The graph $\large{g(x)}$ is the graph of $\large{f(x)=x^{2}}$ shifted 4 units left, vertically stretched by a factor of 3, then shifted 5 units up. What is the function rule for $\large{g(x)}$?

I know how to apply the translation but I need help applying the stretch.

Question

The graph $\large{g(x)}$ is the graph of $\large{f(x)=x^{2}}$ shifted 4 units left, vertically stretched by a factor of 3, then shifted 5 units up. What is the function rule for $\large{g(x)}$?

I know how to apply the translation but I need help applying the stretch.

563blackghost · Answer

Now I currently have $\Large{g(x+4)+5}$ But i don't know on how to apply the stretch of 3.

3mar · Answer

May I help, sister?

563blackghost · Answer

Sure :)

3mar · Answer

Thanks

563blackghost · Answer

Is what I have currently correct? @3mar

3mar · Answer

Yes, without the stretch.

563blackghost · Answer

Great :)

563blackghost · Answer

Would with the stretch applied be like this?

$\huge{y=5g(x+4)+5}$

According to my textbook it says that...

(sorry my internet is not so good at the moment)

3mar · Answer

Never mind.
I think these are the steps.

3mar · Answer

Let's test your result.

563blackghost · Answer

Oooo I forgot the square :(

563blackghost · Answer

$\huge{y=5g(x+4)^{2}+5}$

563blackghost · Answer

I mean't 3g sorry....

563blackghost · Answer

Wait g will not be part of the equation? it would be in place of y?

3mar · Answer

But I think another way.
can you discuss it with me?

563blackghost · Answer

Yes :)

3mar · Answer

Ok. Let's.

563blackghost · Answer

So we would need to find g(x) from $\large{f(x)=x^{2}}$ we would first apply the translation of 4 to the left and 5 up...in which the equation then turns to... $\Large{g(x)=(x+4)^{2}+5}$ Correct?

3mar · Answer

Parent function:
$$f(x)=x^2$$

4 unit shift left :
$$f(x)=(x+4)^2$$
and $$f(x)=g(x+4)$$

3 factors vertical stretch:
$$f(x)=3(x+4)^2$$
and
$$f(x)=g(\sqrt{3}( x+4))$$

finally, 5 units shift up:
$$f(x)=3(x+4)^2+5$$
and
$$f(x)=g(\sqrt{3}(x+4))+5$$

Hope I am right!

3mar · Answer

Take care: "we would first apply the translation of 4 to the left and 5 up"
Order of step is important and it will make a difference for other problems if it does not make her.

563blackghost · Answer

I don't quite understand as to how 3 is rooted.

563blackghost · Answer

Ooo I see my mistake I mean't it as `what i know first` maybe I shouldn't have worded it like that. Thank you for pointing that out @3mar :)

563blackghost · Answer

I seem to understand as how the function is written by studying it quite a bit. Thank you so much @3mar :)

3mar · Answer

Wait, sister!
You said that:"
y=5g(x+4)+5 According to my textbook it says that..."

3mar · Answer

I just wonder how we substitute with (x+4) and then *5?

563blackghost · Answer

No, that was before but I now understand my mistake @3mar thanks to you :)

3mar · Answer

Every thing is ok?

563blackghost · Answer

Yes @3mar Thank you so much :)

3mar · Answer

But I might accept the form:
$$g(x)=3f(x+4)+5$$?
?

It is quiet right. 



But I don't want to get you confused

If you are, just make sure from your teacher.

and kindly let me know which one is more accurate.

563blackghost · Answer

I will @3mar :) I'll make sure of it.

3mar · Answer

If there is more, I won't be late!

563blackghost · Answer

No, that is all for today. Thankfully :) Once again, Thank you @3mar

3mar · Answer

You are welcome, sister!
Be well all the time.