shrink and stretch transformation properties

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1
Chapter 3
Quadratic
Functions and
Equations

2
Copyright © 2014, 2010, 2006 Pearson Education, Inc.
Transformations of
Graphs
♦ Graph functions using vertical and horizontal
shifts
♦ Graph functions using stretching and shrinking
♦ Graph functions using reflections
♦ Combine transformations
♦ Model data with transformations (optional)
3.5

Vertical and Horizontal Shifts
We use these two graphs to demonstrate
shifts, or translations, in the xy-plane.

Vertical Shifts
A graph is shifted up or down. The shape of
the graph is not changed—only its position.
Every point moves
upward 2.

Horizontal Shifts
A graph is shifted right: replace x with (x – 2)
Every point moves
right 2.

Horizontal Shifts
A graph is shifted left: replace x with (x + 3),
Every point moves
left 3.

Vertical and Horizontal Shifts
Let f be a function, and let c be a positive
number.

Combining Shifts
Shifts can be combined to translate a graph of
y = f(x) both vertically and horizontally.
Shift the graph of y = |x| to the right 2 units and
downward 4 units.
y = |x| y = |x – 2| y = |x – 2|  4

Example: Combining vertical and
horizontal shifts
Complete the following.
(a) Write an equation that shifts the graph of f(x) = x2
left 2 units. Graph your equation.
(b) Write an equation that shifts the graph of f(x) = x2
left 2 units and downward 3 units. Graph your
equation.
Solution
To shift the graph left 2 units,
replace x with x + 2.
2
( 2) or ( 2)
y f x y x
   

Example: Combining vertical and
horizontal shifts
(b) Write an equation that shifts the graph of f(x) = x2
left 2 units and downward 3 units. Graph your
equation.
Solution
To shift the graph left 2 units, and downward 3 units,
we subtract 3 from the equation
found in part (a).
2
( 2) 3
y x
  

Vertical Stretching and Shrinking
If the point (x, y) lies on the graph of
y = f(x), then the point (x, cy) lies on the
graph of y = cf(x). If c > 1, the graph of
y = cf(x) is a vertical stretching of the
graph of y = f(x), whereas if 0 < c < 1 the
graph of y = cf(x) is a vertical shrinking of
the graph of y = f(x).

Vertical Stretching and Shrinking

Horizontal Stretching and Shrinking
If the point (x, y) lies on the graph of
y = f(x), then the point (x/c, y) lies on the
graph of y = f(cx). If c > 1, the graph of
y = f(cx) is a horizontal shrinking of the
graph of y = f(x), whereas if 0 < c < 1 the
graph of y = f(cx) is a horizontal stretching
of the graph of y = f(x).

Horizontal Stretching and Shrinking

Example: Stretching and shrinking
of a graph
Use the graph of y = f(x) to sketch the graph
of each equation.
a) y = 3f(x)
b)
y f
1
2
x







of a graph
Solution
a) y = 3f(x)
Vertical stretching
Multiply each y-coordinate on
the graph by 3.
(1, –2  3) = (1, –6)
(0, 1  3) = (0, 3)
(2, –1  3) = (2, –3)

of a graph
Solution continued
b)
Horizontal stretching
Multiply each x-coordinate on
the graph by 2 or divide by ½.
(1  2, –2) = (2, –2)
(0  2, 1) = (0, 1)
(2  2, –1) = (4, –1)
y f
1
2
x







Reflection of Graphs Across
the x- and y-Axes
1. The graph of y = –f(x) is a reflection of
the graph of y = f(x) across the x-axis.
2. The graph of y = f(–x) is a reflection of
the graph of y = f(x) across the y-axis.

Reflection of Graphs Across
the x- and y-axes

Example: Reflecting graphs of
functions
For the representation of f, graph the
reflection across the x-axis and across the y-
axis. The graph of f is a line graph determined
by the table.

functions
Solution
Here’s the graph of
y = f(x).

functions
Solution continued
To graph the reflection of
f across the x-axis, start by
making a table of values for
y = –f(x) by negating each
y-value in the table for f(x) .

functions
Solution continued
To graph the reflection of
f across the y-axis, start
by making a table of
values for y = f(–x) by
negating each x-value in
the table for f(x) .

Combining Transformations
Transformations of graphs can be combined
to create new graphs. For example the graph
of y = 2(x – 1)2
+ 3 can be obtained by
performing four transformations on the graph
of y = x2
.

1. Shift the graph 1 unit right: y = (x – 1)2
2. Vertically stretch the graph by factor of 2:
y = 2(x – 1)2
3. Reflect the graph across the x-axis:
y = 2(x – 1)2
4. Shift the graph upward 3 units:
y = 2(x – 1)2
+ 3

continued
y = 2(x – 1)2
+ 3
Shift to the
left 1 unit.
Shift upward
3 units.
Reflect across
the x-axis.
Stretch vertically
by a factor of 2

The graphs of the four transformations.
2
( 1)
y x
  2
2( 1)
y x
 

The graphs of the four transformations.
2
2( 1)
y x
  2
2( 1) 3
y x
  

Example: Combining
transformations of graphs
Describe how the graph of each equation can be
obtained by transforming the graph of Then
graph the equation.
y  x.
1
a.
2
b. 2 1
y x
y x

   

Example: Combining
Solution
Vertically shrink the graph
by factor of 1/2 then reflect it
across the x-axis.
a. y 
1
2
x

Example: Combining
Solution continued
Reflect it across the
y-axis.
Shift left 2 units.
Shift down 1 unit.
b. 2 1
y x
   

Modeling Data with Transformations
Transformations of the graph of y = x2
can be
used to model some types of nonlinear data.
By shifting, stretching, and shrinking this
graph, we can transform it into a portion of a
parabola that has the desired shape and
location. In the next example we
demonstrate this technique by modeling
numbers of Wal-Mart employees.

Example: Modeling data with a
quadratic function
The table lists numbers of Wal-Mart employees in
millions for selected years.

quadratic function
(a) Make a scatterplot of the data.
(b) Use transformations of graphs to determine
f(x) =a(x – h)2
+ k so that f(x) models the data.
Graph y = f(x) together with the data.
(c) Use f(x) to estimate the number of Wal-Mart
employees in 2010. Compare it with the actual
value of 2.1 million employees.

quadratic function
Solution
Here’s a calculator display of a scatterplot of
the data.

quadratic function
Solution continued
It’s a parabola opening up so a > 0.
Vertex (minimum number of employees) could
be (1987, 0.20): translate graph right 1987
units and up 0.20 unit.
f(x) = a(x – 1987) + 0.20
To determine a, graph the data for different
values of a:
First graph a = 0.001 and a = 0.01.

quadratic function
Solution continued
From this graph we see the value of a is
between 0.001 and 0.01.

quadratic function
Solution continued
Experimenting yields a value of a near 0.005.
So f(x) = 0.005(x – 1987) + 0.20

shrink and stretch transformation properties

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shrink and stretch transformation properties