[Highschool - Math 12] Having trouble with a transformation

[x After Dawn x]

So I was preparing for my math 12 diploma when I came across this question on a practice exam:

"The graph of y = f(x) is transformed by reflecting it in the y-axis and then translating it right by 2 units. The equation of the transformed graph is..."

So I put y = f(-x - 2) because reflecting it over the y-axis means reflecting it horizontally, and translating the graph 2 units to the right means -2. But the answer in the back of the book says the answer is y = -f(x - 2).

I checked the method that the book uses and they say to replace x with x - 2 (as I did), then replace y with -y to get -y = f(x - 2). Then you divide each side by -1 to get y = -f(x - 2).

The problem is that these books have a lot of mistakes (there are PLENTY in my other workbooks) and I'm not sure if this is one of them. Can somebody explain to me why we turn y into -y? Wouldn't that turn the y-value of each coordinate into a a negative, and that would reflect it vertically over the x-axis? And if we multiply each side by -1, wouldn't the - in front of the function symbolize a vertical reflection over the x-axis?

[Patashu]

reflecting it across the y axis:
this means that, for instance, the old 1,foo should now occur at -1,foo
y = f(-x)

translating it right by two units:
this means that, for instance, the old 0,foo should now occur at 2,foo and -2,bar occurs at 0,bar
y = f(-x-2)

Adding a minus sign outside of the function would, indeed, be flipping it around the x axis. Book's wrong you're right

[Zageron]

Math 12 is fun stuff.
Too bad the tests are completely different from the work.
:(

[warriormag17]

Math 12 = Calc? Confused here. lol

[x After Dawn x]

Math 12 = Grade 12 normal math, although I am taking Calculus in grade 12, hehe.

But yeah, I don't even know why I made this thread, I went through all of trigonometry and conics today and there are soooooo many mistakes that they're not even a big deal anymore. Apparently 5 x (-5) = 0...

[QED Stepfiles]

Well so far everyone has been wrong...

Reflecting across the y axis indeed transforms f(x) --> f(-x)

Then proceeding to translate two units to the right means that what originally started as (x,f(x)) is now (-x + 2, f(x)). In other words, calling our original function f and our transformed function g, g(-x+2) = f(x)

and changing variables y = -x + 2 => x = -y + 2...

g(y) = f(-y + 2)

The answer is therefore f(-x + 2), not f(-x - 2)

Another way to do this is just to remember that you have to put parentheses around the argument when you do the translation, and so it's f(-(x-2)) = f(-x+2). Be careful with this.

NB: There's a simple way to do this without thinking it through in the general case. Just check it for the simple line f(x) = x

And indeed you'll find that you should get f(x) = -x+2.

PS: 5 x (-5) is indeed zero if you're over the correct field =)