Question

How do you transform $ y = \dfrac{1}{x} $ to $ y = \dfrac{{(3x - 2)}}{{x + 1}} $ ?
The answer says it’s the reflection in the $ x $ axis, stretching by a scale factor of 5 and translates it by $ ( - 1.3) $ but how do I do it?

Answer 1

Hint: When we transform a function in this context it means that we have to modulate the function in such a way that first function on applying given modulations becomes the second function, but since we are given answer itself that is we first change $ y $ to $ - y $ then multiply whole function by factor of $ 5 $ and then translate it by $ ( - 1.3) $ . Since the question already gives to us the answer we will try to prove how we reached the conclusion that we have we can prove it by remembering the formula for a typical transformation of a function $ y = f(x) $ is as follows:
 $ y = af[b(x - c)] + d $
Where a,b,c ,d are the different values which transform our given function. If we are able to find values of a,b,c and d we will be able to prove that it is in fact the given constants that can transform the given function.

Complete step by step solution:
The transformation for our given function can be written as
 \[y = \dfrac{a}{{b(x - c)}} + d\]
 $ a $ here is the constant that stretches the function up and down
 $ b $ here is the constant that stretches or shrinks the function left to right,
 $ c $ here moves the function from left to right, whereas
 $ d $ here moves the function up and down.
We will now try to write the given function $ y = \dfrac{{(3x - 2)}}{{x + 1}} $ in the above form and find a,b,c,d in the process.
 $ y = \dfrac{{(3x - 2)}}{{x + 1}} $ can be conveniently written as
 \[y = \dfrac{{3(x + 1 - 1) - 2}}{{x + 1}}\]
Which can then be written as
 \[y = \dfrac{{3(x + 1) - 5}}{{x + 1}}\]
Then we cancel $ (x + 1) $ with the first part of the numerator
And in write:
 \[y = \dfrac{{ - 5}}{{x + 1}} + 3\]
And we get our values for all the constants
 \[\
  a = -5 \\
  b = 1 \\
  c = -1 \\
  d = 3 \;
 \]
These are the constants that were given in the question.

Note: For questions like this try to express the given function into the standard form and then find the value of the constants. The values of constants a,b,c,d can be found out by trying to write the given function into the form of the following formula:
 $ y = af[b(x - c)] + d $