Question

f(x) = 0.2^x is transformed 9 units right, compressed vertically by a factor of 1/6 and reflected across the x-axis. Write the transformed function g(x)

Answer 1

g(x) = -⅙(0.2)^(x – 9)

Explanation:

F(x) = 0.2^x

(a) Transformation to the right

You transform a function 9 units to the right by subtracting 9 from the value of x. Thus, the function becomes

F₁(x) = 0.2^(x – 9)

(b) Vertical compression

To compress a function vertically by a factor of 6, you divide the whole function by 6. Thus, the function becomes

F₂(x) = ⅙F₁(x) = ⅙(0.2)^(x – 9)

(c) Reflection across x-axis

When you reflect a point (x, y) across the x-axis, the x-coordinate remains the same, but the y-coordinate gets the opposite sign. Thus,

g(x) = -⅙(0.2)^(x – 9)

The image below shows F(x) and its appearance after each of the operations.

Answer 2

The function g(x) is defined as
`g(x)=-(1)/(6)(0.2)^(x-9)`.

Explanation:

The given function is

`f(x)=-(0.2)^x`

The function f(x) transformed 9 units right, compressed vertically by factor of 1/6 and reflected across the x-axis.

The transformation of function is defined as

`g(x)=kf(x+b)+c`

Where, k is vertical stretch, b is horizontal shift and c is vertical shift.

If b>0, then the graph of f(x) shifts b units left and if b>0, then the graph of f(x) shifts b units right.

If c>0, then the graph of f(x) shifts c units upward and if c>0, then the graph of f(x) shifts c units downward.

The value of b is -9 because the graph shifts 9 units right. The value of k is 1/6. If the graph of function f(x)reflect across x-axis, therefore the function is defined as -f(x).

`g(x)=(1)/(6)(-f(x-9))`

`g(x)=-(1)/(6)(0.2)^(x-9)`
`[\because f(x)=-(0.2)^x]`

Therefore the function g(x) is defined as
`g(x)=-(1)/(6)(0.2)^(x-9)`.