (1/b x, y)

Observations Notice that if b is with the x, it does the opposite of what you would expect Think of it as if you were solving what was in the parenthesis Usually these transformations will happen more than one at a time

Example x y -3 -2 2 1 4 Label each (x,y) point and make a table

Example cont. Now graph each of the following transformations a. f(x+3) Left 3 (x-3,y) x-3 y -6 -2 -5 2 -3 1

Example cont. Now graph each of the following transformations b. f(x)+3 up 3 (x,y+3) x y+3 -3 1 -2 5 3 4

Example cont. Now graph each of the following transformations (x,-3y) c. -3f(x) Reflect in x axis, vertical stretch by 3 x -3y -3 6 -2 -6 1 4

Example cont. Now graph each of the following transformations (- ½ x,y) d. f(-2x) Reflect in y axis, hor. compression by 1/2 - ½ x y 1.5 -2 1 2 - ½

Example cont. Now graph each of the following transformations (- x,y+3) e. f(-x)+3 Reflect in y axis, up 3 - x y+3 3 1 2 5 - 1 -4

Example cont. Now graph each of the following transformations (1/3x-2,2y-4) f. 2f(3x+6)-4 vertical stretch 2, horizontal compression 1/3, left 2, down 4 1/3x-2 2y-4 -3 -8 -2.7 -2 -4 - 1.7 - .7 Careful – this one is tricky!!

Example cont. Now graph each of the following transformations (-x -2,-y-1) g. -f(-x-2)-1 Reflect in x axis, reflect in y axis, left 2, down 1 -x-2 -y-1 1 -3 -2 -1 - 6 One more tricky one!!

Parent graphs There are many common functions that we call parent graphs that we need to know You will need to memorize all their shapes and which points make sense to plug in Today we will graph all of them Tomorrow we will apply the tranformations we know to these parent graphs

Quadratic x y -2 -1 1 2 4 1 1 4

Cubic x y -2 -1 1 2 -8 -1 1 8

Square Root x y 1 4 1 2

Absolute Value x y -2 -1 1 2 2 1 1 2

Rational x y -2 -1 1 2 -1/2 -1 error 1 1/2 Has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0 x y -2 -1 1 2 -1/2 -1 error 1 1/2

Has a horizontal asymptote at y = 0 Exponential Growth Has a horizontal asymptote at y = 0 Can be any number greater than 1 x y 1 1 2 Will match the base in the equation

Has a horizontal asymptote at y = 0 Exponential Decay Has a horizontal asymptote at y = 0 x y -1 Can be any number less than 1 1 2 Will match the base in the equation

Has a vertical asymptote at x = 0 Log Has a vertical asymptote at x = 0 x y 1 6 Can be any number except 0 1 Will match the base in the equation

Has a vertical asymptote at x = 0 Log Has a vertical asymptote at x = 0 x y 1 6 Can be any number except 0 1 Will match the base in the equation

Has a vertical asymptote at x = 0 Natural Log Has a vertical asymptote at x = 0 x y 1 e 1 Equals approximately 2.7

sin x y 1 -1

cos x y 1 -1 1