Write an equation for a function whose graph of fxx is shift

Write an equation for a function whose graph of f(x)=|x| is shifted four units right, compressed horizontally by a factor of 2, reflected in the x-axis, and shifted down 3 units.

Solution

By shifting the graph f(x) = |x| by 4 units towards right we get the transformed graph ahter this step as

f(x) = |x-4| [ Note that f(x) was earlier zero at x = 0, but now it is zero at x = 4. ]

Now let us compress the graph by a factor 2. Let new value is represented by y then y should be equal to x/2 or in terms of compressed variable x = 2y.

Hence, after the application of above two operation, the new equation of the function will be

f(y)=|2y-4| or f(x) = |2x-4| [ Using formal variable x]

Furhter, reflecting it about x - axis, the new function will be

f(x) = - |2x - 4|

Finally, shifting it down by 3 units, the final form of the function is

f(x) = - |2x - 4| - 3