Question

Write the equation of the graph obtained when the graph of y=1/x is compressed vertically by a factor of 0.25, translated 4 units right, and then translated 3 units up.

Answer 1

g ( x ) = 0.25 / ( x - 4 ) + 3

Explanation:

Solution:-

- The given function is defined as an inverse function:

f ( x ) = 1 / x

- The transformation on the function of f ( x ) are of three kinds. We will use a general form to look at the effect of each transformation.

`f ( x ) --- > g ( x ) = [ (a)/(( x +/- b)) ] + c`

Where a,b, and c are constants for transformation.

- The effect of constant a:

a > 1 ------ > Flattens the graph

0 < a < 1 -------- > Compresses the graph vertically

- The effect of constant b:

b > 0 ------ > Horizontal shift to left

b < 0 -------- > Horizontal shift to right

- The effect of constant c:

c > 0 ------ > Vertical shift up

c < 0 -------- > Vertical shift down

- Now we will determine the result of function after each transformation.

• Compressed vertically by a factor of 0.25. Hence, a = 0.25 [ 0 , 1 ]

k ( x ) = 0.25 / x

• Translated 4 units right. The value of b = -4 , right shift : b < 0.

j ( x ) = 0.25 / ( x - 4 )

• translated 3 units up. The value of c = 3, up shift. c > 0

g ( x ) = 0.25 / ( x - 4 ) + 3

Hence, the resulting function of all the mentioned transformations of original function f ( x ) is given as g ( x ):

g ( x ) = 0.25 / ( x - 4 ) + 3

Answer 2

Answer: The equation is y = 0.25/(x - 4) + 3

Explanation:

If we have a function y = f(x)

A compression means that we multiply the function by a factor smaller than 1.

Then the vertical compression by a factor of 0.25 is:

y = 0.25*f(x)

a translation by A units to the right, means that we need to valuate the function in x - A.

So now we have:

y = 0.25*f(x - 4)

A translation in the y-axis means that we need to add a constant to the equation, then if we tralate it by 3 units up, we have:

y = 0.25*f(x - A) + 3

Then, if f(x) = 1/x

Our new equation is:

y = 0.25/(x - 4) + 3