Question

Let f(x)=3^x .

Which function represents a transformation of f(x) by a vertical stretch with factor 6?

g(x)=1/6 ⋅ 3^x

g(x)=3^1/6x

g(x)=3^6x

g(x)=6 ⋅ 3^x


Let f(x)=5^x .

Let g(x)= 5^x − 7 .

Which statement describes the graph of g(x) with respect to the graph of f(x) ?

A. g(x) is translated 7 units left from f(x) .

B. g(x) is translated 7 units up from f(x) .

C. g(x) is translated 7 units right from f(x) .

D. g(x) is translated 7 units down from f(x) .

Answer 1

Most vertical stretching occurs with y = 3^(6x)
The other function g(x) is just translated down 7 units

Answer 2

Ques 1)

`g(x)=6\cdot 3^x`

Ques 2)

D. g(x) is translated 7 units down from f(x) .

Explanation:

Ques 1)

We know that the transformation of the type:

f(x) → a f(x)

is a vertical compression or stretch depending on a.

If a>1 then we have a vertical stretch by a factor of a.

If 0<a<1 then we have a vertical compression.

Here we have a function f(x) as:

`f(x)=3^x`

Now, if it is vertically stretched by a factor of 6 then the transformed function g(x) is given by:

`g(x)=6\cdot 3^x`

Ques 2)

We know that the transformation of the type:

f(x) → f(x)+a

is a shift i.e. translation of the function f(x) either a units up or a units down depending on a.

If a>0 then the shift is a units up.

if a<0 then the shift is a units down.

Here the function f(x) is:

`f(x)=5^x`

and the transformed function g(x) is:

`g(x)=5^x-7`

Since a= -7<0

Hence, the shift is 7 units down.