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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.