Question

The graph of f(x)=sin(x) is transformed into a new function, g(x) , by stretching it vertically by a factor of 4 and shifting it 3 units to the right. What is the equation of the new function g(x) ?

Answer 1

g(x) = 4 sin(x - 3)

Explanation:

The given function is:

f(x) = sin(x)

Stretching by a factor of 4:

A vertical stretch or compression of f(x) can be expressed as:

cf(x), where c is any constant

if c > 1, it indicates a stretch in vertical direction and if 0 < c < 1, it indicates a compression by a factor of c in vertical direction.

Since, given function is being stretched by a factor of 4, the transformed function will become:

4f(x)

Shifting by 3 units right:

A shift of f(x) to right can be obtained by subtracting c from every occurrence of x in f(x) , where c is the number of units f(x) is being shifted in right direction. Since in this case f(x) is shifted 3 units to right, the transformed function will be:

f(x - 3)

Applying both the transformations to f(x), we get g(x) as:

g(x) = 4 f(x - 3)

g(x) = 4 sin(x - 3)

Answer 2

g(x) = 4 sin(x - 3)

Explanation:

* Lets revise some transformation

- If the function f(x) translated horizontally to the right by h units, then

the new function g(x) = f(x - h)

- If the function f(x) translated horizontally to the left by h units, then

the new function g(x) = f(x + h)

- If the function f(x) translated vertically up by k units, then the new

function g(x) = f(x) + k

- If the function f(x) translated vertically down by k units, then the

new function g(x) = f(x) – k

- A vertical stretching is the stretching of the graph away from the

x-axis

- If k > 1, the graph of y = k • f(x) is the graph of f(x) vertically

stretched by multiplying each of its y-coordinates by k.

- A vertical compression is the squeezing of the graph toward

the x-axis.

- If 0 < k < 1 (a fraction), the graph of y = k • f(x) is the graph of f(x)

vertically compressed by multiplying each of its y-coordinates by k.

* Lets solve the problem

∵ f(x) = sin(x)

∵ f(x) is stretching vertically by a factor of 4

∴ f(x) multiplied by 4

∵ f(x) is shifting 3 units to the right

∴ We will change x to (x - 3)

∵ The new function is g(x)

∴ g(x) = 4 sin(x - 3)

- The attached graph for more understand

f(x) is the red graph

g(x) is the blue graph