Question Regarding Translation of Radical Functions

Question

anonymous · Answer

So I have been taught that $$y - k = f(x - h) $$ and the h signifies the translation of a graph horizontally. But I tested this out on a graphing calculator and have some problems.  Say there was an equation like $$y = \sqrt{x - 2}$$ That means that the base equation is translated 2 units right. But if the equation is something like $$y = \sqrt{-x - 2}$$ Why is it that the graphing calculator shows that it is translated 2 units left and reflected on y axis? Should it not be 2 units right and reflected on y axis like the initial equation is.

anonymous · Answer

@jim_thompson5910

jim_thompson5910 · Answer

start with 

$$\Large y = \sqrt{x}$$

reflect it over the y axis by replacing each 'x' with '-x'

$$\Large y = \sqrt{-x}$$

if you want to move it 2 units to the left, you replace x with "x+2"

$$\Large y = \sqrt{-(x+2)}$$

$$\Large y = \sqrt{-x-2}$$


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An alternative is to start with 

$$\Large y = \sqrt{x}$$

then shift the graph 2 units to the right by replacing each 'x' with 'x-2'

$$\Large y = \sqrt{x-2}$$

then you reflect over the y axis by replacing each 'x' with '-x'

$$\Large y = \sqrt{-x-2}$$

and you get the same result as before

jim_thompson5910 · Answer

the portion on the left that you see is the result AFTER the reflection over the y axis

anonymous · Answer

but in the base equation for reflection in the y axis is $$y = f \left( -x \right)$$ this equation only affects x why is the h is affected

jim_thompson5910 · Answer

h doesn't change though

jim_thompson5910 · Answer

h = 2 the whole time

in y = sqrt(x-2), it tells you to go right 2 units

when it comes to y = sqrt(-x-2), you just flip everything. What is right is now left and vice versa

anonymous · Answer

you said $$y = \sqrt{- (x - h)}  $$ but isn't it $$y = \sqrt{-x - h}$$ if you want to reflect in y axis

jim_thompson5910 · Answer

well notice how I first reflected and then went left

so I had -(x+2) and then it turns into -(x+2) ---> -x - 2

jim_thompson5910 · Answer

in the end though, h is still 2

anonymous · Answer

yes therefore if h is = 2 then that means it is reflection to the right not left

anonymous · Answer

but graph shows:

|dw:1415069039872:dw|

jim_thompson5910 · Answer

let's start with sqrt(x)

|dw:1415069123903:dw|