# lesson-7-combinations-of-transformations

```Combinations of Transformations
MCR3U
Stretched vertically by
a factor of đ
ï· If |đ| &gt; 1, the graph is
expanded
ï· If 0 &lt; |đ| &lt; 1 the
graph is compressed
ï· If đ &lt; 0, the graph is
reflected in the đ„-axis
Translated
vertically đ units
y ïœ af ïš k ïš x ï­ d ï© ï© ï« c
Stretched horizontally
by a factor of đ/đ
ï· If |đ| &gt; 1, the graph is
compressed
ï· If 0 &lt; |đ| &lt; 1 the graph
is expanded
ï· If đ &lt; 0, the graph is
graph is reflected in the
đŠ-axis
ï· If đ &gt; 0, the graph
shifts up
ï· If đ &lt; 0, the graph
shifts down
Translated horizontally đ
units
ï· If the sign is negative, the
graph shifts to the right
ï· If the sign is positive, the
graph shifts to the left
The đ affects the graph y ïœ f ïš x ï© by stretching or compressing ___________________ by
a factor of đ.
If the đ is negative, there is a reflection about the __________________.
The đ affects the graph y ïœ f ïš x ï© by translating ______________________ đ units.
The đ affects the graph y ïœ f ïš x ï© by translating ______________________ đ units.
The đ affects the graph y ïœ f ïš x ï© by stretching or compressing ____________________ by
a factor of
1
.
k
If the đ is negative, there is a reflection about the _________________.
Does the order of transformations matter?
Graphing Point by Point
1) Start by graphing the base function ( y ïœ x 2 , y ïœ x , y ïœ x3 , y ïœ x , y ïœ
1
).
x
2) Graph đ by taking the significant point (đ„, đŠ) and multiply only the đŠ-values by đ so that the
point (đ„, đŠ) becomes (đ„, đđŠ).
3) Graph đ by taking the significant point (đ„, đŠ) and multiply only the đ„-values by
1
so that the
k
ïŠx ï¶
point (đ„, đŠ) becomes ï§ , y ï· .
ïšk ïž
4) Graph đ by taking the significant point (đ„, đŠ) and adding đ to the đ„-value so that the point
(đ„, đŠ) becomes (đ„ + đ, đŠ).
5) Graph đ by taking the significant point (đ„, đŠ) and adding đ to the đŠ-value so that the point
(đ„, đŠ) becomes (đ„, đŠ + đ).
Example 1 – Sketching Graphs of Transformed Functions
1. Given the graph of đ(đ„) = |đ„|, describe how you would
graph đ(đ„) = −đ(đ„ − 3) + 4 using transformations.
2. a) Using transformations, sketch the graph of đŠ =
√2đ„ + 4 + 1.
Hint: Rewrite 2đ„ + 4 in factored form to determine the
horizontal translation.
Example 2 – Writing Equations of Transformed Functions
1. The function đŠ = đ(đ„) has been transformed into đŠ = đđ(đ(đ„ − đ)) + đ. Write the
following in the appropriate form:
1
(a) a vertical compression by a factor of 2 , a reflection in the đ„-axis and a translation 3 units
right.
(b) a vertical stretch by a factor of 3, a horizontal stretch by a factor of 2, a translation left 5
and up 4, and a reflection in the đŠ-axis.
Practice Transformations Given an Equation
Graph each of the following functions by:
1ï¶
ïŠ
a) Graphing the base function first. ï§ y ïœ x 2 , y ïœ x , y ïœ x3 , y ïœ x , y ïœ ï·
xïž
ïš
b) Listing the transformations.
c) Applying the transformations to the base function.
1) y ïœ 2 ïš x ï« 1ï© ï­ 1
2
2) y ïœ
2
ï«1
xï«2
3) y ïœ ïš 2 x ï­ 2 ï© ï­ 1
3
4) y ïœ ï­ x ï­ 2 ï« 1
7) y ïœ
1
1
xï­
2
2
6) y ïœ 3 x ï« 1 ï« 1
5) y ïœ 2 3 x
ïŠ 1
ï¶
8) y ïœ ï­ ï§ ï­ ïš x ï« 1ï© ï·
ïš 2
ïž
3
9) y ïœ 2 2 x ï« 2 ï« 2
Practice Transformations Given a Graph
List the transformations.
Apply the transformations to key points on the graph.
1) y ïœ 3g ïš ï­2 ïš x ï­ 1ï© ï© ï« 1
3) y ïœ
1
h ïš x ï« 1ï© ï­ 2
2
2) y ïœ ï­ f ïš 2 ïš x ï« 1ï© ï©
4) y ïœ ï­
1
f ïš 2x ï« 4ï© ï­1
2
5) y ïœ f (3x ï­ 6)
6) y ïœ f (ï­2 x ï« 4)
7) y ïœ ï­2 f ( x ï­ 3) ï« 1
8) y ïœ
1 ïŠ1
ï¶
f ï§ x ï­ 2ï·
2 ïš3
ïž
```