axis.’ reflection across ‘x axis.’ reflection across ‘y axis.’€¦ · transformations...

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)( =