Finding coordinates of a point of a function on a graph. Let $f(x) = \dfrac{1}{3}x^3 - x^2 - 3x.$ Part of the graph $f$ is shown below. There is a maximum point at $A$ and a minimum point at $B(3,-9)$. 
 
a) Find the coordinates of $A$.  
I already did this part and found it to be $(-1, 5/3)$.   
b) Write down the coordinates of:
i. the image of $B$ after reflection in the $y-axis$.   
would this simply be the opposite? Like, (-3, -9).   
ii. the image of B after translation by the vector (-2, 5).   
Would this be (1, -4)?   
iii. the image of B after reflection in the x-axis followed by a horizontal stretch with a scale factor of 1/2.   
Would this be $(3, 9)$ multiplied by 1/2?
 A: All your stated answers (to the problems before (b)(iii)) are correct. For (b)(iii), do the calculation in steps:


*

*Reflect $B$ in the $x$-axis. This sends $(3,-9)$ to the point $(3,9)$, negating the $y$-coordinate.

*Stretch horizontally with a scale factor of $\frac12$. This corresponds to multiplying the $x$-coordinate of the point we obtained after the first transformation by that scale factor of $\frac12$, so we get the final answer of $\left(\frac32,9\right)$.

A: In regards to part b i, the reflection in the $y$-axis would be basically multiplying the $x$-coordinate of $B$ by $-1$. So you did that correctly.
In regards to part b ii, translation by the vector $(-2,5)$ basically amounts to adding $-2$ to $B$'s $x$-coordinate, and $5$ to to its $y$-coordinate. This part you did right as well.
As for part b iii, a reflection in the $x$-axis basically amounts to multiplying the $y$-coordinate by $-1$. The horizontal stretch factor of $1/2$ is what you would then multiply the $x$-coordinate by.

Intuitively, you can think of translations as just sliding the graph around, reflections as a way of mirroring it or flipping it, and stretching as changing various distances by a scaled factor. (For example, a scale factor of $1/2$ on the horizontal squashes everything so that horizontal distances are shorter: the interval $[0,1]$ magically becomes $[0,1/2]$ because of that.)
