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Transformations can be applied to any function (not just a
parent function).
In Math-2 and Math-3 we talked about transformations
applied to the parent function.
In Session 5 we proved whether any function is even or odd by
rewriting the equation to reflect it across either the ‘x’ or ‘y-axis.’
If g(x) is a function, then: (-1)*g(x) → -g(x)
→ reflection across ‘x-axis.’
If k(x) is a function, then: k(-1*x) → k(-x)
→ reflection across ‘y-axis.’
Math-1050 Session 7 (Textbook 3.5) Graphical Transformations
Transformation: an adjustment made to any function that
results in a change to the graph of the function.
shifting (“translating”) the graph up or down,
“translating” the graph left or right
vertical stretching
Reflecting across x-axis or y-axis
horizontal stretching (or “compression”)
Types of transformations
(our textbook also discusses vertical “compression”)
𝑓(𝑥) = 𝑥2
Multiplying the parent function
by -1 changes the sign of every
y-value of the parent function.
𝑔(𝑥) = −𝑥2
x f(x)
-2
-1
0
1
2
x f(x)
-2
-1
0
1
2
4
1
0
1
4
-4
-1
0
-1
-4
The function has been
reflected across the x-axis.
This example transforms the
parent function.
To apply this transformation
to any function we write:
𝑓 𝑥 → −𝑓(𝑥)
𝑓 𝑥 = 2𝑥 − 1
−𝑓 𝑥 = − 2𝑥 − 1
𝑔 𝑥 = − 2𝑥 − 1
𝑔 𝑥 = −𝑓(𝑥)
g(x) is f(x) reflected
across the x-axis.
22 += xy2xy =
x y
-2
-1
0
1
2
4
1
0
1
4
x y
-2
-1
0
1
2
6
3
2
3
6
Vertex: (0, 0)Vertex: (0, 2)
Parent function has
been moved up 2.
This example transforms
the parent function.
To transform any function
we write:
𝑓 𝑥 → 𝑓(𝑥) + 2
𝑓 𝑥 = 2𝑥 − 1
𝑓 𝑥 + 2 = 2 + 2𝑥 − 1
𝑔 𝑥 = 2 + 2𝑥 + 3
𝑔 𝑥 = 𝑓 𝑥 + 2
g(x) is f(x) translated up two.
xxf =)(
x y
-2
-1
0
1
2
x y
-2
-1
0
1
2
2
1
0
1
2
4
2
0
2
4
xxg 2)( =
Multiplying the function by 2
multiplies the y-values by 2
Parent function has been
vertically stretched by a
factor of 2 (VSF=2)
This example transforms
the parent function.
To transform any function
we write:
𝑓 𝑥 → 2𝑓(𝑥)
𝑓 𝑥 = 2𝑥 − 1
2𝑓 𝑥 = 2 2𝑥 − 1
𝑔 𝑥 = 2 2𝑥 + 3
𝑔 𝑥 = 2𝑓 𝑥
g(x) is f(x) vertically stretched
by a factor of two.
xxf =)(
x y
-2
-1
0
1
2
x y
-2
-1
0
1
2
2
1
0
1
2
1
0.5
0
0.5
2
𝑔(𝑥) =1
2𝑥
Parent function has been vertically
stretched by a factor of ½.
Our textbook calls this a
‘vertical compression.’
Vertical stretch
𝑎𝑓 𝑥 𝑓𝑜𝑟 𝑎 > 1
Is there such a thing as a
negative vertical stretch or
compression?
Vertical compression
𝑎𝑓 𝑥 𝑓𝑜𝑟 0 < 𝑎 < 1
No! → -a*f(x) is both stretching
(or compression) and reflection
across x-axis.
2xy =Multiplying the parent function by 3, makes it look “steeper”
23xy =
Right 1
Up 1
Right 1
Up 3
Parent: right 1
Up 1 from vertex.
Transformation: right
1 up 3 from vertex.
𝑓(𝑥) = 𝑥
x y
-1
0
1
4
9
x y
-1
0
1
2
3
6
--
0
1
2
3
--
--
--
0
1
𝑔(𝑥) = 𝑥 − 2
2
Replacing ‘x’ with
‘x – 2’ in the
parent function
moves it right 2.
This example transforms
the parent function.
To transform any function we write
𝑓 𝑥 → 𝑓(𝑥 − 2)
𝑓 𝑥 = 2𝑥 − 1
𝑓 𝑥 − 2 = 2(𝑥 − 2) − 1
𝑔 𝑥 = 2𝑥 − 5
𝑔 𝑥 = 𝑓 𝑥 − 2g(x) is f(x) translated right two.
𝑓 𝑥 − 2 = 2𝑥 − 5
The x-intercepts of y = f(x) are -3 and 5.
What are the x-intercepts of y = f(x – 2)?
f(x – 2) is f(x) shifted right 2
x-intercepts of y = f(x – 2) are (-3 + 2) and (5 + 2).
x-intercepts of y = f(x – 2) are (-1) and (7).
y = g(x) is increasing on the interval: x = (2, 4)
On what interval is y = g(x + 3) increasing?
y = g(x) is increasing on the interval: x = ((2-3), (4-3))
y = g(x) is increasing on the interval: x = (-1, 1)
Interpret the transformation then graph the function
g(x) = -2(𝑥 - 3)2 + 4k(x) = (𝑥 + 2)2 − 3
What is the equation that has been graphed?
