Dilation of Graphs

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Dilation of Graphs
Created by Kenny Kong HKIS 200
3
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Dilation of Graphs

• Stretching and shrinking of graphs are called
  dilation.

• The coefficient factor which multiplies the x
  variable term is the dilation factor.

E.g. y ax 1 ? y 3x 1
y a?x? 2 ? y 3?x? 2
y ax2 3? y 3x2 3
y a2x 4 ? y 3 ? 2x 4
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Dilation of Graphs

• If dilation factor is greater than 1, it is a
  vertical stretch.

i.e. Points will go farther away from the
x-axis.

• If dilation factor is between 0 and 1, it is a
  vertical shrink.

i.e. Points will go closer to the x-axis.

• If dilation factor is negative, it can be a
  vertical stretch or shrink, but reflected across
  the x-axis.

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Dilation of Figures

• The pink figure is dilated with a stretch factor
  a of 2 and 2.5 respectively.

C
a 2 forms the blue figure
C
No change when y is on the x-axis.
C
A
A
Here y is stretched by a factor of 2 and 2.5
respectively.
B
A
B
a 2.5 forms the orange figure
B
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Dilation of Absolute Functions

• The pink figure is dilated with a factor a of 2
  and 0.5 respectively.

The blue figure y 2?x? has been the
result of a stretch factor of 2.
B
A
B
A
B
The orange figure y 0.5?x? has been
the result of a shrink factor of 0.5.
A
a 2 forms the blue figure
a 0.5 forms the orange figure
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Dilation of Quadratic Functions

• y ax2

y x2
y 1/2 x2
y 2x2
y 1/3 x2
y -x2
y -1/2 x2
y -2x2
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Tips

• The negative sign added just in front of the x in
  any functions suggests a flip over the y-axis.

E.g. y x 1 ? y -x 1
y ?x? 2 ? y ?-x? 2
y x2 3 ? y (-x)2 3
y 2x 4 ? y 2-x 4

• The negative sign distributed to all the terms on
  the x side of functions suggests a flip over the
  x-axis.

E.g. y x 1 ? y -(x 1) -x ? 1
y ?x? 2 ? y -(?x? 2) -?x? ? 2
y x2 3 ? y -((x)2 3) -(x)2 ? 3
y 2x 4 ? y -(2x 4) -2x ? 4