Exercise 140 Page 575

a Using the function f(x), we want to find the expression f(x)+2. To do this, we will add 2 to the given function.

f(x)=2x^2

LHS+2=RHS+2

f(x)+ 2=2x^2+ 2

Now, let's look at all of the possible transformations so that we can more clearly identify what is happening to our function.

Transformations of f(x)
Vertical Translations	Translation up k units, k>0 y=f(x)+ k
Vertical Translations	Translation down k units, k>0 y=f(x)- k
Horizontal Translations	Translation right h units, h>0 y=f(x- h)
Horizontal Translations	Translation left h units, h>0 y=f(x+ h)
Vertical Stretch or Compression	Vertical stretch, a>1 y= af(x)
Vertical Stretch or Compression	Vertical compression, 0< a< 1 y= af(x)
Horizontal Stretch or Compression	Horizontal stretch, 0< b<1 y=f( bx)
Horizontal Stretch or Compression	Horizontal compression, b>1 y=f( bx)
Reflections	In the x-axis y=- f(x)
Reflections	In the y-axis y=f(- x)

As we can see, we can describe this transformation as a vertical translation 2 units up. It means that the graph of the given function was shifted 2 units up.

b Using the function f(x), we want to find the expression 2 * f(x). To do this, we will multiply the given function by 2 and simplify.

f(x)=2x^2

MultEqn

LHS * 2=RHS* 2

2 * f(x)= 2 * 2x^2

Multiply

2 * f(x)=4x^2

We can describe this transformation as a vertical stretch by a factor of 2.

c Using the function f(x), we want to evaluate for the given value, f( x+2). To do this we need to substitute x+2 for x in each instance of the x-variable.

f(x)=2x^2

Substitute

x= x+2

f( x+2)=2( x+2)^2

We can describe this transformation as a horizontal translation 2 units left. It means that the graph of the given function was shifted 2 units left.

d Using the function f(x), we want to evaluate for the given value, f( 2x). To do this we need to substitute 2x for x in each instance of the x-variable and simplify.

f(x)=2x^2

Substitute

x= 2x

f( 2x)=2( 2x)^2

PowProd

(a * b)^m=a^m* b^m

f(2x)=2 (2^2x^2 )

CalcPow

Calculate power

f(2x)=2 ( 4x^2 )

Multiply

f(2x)=8x^2