Families of Functions Lesson 2-6 Part 1 Algebra 2 Families of Functions Lesson 2-6 Part 1

Goals Goal Rubric To analyze transformations of functions. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Essential Question Big Idea: Function What is a translation of a function?

Vocabulary Parent Function Transformation Translation

Definition Similar to the way that numbers are classified into sets based on common characteristics, functions can be classified into families of functions. The parent function is the simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function.

Parent Functions Parent Functions

Definition A transformation is a change in the position, size, or shape of a figure or function. A translation, or slide, is a transformation that moves each point in a figure or function the same distance in the same direction.

Types of Transformations Reflections: These are like mirror images as seen across a line or a point. Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure. Dilations: This reduces or enlarges the figure to a similar figure.

Translations (slides) If a figure is simply moved to another location without change to its shape or direction, it is called a translation (or slide). If a point is moved “a” units to the right and “b” units up, then the translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the translated point will be at (x - a, y - b). A Example: Image A translates to image B by moving to the right 3 units and down 8 units. B A (2, 5)  B (2+3, 5-8)  B (5, -3)

Example: Perform the given translation on the point (–3, 4). Give the coordinates of the translated point. 5 units right 5 units right   (-3, 4) (2, 4) Translating (–3, 4) 5 units right results in the point (2, 4).

Example: Perform the given translation on the point (–3, 4). Give the coordinates of the translated point. (–5, 2)  2 units 3 units 2 units left and 3 units down (–3, 4)  Translating (–3, 4) 2 units left and 3 units down results in the point (–5, 1).

Your Turn: Perform the given translation on the point (–1, 3). Give the coordinates of the translated point. 4 units right 4 units  (3, 3) Translating (–1, 3) 4 units right results in the point (3, 3).  (–1, 3)

Your Turn: Perform the given translation on the point (–1, 3). Give the coordinates of the translated point. 1 unit left and 2 units down 1 unit   (–2, 1) (–1, 3) Translating (–1, 3) 1 unit left and 2 units down results in the point (–2, 1). 2 units

Horizontal Translation Notice that when you translate left or right, the x-coordinate changes, and when you translate up or down, the y-coordinate changes. Translations Horizontal Translation Vertical Translation

Translations of Functions A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation. How does the graph of a function change if you add a constant to the input of a function (x)? To the output of a function (f(x))? For a positive constant k and a parent function f(x), f(x) ± k is a vertical translation. For a positive constant h and a parent function f(x), f(x ± h) is a horizontal translation.

Horizontal Translations Start with f(x) Add or subtract to the input of the function. Translates the function horizontally (+ left, - right)

Horizontal Translations Given the graph you get the graph by moving the first graph left k units (for k>0) Adding a constant to the input of the function moves the graph of the function left.

Horizontal Translations (Continued) Given the graph you get the graph by moving the first graph right k units (for k>0) Subtracting a constant from the input of the function moves the graph of the function right.

Horizontal Shifts If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. x y -c +c If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units. y = f (x + c) y = f (x) y = f (x – c)

Horizontal Translating

Horizontal Translating (Continued)

Example: Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3 . y -4 4 f (x) = x3 h(x) = (x + 4)3 g(x) = (x – 2)3

Example y = x2 + 3x + 1

Example y = (x-3)2 + 3(x-3) + 1 y = x2 + 3x + 1

Horizontal Translation Rules For any graph y = f(x) The translation y = f(x - a) moves it ‘a’ units to the right The translation y = f(x + a) moves it ‘a’ units to the left

Your Turn: f(x) = 4x + 3 f(x) = 4(x+5) + 3 Describe the transformation Translation of -5 units in the ‘x’ direction

Vertical Translations Start with f(x) Add or subtract to the output of the function. Translates the function vertically (+ up, - down)

Vertical Translations Given the graph you get the graph by moving the first graph up k units (for k>0) Adding a constant to the output of the function moves the graph of the function up.

Vertical Translations (Continued) Given the graph you get the graph by moving the first graph down k units (for k>0) Subtracting a constant from the output of the function moves the graph of the function down.

Vertical Shifts If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units. If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units. x y f (x) + c f (x) +c f (x) – c -c

Vertical Translating

Vertical Translating (Continued)

Example y = x2 + 3x + 1

Example y = x2 + 3x + 1 - 4 y = x2 + 3x + 1 y = x2 + 3x + 1

Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4. y -4 4 8 g(x) = |x| + 3 f (x) = |x| h(x) = |x| – 4

Vertical Translation Rules For any graph y = f(x) The translation y = f(x) + b moves it ‘b’ units up … this can be considered as y - b = f(x) The translation y = f(x) - b moves it ‘b’ units down … this can be considered as y + b = f(x)

Your Turn: f(x) = 4x + 3 f(x) = 4x + 7 Describe the transformation Translation of 4 units in the ‘y’ direction

Combined Vertical and Horizontal Translations

Example: Graph the function using the graph of . First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. -4 x y 4 -4 y 4 x (4, 2) (0, 0) (4, –2) (–1, –2) (0, – 4) (– 5, –4)

Your Turn: f(x) = x2 - 3 f(x) = (x-2)2 + 2 Describe the transformation Translation of 5 units in the ‘y’ direction and 2 units in the ‘x’ direction

Problems

Your Turn: Describe the translation in y = (x + 1)2. Then graph the function. Answer: The graph of the function y = (x + 1)2 is a translation of the graph of y = x2 left 1 unit.

Your Turn: Describe the translation in y = |x – 4|. Then graph the function. A. translation of the graph y = |x| up 4 units B. translation of the graph y = |x| down 4 units C. translation of the graph y = |x| right 4 units D. translation of the graph y = |x| left 4 units A B C D

Example: Use a table to perform each transformation of y=f(x). Use the same coordinate plane as the original function. translation 2 units up

Example: x y y + 2 translation 2 units up Identify important points from the graph and make a table. x y y + 2 –5 –3 –3 + 2 = –1 –2 0 + 2 = 2 –2 + 2 = 0 2 5 Add 2 to each y-coordinate. The entire graph shifts 2 units up.

Your Turn: x y x + 3 –2 4 –2 + 3 = 1 –1 –1 + 3 = 2 2 0 + 3 = 3 Use a table to perform the transformation of y = f(x). Use the same coordinate plane as the original function. translation 3 units right x y x + 3 –2 4 –2 + 3 = 1 –1 –1 + 3 = 2 2 0 + 3 = 3 2 + 3 = 5 Add 3 to each x-coordinate. The entire graph shifts 3 units right.

Essential Question Big Idea: Function What is a translation of a function? A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation. To translate the parent function f(x) horizontally, add a constant h to the input. To translate the parent function vertically, add a constant k to the output.

Assignment Section 2-6 part 1, pg 112 – 113; #1 – 3 all, 4 – 20 even.