can some one please tell me how compressed by 1/2 effect the whole equation?

Question

If the parabola y=x^2, undergoes the following transformations: reflected over the x-axis, translated 5 units left and 2 units down, and compressed vertically by a factor of 1/2 , what is its new equation?

Answer 1

This is a great problem because it reviews the ideas of transformation which are essential. I would highly recommend using a graphing calculator or software to see these transformation in action because you will need them in the future.

If you begin with x^2, a reelection around the x-axis turns all negative outputs to positive and all positive to negative. Hence putting a negative sign in front of the function does this. So we have f(x) = -x^2.

To translate to the left, we need to ADD 5 to the input (I know this seems counterintuitive, but try it on a graphing calculator. So f(x) = -(x+5)^2. Incidentally, if you subtracted 5 instead of adding it would shift to the right.

To move 2 units down we simply have to remove 2 from the output so f(x) = -(x+5)^2 - 2.

To compress vertically, we need to shorten each output by a factor of 0.5. So we multiply by 0.5. Your final function: f(x) = 1/2*(-(x+5)^2 - 2).