Transcript Warm-up (6 min.)

1.5 Graphical Transformations
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Represent translations algebraically and
graphically
Consider this…
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How is the graph (x – 2)2 + (y+1)2 = 16
related to the graph of x2 + y2 = 16?
Some change is good!! 
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Transformations - functions that map real
numbers to real numbers
Rigid transformations – leave the size and
shape of a graph unchanged, include
horizontal translations, vertical translations,
reflections or any combination of these.
Non-rigid transformations – generally distort
the shape of a graph, include horizontal or
vertical stretches and shrinks.
Vertical and Horizontal Translations
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Vertical translation – shift of the graph up or
down in the coordinate plane
Horizontal translation – shift of the graph left
or right in the coordinate plane
Exploration #1
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Complete the activity on p. 132
No talking – first 4 min.
You will be able to discuss with classmates
the last 2 min.
Translations
Let c be a positive real number. Then the following
transformations result in translations of the graph of
y = f(x)
 Horizontal translations
y = f(x – c) a translation to the right by c units
y = f(x + c) a translation to the left by c units
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Vertical translations
y = f(x) + c a translation up by c units
y = f(x) – c a translation down by c units
Ex 1 Describe the graph of y = |x| can
be transformed to the graph of the
given function:
a) y = |x – 4|
b) y = |x| + 2
Reflections, Stretches, and
Shrinks
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Represent reflections, stretches, and shrinks
of functions algebraically and graphically
Graph in the Mirror!! 
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Reflections – the graphs of two functions are
symmetric with respect to some line
Complete Exploration #2 on p. 134
First 6 min (No Talking)
Last 2 min (Discuss with a neighbor)
Reflections
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Over the x-axis – flips the graph of a function
over the x-axis
–
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Over the y-axis – flips the graph of a function
over the y-axis
–
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Symbolically (x,y)  (x,-y)
Symbolically (x,y)  (-x,y)
Over the line y = x – flips the graph of a
function over the line y = x
–
Symbolically (x,y) (y,x)
Ex 1 Find an equation for the reflection of
5 x  9 across each axis
f ( x) 
x2  3
Tonight’s Assignment
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P. 139 – 140 Ex # 3-24 m. of 3
Ex: Express h(x) so that it represents the
graph of f(x) = x2 – 3 reflected over the xaxis? y-axis?
Stretching & Shrinking
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Complete the exploration on p. 136
First 8 min. No talking
Last 8 min. you can discuss with a neighbor
Stretches and Shrinks
Let c be a positive real number. The following
transformations result in stretches or shrinks
of the graph of y = f(x).
 Horizontal stretches or shrinks
y = f(x/c) a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1
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Vertical stretches or shrinks
y = cf(x) a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1
Ex 3 Transform the given function by (a) a vertical
stretch by a factor of 2 and (b) horizontal shrink by
a factor of 1/3.
a) f(x) = |x + 2|
b) f(x) = x2 + x - 2
Combining Transformations
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The order in which transformations are performed often
affect the graph that results
Ex 4
Use f(x) = x2 to perform each transformation.
Write the formula for the resulting function.
a)
A horizontal shift 2 units to the right, a vertical stretch
by a factor of 3, and vertical translation 5 units up
Apply the transformations in the reverse order
Are the graphs the same? Are the formulas the same?
b)
c)