Stretch about the x axis

[markl77]

Okay so I came across this question asking me to find the equation of the stretched graph ... The first graph is y=x^2
In the diagram you can see that every value on the new graph for x is 2 times the number.
For example , at (3,9) the new coordinate is (6,9)
So I assumed the equation would be y=(1/2)x^2. But that is wrong.. is there something I'm missing? I keep running into the similar problem on other questions as well..

 

[ksdhart2]

You're very very close to the correct answer. We're told that the original graph is y = x², so the new graph is going to be some variant of that. The given transformation also only changes the x-coordinates and leaves the y-coordinates alone. So, something's going to happen to the x term in the equation... but what is that something? Let's think about how the coordinates of the point in the example are related, and see if we can't figure it out.

The example x-coordinate is 6. 6² = 36. But we want f(6) = 9. So, how is 36 related to 9? What would you multiply 36 by to get 9? And how is that transformation related to multiplying by 1/2? As a hint, consider that (1/x)² = 1/x².

 

[markl77]

You're very very close to the correct answer. We're told that the original graph is y = x², so the new graph is going to be some variant of that. The given transformation also only changes the x-coordinates and leaves the y-coordinates alone. So, something's going to happen to the x term in the equation... but what is that something? Let's think about how the coordinates of the point in the example are related, and see if we can't figure it out.

The example x-coordinate is 6. 6² = 36. But we want f(6) = 9. So, how is 36 related to 9? What would you multiply 36 by to get 9? And how is that transformation related to multiplying by 1/2? As a hint, consider that (1/x)² = 1/x².

Okay, thankyou very much. I figured out the answer (1/4)x^2 but I'm still confused...
in my notes it says that the rule is to just find the number and put it in for all x values,,, an example:

If y=f(x)=sqrt4-x^2 and I'm graphing y=f(4x) , I just multiply every x value point by (1/4) right? Why doesn't the graph just multiply by (1/2) using this same logic in the first question?

 

[ksdhart2]

Okay, thankyou very much. I figured out the answer (1/4)x^2 but I'm still confused...
in my notes it says that the rule is to just find the number and put it in for all x values,,, an example:

If y=f(x)=sqrt4-x^2 and I'm graphing y=f(4x) , I just multiply every x value point by (1/4) right? Why doesn't the graph just multiply by (1/2) using this same logic in the first question?

The answer of y = (1/4)x² is technically correct, but the better way to phrase it is y = (1/2x)². This results in the exact same function, but the placement of the grouping symbols is very very important here. In the former, only the x term is squared, but in the latter, the x term and the (1/2) term are both squared. What you're doing here is changing the function from f(x) to f(1/2x). If f(x) = x², then f(1/2x) = (1/2x)². In general \(\displaystyle kx^2 \ne (kx)^2\).

In the same vein, if you started with the function \(\displaystyle f(x) = \sqrt{4-x^2}\), then you'd find that \(\displaystyle f(4x)=\sqrt{4-(4x)^2}=\sqrt{4-16x^2}\)