Math Notation: x^2 means x raised to the second power or x squared.

Explain how to translate the graph of y = -x^2 to produce the graph y = -x^2 + 3. What would the graph look like?

You should know what

y = x^2 looks like, sketch it

reflect that graph in the y-axis to get
y = -x^2

now move every point up 3 units to get
y = -x^2 + 3

I believe Reiny meant to reflect the y = x^2 graph about the x axis, in the second step of his answer.

drwls, thanks for pointing that out , you are right.

(another case of "oldtimers" syndrome)

To translate the graph of y = -x^2 to produce the graph y = -x^2 + 3, you need to shift the entire graph vertically upward by 3 units. This is because the "+3" term is added to the equation, affecting the y-coordinate of each point on the graph.

To visualize the translation, you can follow these steps:

1. Start with the graph of y = -x^2, which is a downward-opening parabola centered at the origin.
2. Consider any point on the original graph, such as (1, -1). This point has an x-coordinate of 1 and a y-coordinate of -1.
3. To shift the graph vertically upward by 3 units, add 3 to the y-coordinate of this point. Hence, the new coordinates on the translated graph would be (1, -1 + 3) = (1, 2).
4. Repeat this process for other points on the original graph, and calculate their new coordinates by adding 3 to their y-coordinates.
5. Plot these new set of points on the coordinate plane, and connect them to form the new graph.

In the case of y = -x^2 + 3, the translated graph would be the same shape as the original parabola but upwards shifted by 3 units. It will still be a downward-opening parabola, but the vertex (the highest or lowest point on the graph) will now be located at (0, 3).