1.6 GRAPH TRANSFORMATIONS

Shelby Sell

Sammie Meddaugh

Emily Wojahn

INTRODUCTION

HTTP://WWW.YOUTUBE.COM/WATCH?V=WXLLL46ADAA

VOCABULARY Transformation: Functions that map

real number to real numbers.

Rigid Transformations: Leave the side and shape of the graph unchanged (horizontal and vertical translations, reflections).

Non Rigid Transformations: Distort the shape of a graph (horizontal or vertical stretches and shapes).

TRANSLATIONS If c is a positive real number: Horizontal

y= f(x-c) A translation to the right by c units

y= f(x+c) A translation to the right by c units

Vertical

y= f(x) + c A translation up by c units y= f(x) - c A translation down by c units

EXAMPLES

y= x2 + 3

y= abs(x – 2)

2

y= abs(x)

y= x2

REFLECTIONS

Across the x-axis (x,y) (x,-y) y= -f(x)

Across the y-axis (x,y) (-x,y) y= f(-x)

Through the origin (x,y) (-x.-y) y= -f(-x)

EXAMPLES

Reflection over y axis

Reflection over y= x axis

Reflection over x axis

STRETCHES/SHRINKS

Horizontal Stretch/Shrink A stretch by a factor of c if c>1

A shrink by a factor of c if c<1

Vertical Stretch/Shrink A stretch by a factor of c if

c> 1 A shrink by a factor of c if c<1

y= f (x/c)

y= c f(x)

EXAMPLES VERTICAL STRETCHES

Graph of y = x²

Multiply all red values by 3 to get coordinates for the new graph.

Graph y = 3x²

y = x²

"Transformations on the Basic Parabola." W.A.E.C.E. Math Help. N.p., n.d. Web.

SOLUTION TO Y = 3X²

Graph of y = x² with a stretch of 3.

COMBINING TRANSFORMATIONS IN ORDER

1.) Horizontal shift 2 units to the right y=(x-2) ²

2.) Stretch Vertically by factor 3 y=3(x-2) ² 3.) Vertical Translation 5 units up y=3(x-2) ²

+5

Given y=x²

1.) HORIZONTAL SHIFT 2 UNITS TO THE RIGHT Y=(X-2) ²

2.) STRETCH VERTICALLY BY FACTOR 3 Y=3(X-2) ²

3.) VERTICAL TRANSLATION 5 UNITS UP Y=3(X-2) ² +5

ABSOLUTE VALUE- DISTANCEY= f(x) Entire functions

absolute value (change negative y values to

positive)

Y= - f(x) Only Negative y values

MORE ABSOLUTE VALUE

1.) ASSESSMENT

A) y = ½x2 + 2

B) y = 3x2 + 2

C) y = 3(x + 2)2

D) y = ½(x + 2)2

2

2.)

A) y = ½(x - 3)2

B) y = ½(x + 3)2

C) y = 2(x - 3)2

D) y = 2(x + 3)2

3

3.)

A y = 0.5(x + 1)4

B y = 0.5(x - 1)4

C y = 2(x + 1)4

D y = 2(x - 1)4

4.Describe how the graph of y= x² can be transformed to the graph of the given equation

A) Vertical translation up 3 unitsB) Horizontal translation to the right 3 unitsC) Horizontal translation to the left 3 unitsD) Vertical translation down 3 units

Y=x²-3

5.) Given function f, which of the following

represents a vertical stretch by a factor of 3.

C) y=f(9x/3)D)y=f(x)/3

A) y=f(3x)B) y=3f(x)

6.) Given a function f, which of the following

represents a vertical translation of 2 units upward, followed by a reflection across the y-axis.

A) y=f(-x) + 2 C) y= -f(x-2)

B) y= 2-f(x) D) f(x) -2

7.) TRUE OR FALSE?

The function y=f(x+3) represents a translation to the right by 3 units of the graph of y = f(x).

8.) TRUE OR FALSE?

The function of y=f(x)-4 represents a translation down 4 units of the graph of y=f(x)

9.) Write an equation whose graph is

Y=x ²; a vertical stretch by a factor of 3, then shift right 4 units

A) y=3(x-4) ² C) y=3x ² -4

B) y=-3x ² +4 D) y=3(x+4) ²

10.) Write an equation whose graph is Y= x ; a shift left 2 units, then a vertical

stretch y a factor of 2, and finally a shift down 4 units.

A) 2 x+2 -4 C) 2 x-2 +4

B) 2(x+2) -4 D) 3(x-2) +4

ANSWERS 1.) B 2.) A 3.) A 4.) D 5.) B

6.) A7.) False, it is translated left. 8.)True9.) A10.)A

SOURCEShttp://www.mathopolis.com/questions/q.php?id=555&site=1&ref=/sets/function-transformations.html&qs=555_556_557_558_1191_2440_1192_2441_2442

http://departments.jordandistrict.org/curriculum/mathematics/secondary/impact/Algebra%20II/Ready%20for%20web%20site/zTransformationMatchingGame.pdf

http://tutorial.math.lamar.edu/Classes/Alg/Transformations.aspx

Pre calculus- Eighth edition book

http://cheezburger.com/6321270784

"Transformations on the Basic Parabola." W.A.E.C.E. Math Help. N.p., n.d. Web.