# Thread: More problems with analytical geometry

1. ## More problems with analytical geometry

Hello

I have a problem with translating a parabola after a series of other transformations.
Given the parabola y= (x-2)(x-5) find the turning points. I have done this ( 3.5, -2.25).
Reflect this about a horizontal line passing through its turning point. I have also done this, with the equation now being y= (-x^2 -4.5)+7x -10

1.The function is now translated 4 units to the left.

At this point I am lost as to where I insert the next translations into the equation.

I can hand graph this onto paper but the equation escapes me.

2. The function is now stretched by a factor of 2.

I am also lost at this point.

Any assistance would be appreciated.

Thank you.

2. Originally Posted by richie
Hello

I have a problem with translating a parabola after a series of other transformations.
Given the parabola y= (x-2)(x-5) find the turning points. I have done this ( 3.5, -2.25).
Reflect this about a horizontal line passing through its turning point. I have also done this, with the equation now being y= (-x^2 -4.5)+7x -10

1.The function is now translated 4 units to the left.

At this point I am lost as to where I insert the next translations into the equation.

I can hand graph this onto paper but the equation escapes me.

2. The function is now stretched by a factor of 2.

I am also lost at this point.

Any assistance would be appreciated.

Thank you.
Transforming a parabola is always easiest if you put it in general form

$y = a(x - h)^2 + k$

where $a$ is the dilation factor, $(h, k)$ is the turning point, and also represents the translations.

You have

$y = (x - 2)(x - 5)$

$= x^2 - 5x - 2x + 10$

$= x^2 - 7x + 10$

$= x^2 - 7x + \left(-\frac{7}{2}\right)^2 - \left(-\frac{7}{2}\right)^2 + 10$

$= \left(x - \frac{7}{2}\right)^2 - \frac{9}{4}$.

So the turning point is $(x, y) = \left(\frac{7}{2}, -\frac{9}{4}\right)$.

To reflect this, you need to negate the dilation factor, while keeping the turning point the same.

So if the function is reflected about the turning point

$y = -\left(x - \frac{7}{2}\right)^2 - \frac{9}{4}$.

To translate the function 4 units to the left, that means that the turning point will move to $\frac{7}{2} - 4 = -\frac{1}{2}$.

$y = -\left(x + \frac{1}{2}\right)^2 - \frac{9}{4}$.

Finally, to STRETCH a function by a factor, divide your dilation factor by that factor.

So since you're stretching by a factor of 2, that means you divide your dilation factor by 2.

The function finally becomes

$y = -\frac{1}{2}\left(x + \frac{1}{2}\right)^2 - \frac{9}{4}$.

3. ## more problems with analytical geometry

Dear Prove It

Thanks for your help and assistance I appreciate it.

Thanks.