Transformation of Functions College Algebra Section 1.6

Three kinds of Transformations Horizontal and Vertical Shifts A function involving more than one transformation can be graphed by performing transformations in the following order: Horizontal shifting Stretching or shrinking Reflecting Vertical shifting Expansions and Contractions Reflections

How to recognize a horizontal shift. Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function has been replaced by has been replaced by

How to recognize a horizontal shift. Basic function Transformed function Recognize transformation The inside part of the function has been replaced by

The effect of the transformation on the graph Replacing x with x – number SHIFTS the basic graph number units to the right Replacing x with x + number SHIFTS the basic graph number units to the left

The graph of Is like the graph of SHIFTED 2 units to the right

The graph of Is like the graph of SHIFTED 3 units to the left

How to recognize a vertical shift. Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function remains the same The inside part of the function remains the same 2 is THEN subtracted 15 is THEN subtracted Original function Original function

How to recognize a vertical shift. Basic function Transformed function Recognize transformation The inside part of the function remains the same 3 is THEN added Original function

The effect of the transformation on the graph Replacing function with function – number SHIFTS the basic graph number units down Replacing function with function + number SHIFTS the basic graph number units up

The graph of Is like the graph of SHIFTED 3 units up

The graph of Is like the graph of SHIFTED 2 units down

How to recognize a horizontal expansion or contraction Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function Has been replaced with Has been replaced with

How to recognize a horizontal expansion or contraction Basic function Transformed function Recognize transformation The inside part of the function Has been replaced with

The effect of the transformation on the graph Replacing x with number*x CONTRACTS the basic graph horizontally if number is greater than 1. Replacing x with number*x EXPANDS the basic graph horizontally if number is less than 1.

The graph of Is like the graph of CONTRACTED 3 times

The graph of Is like the graph of EXPANDED 3 times

How to recognize a vertical expansion or contraction Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function remains the same The inside part of the function remains the same 2 is THEN multiplied 4 is THEN multiplied Original function Original function

The effect of the transformation on the graph Replacing function with number* function EXPANDS the basic graph vertically if number is greater than 1 Replacing function with number*function CONTRACTS the basic graph vertically if number is less than 1.

The graph of Is like the graph of EXPANDED 3 times vertically

The graph of Is like the graph of CONTRACTED 2 times vertically

How to recognize a horizontal reflection. Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function has been replaced by has been replaced by The effect of the transformation on the graph Replacing x with -x FLIPS the basic graph horizontally

The graph of Is like the graph of FLIPPED horizontally

How to recognize a vertical reflection. Basic function Transformed function Recognize transformation The inside part of the function remains the same The function is then multiplied by -1 Original function The effect of the transformation on the graph Multiplying function by -1 FLIPS the basic graph vertically

The graph of Is like the graph of FLIPPED vertically

Summary of Graph Transformations Vertical Translation: y = f(x) + k Shift graph of y = f (x) up k units. y = f(x) – k Shift graph of y = f (x) down k units. Horizontal Translation: y = f (x + h) y = f (x + h) Shift graph of y = f (x) left h units. y = f (x – h) Shift graph of y = f (x) right h units. Reflection: y = –f (x) Reflect the graph of y = f (x) over the x axis. Reflection: y = f (-x) Reflect the graph of y = f(x) over the y axis. Vertical Stretch and Shrink: y = Af (x) A > 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by A. 0 < A < 1: Shrink graph of y = f (x) vertically by multiplying each ordinate value by A. Horizontal Stretch and Shrink: y = Af (x) A > 1: Shrink graph of y = f (x) horizontally by multiplying each ordinate value by 1/A. 0 < A < 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by 1/A.

Sequence of Transformations A function involving more than one transformation can be graphed by performing transformations in the following order. (PEMDAS) 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflecting 4. Vertical shifting

Examples! Use the graph of y = x2 to obtain the graph of y = x2 + 4.

Example cont. Use the graph of y = x2 to obtain the graph of y = x2 + 4. Step 1 Graph f (x) = x2. The graph of the standard quadratic function is shown. Step 2 Graph g(x) = x2+4. Because we add 4 to each value of x2 in the range, we shift the graph of f vertically 4 units up.

g(x) Write the equation of the given graph g(x). The original function was f(x) =x2 (a) (b) (c) (d)

Example Use the graph of f(x) = x3 to graph g(x) = (x+2)3 - 6

Card Activity In your self chosen groups (no larger than 3 people) you will begin by cutting the functions represented algebraically, verbally, and as a graph. You will match the three corresponding cards. Use this as a study guide!! (There will be a quiz Friday!)

Quiz Content: Domain 4 ways to represent a function Graphing- Graphing Essential Graphs - Piecewise - Even/ Odd Graph Transformations

HW Pg 187 1,2, 4, 7,8,13, 19d, 20c