Consider a parabola P that is congruent (has the same shape) to y=x^2 , opens upward, and has vertex (2,-3). Now find the equation of a new parabola that results if P is Compressed to a factor of 1/2

Translated 2 units to the left
Translated 3 units up
Reflected in the x-axis and translated 2 units to the right and 4 units down

vertex at (2,-3) y = (x-2)^2 - 3

if compressed vertically, then 2y = (x-2)^2-3
if compressed horizontally, y = (2x-2)^2-3

Step 1: Compressing the parabola to a factor of 1/2:

When compressing a parabola, we need to divide the x-coordinate of each point on the original parabola by the compression factor squared. Since the compression factor is 1/2, we will divide the x-coordinate of each point by (1/2)^2 which is equal to 1/4.

The equation of the compressed parabola will be y = (1/4)x^2.

Step 2: Translating the parabola 2 units to the left:
To translate a parabola horizontally, we need to add or subtract the translation amount from the x-coordinate of each point on the graph. In this case, we need to subtract 2 from the x-coordinate of each point.

The equation of the translated parabola will be y = (1/4)(x + 2)^2.

Step 3: Translating the parabola 3 units up:
To translate a parabola vertically, we need to add or subtract the translation amount to the y-coordinate of each point on the graph. In this case, we need to add 3 to the y-coordinate of each point.

The equation of the translated parabola will be y = (1/4)(x + 2)^2 + 3.

Step 4: Reflecting the parabola in the x-axis:
To reflect the parabola in the x-axis, we need to negate the y-coordinate of each point on the graph.

The equation of the reflected parabola will be y = -(1/4)(x + 2)^2 - 3.

Step 5: Translating the parabola 2 units to the right and 4 units down:
To translate a parabola horizontally and vertically, we need to add or subtract the translation amount from the x-coordinate and y-coordinate of each point on the graph, respectively. In this case, we need to subtract 2 from the x-coordinate and subtract 4 from the y-coordinate.

The equation of the final parabola will be y = -(1/4)(x - 2)^2 - 7.

Therefore, the equation of the new parabola after all the transformations is y = -(1/4)(x - 2)^2 - 7.

To find the equation of the new parabola resulting from the transformations mentioned, we will apply each transformation step by step.

Step 1: Compress to a factor of 1/2
When a parabola is compressed by a factor of 1/n, the equation can be achieved by replacing x with nx. In this case, we need to replace x with 2x because we want to compress the parabola to a factor of 1/2.
The equation becomes: y = (2x)^2

Step 2: Translate 2 units to the left
To translate a parabola to the left, we subtract the desired amount from x. In this case, we want to translate 2 units to the left.
The equation becomes: y = (2(x + 2))^2

Step 3: Translate 3 units up
To translate a parabola upwards, we add the desired amount to y. In this case, we want to translate 3 units up.
The equation becomes: y + 3 = (2(x + 2))^2

Step 4: Reflect in the x-axis and translate 2 units to the right and 4 units down
To reflect a parabola in the x-axis, we multiply y by -1. Then, we translate it 2 units to the right and 4 units down.
The equation becomes: -(y + 3) = (2(x + 2 - 2))^2 - 4

Simplifying this equation, we get:
-(y + 3) = (2x)^2 - 4

Expanding (2x)^2 gives us:
-(y + 3) = 4x^2 - 4

Finally, rearranging the equation, we have:
y + 3 = -4x^2 + 4

To obtain the final equation, we isolate y:
y = -4x^2 + 4 - 3

Simplifying further, we get:
y = -4x^2 + 1

So, the equation of the new parabola after the given transformations is y = -4x^2 + 1.