Reciprocal Function General Transformation Equation
khx
axf +
−
−=
)1()(
Vertical stretch factor.Reflection
across x-axis
Vertical shift
Horizontal shift
(Vertical Asymptote)
(Horizontal Asymptote)
(ℎ, 𝑘) The point of intersection of the
vertical and horizontal asymptotes.
hxDomain : kyRange :
a) Describe the transformations of the reciprocal function.
b) What is the intersection of the asymptotes?
c) What is the horizontal asymptote?
d) What is the vertical asymptote?
e) What is the domain?
f) What is the range?
𝑔(𝑥) =1
𝑥+ 7 ℎ(𝑥) =
5
(𝑥 − 2)𝑓 𝑥 =
−3
𝑥 + 3− 5
(a) Up 7
(b) (0, 7)
(c) x = 0
(d) y = 7
(e) x ≠ 0
(f) y ≠ 7
(a) VSF=5, right 2
(b) (2, 0)
(c) x = 2
(d) y = 0
(e) x ≠ 2
(f) y ≠ 0
(a) Reflect (x-axis),
left 3, down 5
(a) (-3, -5)
(b) x = -3
(c) y = -5
(d) x ≠ -3
(e) y ≠ -5
x = 3
xxf
1)( =
y = 2
𝑔 𝑥 =1
𝑥 − 3+ 2
What is the equation of the graph?
Right 1
Up 1
x = -4
xxf
1)( =
y = -3
𝑔 𝑥 =−2
𝑥 + 4− 3
Right 1
Down 2
𝑓(𝑥) = 𝑥2
𝑓 2𝑥 = ?
Horizontal compression by ½
(multiply x-value of point by ½)
Looks like vertical stretch by 4
(multiply y-value of point by 4).
→ f(2x) is f(x) horizontally
compressed by a factor of ½
The graph of f(2x)
looks like 4f(x)
𝑔 𝑥 = 4𝑓(𝑥)
𝑔(𝑥) = 4𝑥2
g(x) is f(x) vertically
stretched by a factor of 4
𝑓 2𝑥 = 2𝑥 2
𝑓 2𝑥 = 4𝑥2
(1,1)
(1,4)
(1,1) → (1,4)
(2,4)
Notice that (2,4) → (1,4)
𝑓(𝑥) = 𝑥 𝑓(𝑥) = 𝑥2
For which of the following could a vertical stretch also look
like a horizontal compression?
𝑔(𝑥) = 3𝑥 VSF=3
𝑘(𝑥) = (3𝑥) HSF= 1/3
𝑔(𝑥) = 4𝑥2 VSF=4
HSF= ½ 𝑘(𝑥) = 2𝑥 2
𝑓(𝑥) = 𝑥
For which of the following could a vertical stretch also look
like a horizontal compression?
𝑔(𝑥) = 2 𝑥 VSF=2
𝑘(𝑥) = 4𝑥 HSF= 1/4
𝑓(𝑥) = 𝑥
𝑔(𝑥) = 3 𝑥 VSF=3
𝑘(𝑥) = 3𝑥 HSF= 1/3
𝑓 𝑥 = 𝑥3 𝑓 𝑥 = 3 𝑥
(Graph is shown to the right side of zero only)
(1,1)
(1,8) (2,8)
𝑔 𝑥 = 8𝑥3
𝑘(𝑥) = 2𝑥 3
VSF= 8
HSF= 1/2
(1,1)
(1,2) (2,8)
VSF= 2
HSF= 1/8
𝑔 𝑥 = 23 𝑥
𝑘 𝑥 =38𝑥
𝑓(𝑥) =1
𝑥
(Graph is shown to the right side of zero only)
(1,1)
(0.5 ,2)(1,2)
VSF= 2
HSF= 2
𝑔(𝑥) =2
𝑥
𝑘(𝑥) =1
12𝑥
𝑓(𝑥) = 2𝑥
For which of the following could a vertical stretch also look
like a horizontal compression?
𝑔 𝑥 = 3(2𝑥)VSF=3
Not a “nice”
horizontal
compression.
(2,4)
(2,12) (??,12)
𝑓(𝑥) = sin 𝑥
VSF= 2
𝑔(𝑥) = 2sin 𝑥 𝑘(𝑥) = sin 2𝑥
HSF= 1/2
(90,1)
(90, 2)
(180,0)
(360,0)
Summary: vertical stretch and horizontal compression are
indistinguishable for the following functions.
xxf =)(
xxf =)(
2)( xxf =
xxf =)(
𝑓(𝑥) = 𝑥3 3)( xxf =
𝑓(𝑥) =1
𝑥
Summary: vertical stretch and horizontal compression are
not that same thing for the following functions.
xxf sin)( =𝑓(𝑥) = 𝑙𝑜𝑔2𝑥xxf 2)( =