Two Questions, translations and functions related

[GuavaEater]

Hey there, I'm struggling with two functions questions. They are in the picture attached, as well as the work I've attempted doing on both questions.

The first question is "The function f(x) = √x + 5 is transformed by a translation of 2 units down and 4 units to the left. The transformation function passes through the point (20,y). To the nearest tenth, the value of y is _________."

What I attempted to do is solve by adding the movements into the function, which looked like this: f(x) = (√x - 4) + 3, which when using the number 20 for x. I then used my graphing calculator and plugged that in, went to the graphing table, and found where x = 20, and took the y co-ordinate from that point.

The second question is "A vertical translation is applied to y = x². The translated function passes through (3,5). What would the vertical translation be?"

This one I did less on, as I'm not sure where to start. I just assumed that I could state (x,y) translates to (x,y+5), but that clearly isn't right. I also tried y= (x² - 3) + 5, to insert the co-ordinates into the original function, but that seems wrong as well.

Any advice would be appreciated!

 

[ksdhart2]

You're very close on the first one, but I see one minor error. The problem text tells you that the transformation moves the graph 4 units to the left. Try graphing the original function \(\displaystyle f(x)=\sqrt{x}+5\) and \(\displaystyle g(x)=\sqrt{x-4}+5\). Notice that this transformation moved the graph to the right. Do you see why this is? Do you see how you might fix the error?

Now for the second question, let's carefully reread the problem text and see if we can't parse out what it's asking us to find. It tells us that the function transformation is only a vertical shift, meaning there's no horizontal shift. The original function is x². When x = 3, what is the corresponding y-coordinate? We're told that the new, vertically shifted, function passes through the point (3,5). What does that make the new corresponding y-coordinate to x = 3? How much is that off by, and in what direction? What does that suggest you need to do to the function to get the expression for the translated function?

 

[Harry_the_cat]

Hey there, I'm struggling with two functions questions. They are in the picture attached, as well as the work I've attempted doing on both questions.

The first question is "The function f(x) = √x + 5 is transformed by a translation of 2 units down and 4 units to the left. The transformation function passes through the point (20,y). To the nearest tenth, the value of y is _________."

What I attempted to do is solve by adding the movements into the function, which looked like this: f(x) = (√x - 4) + 3, which when using the number 20 for x. I then used my graphing calculator and plugged that in, went to the graphing table, and found where x = 20, and took the y co-ordinate from that point.

The second question is "A vertical translation is applied to y = x². The translated function passes through (3,5). What would the vertical translation be?"

This one I did less on, as I'm not sure where to start. I just assumed that I could state (x,y) translates to (x,y+5), but that clearly isn't right. I also tried y= (x² - 3) + 5, to insert the co-ordinates into the original function, but that seems wrong as well.

Any advice would be appreciated!

First question: Function should be \(\displaystyle y=\sqrt{x+4}+3\). Your function moves it to the RIGHT rather than the left. (Note also how the x+4 is all under the square root sign.)

Second question: Let the vertical translation be k. The new function is \(\displaystyle y=x^2 + k\).
This new function passes through (3, 5) so this point must satisfy the equation \(\displaystyle y=x^2 + k\).
Use that fact to find k.

 

[GuavaEater]

Hey, so I think I figured out what I did wrong in the first one. I ended up with -4, instead of +4. So my inverse function in it's corrected form looked like this: f(x)= (√x + 4) + 3, which led to me getting y = 9.5 (rounded to nearest tenth)

For the second one, I used what ksdhart2 suggested, in solving for k, which looked like this when I did it:
y=x²+ k
5=2³+ k
5=9+k
-4=k

Let me know if I solved them correctly, and thanks for all the helps guys

 

[ksdhart2]

Aside from one minor typo, where you wrote 2³ instead of 3² (you used the correct value of 9 in the next line, so I'm assuming this was just a slip of the fingers), everything looks good to me. In the future, if you're ever unsure of an answer, you can always check it yourself by plugging it back in and seeing it meets the given criteria. In this case, your solution is g(x) = x²-4. When x = 3, we have g(3) = 3² - 4 = 5. This means the function passes through the point (3,5) as it was supposed to, so your answer is correct.

 

[Harry_the_cat]

Hey, so I think I figured out what I did wrong in the first one. I ended up with -4, instead of +4. So my inverse function in it's corrected form looked like this: f(x)= (√x + 4) + 3, which led to me getting y = 9.5 (rounded to nearest tenth)

The question is:
The transformation function passes through the point (20,y). To the nearest tenth, the value of y is _________."

Not sure how you got 9.5 when you sub in x=20??? (I think you put x=20 into the original function, rather than the transformed one.)

Also, what you have written as the transformed function is incorrect. The square root extends over the x+4 not just the x. You could write it as: f(x)= √(x + 4) + 3,

For the second one, I used what ksdhart2 suggested (ahem!) , in solving for k, which looked like this when I did it:
y=x²+ k
5=2³+ k
5=9+k
-4=k

Correct! The question was: What would the vertical translation be? Now interpret your answer: The vertical translation is 4 units downwards. (There was no k in the question, so there should be no k in the answer!)

Let me know if I solved them correctly, and thanks for all the helps guys

see comments above

 

[GuavaEater]

Alright, I took a second look at what I was doing, and yeah, I didn't put the -4 under the square root. My answer after clearing that up was 7.9. Again, thank you all for all the help, each comment definitely added to my understanding of functions