Unit 4: Transformations and piecewise functions Final Exam Review

Topics to cover Domain and Range Transformations of Functions Transformations of Points Toolkit Functions Piecewise Functions

Domain and range Domain and Range are important to be able to recognize in the graph of a function Domain X values LEFT to RIGHT Range Y values BOTTOM to TOP Use Interval Notation If there is an OPEN CIRCLE, use a PARENTHESIS If there is a CLOSED CIRCLE, use a BRACKET If there is an ARROW, use −∞ or ∞ and a PARENTHESIS

Domain and range Example: Domain: [-7, 5) Range: [-3, 1)

Domain and range Now you try: 1. 2.

Transformations of functions You can move a function anywhere on a graph using transformations You can go: UP AND DOWN LEFT AND RIGHT STRETCH AND SHRINK REFLECT OVER THE X AXIS AND Y AXIS

Transformations UP AND DOWN Use these rules when translating a function up and down Up: f(x) + c Example: y = f(x) + 4 will move a function Up 4 Down: f(x) – c Example: y = f(x) – 5 will move a function Down 5

Transformations left and right Use these rules when translating a function left and right Left: f(x + c) Example: y = f(x + 2) will move a function Left 4 Right: f(x – c) Example: y = f(x – 6) will move a function Right 6

Stretch and Shrink Use these rules when stretching and shrinking c ∙ f(x) when c > 1 Example: y = 3 f(x) will stretch a function by 3 Shrink: c ∙ f(x) when 0 < c < 1 Example: y = 𝟐 𝟑 f(x) will shrink a function by 𝟐 𝟑

Reflections over the x axis and y axis Use these rules when reflecting a functions over the x axis or the y axis X axis: -f(x) The negative is OUTSIDE of the function Y axis: f(-x) The negative is INSIDE of the parenthesis with the x

Given the function, write the transformations Example y = 5 f(x – 3) y = -f(x) + 2 y = 1 3 f(−x+7) - 8 Answers Stretch 5, Right 3 Reflect over the X axis, Up 2 Shrink 𝟏 𝟑 , Reflect over the y axis, Left 7, Down 8

Given the function, write the transformations Now you try: y = f(x – 4) – 3 y = 6 f(x + 2) y = − 1 3 f(x) + 6

Given the transformations, write the equation Example Left 1, Up 9 Shrink 3 4 , Reflect over the x axis Stretch 10, Right 4, Down 2 Answers y = f(x + 1) + 9 y = - 𝟑 𝟒 f(x) y = 10 f(x – 4) – 2

Given the transformations, write the equation Now you try: Right 7, Reflect over the y axis Stretch 4, Left 2, Down 5 Reflect over the x axis, Shrink 4 5 , Up 8

Transforming points When transforming points, you must figure out: If the transformation is affecting the X VALUE or the Y VALUE Then ADD, SUBTRACT, or MULTIPLY the point by the correct value UP AND DOWN will affect the y value LEFT AND RIGHT will affect the x value STRETCHING AND SHRINKING will affect the y value

Transforming points Example Move the point (5, -2) using the transformations in the function y = f(x – 4) Answer: (9, -2) *This transformation moves the function right 4, so the x value is moved right 4 spaces

Transforming points Now you try: Move the point (-4, 2) using the functions: y = f(x) – 5 y = f(x + 1) y = 3 f(x) y = f(x – 2) + 9

Toolkit functions Toolkit functions are the basic functions that can be transformed. They are: LINEAR FUNCTION y = x QUADRATIC FUNCTION y = x2 CUBIC FUNCTION y = x3 SQUARE ROOT FUNCTION y = 𝑥 CUBE ROOT FUNCTION y = 3 𝑥 RECIPROCAL FUNCTION y = 1 𝑥 ABSOLUTE VALUE FUNCTION y = 𝑥 EXPONENTIAL FUNCTION y = ex

Toolkit functions Example: Write the correct toolkit function and the transformations that happened y = 4 (x + 3)2 Toolkit Function: y = x2 Transformations: Stretch 4, Left 3

Toolkit functions Now you try: Write the correct toolkit function and the transformations that happened to it. y = - (x + 1)3 y = 1 𝑥+4 −3 y = 6 𝑥 −2 +5

Piecewise functions Piecewise Functions are functions that are in SEVERAL pieces Each function has a restriction on its DOMAIN Use this restriction in order to EVALUATE a function at certain points Example: 𝑓 𝑥 = 2𝑥+4, 𝑥<−3 &𝑥 −5, 𝑥≥−3

Piecewise functions Example: 𝑓 𝑥 = 9𝑥−4, 𝑥<1 &𝑥+2, 𝑥≥1 Evaluate f(4) Since 4 is greater than or equal to 1, you should plug 4 into the SECOND equation Answer: f(4) = 4 + 2 = 6

Piecewise functions Now you try Example: 𝑓 𝑥 = 5𝑥+3, 𝑥<3 &2𝑥 −7, 𝑥≥3 Evaluate 1. f(-4) 2. f(6) 3. f(3)

ALL DONE