Final answer:
The originally given parabola y = x^2 has been vertically compressed by a factor of 0.5 and shifted up by 3 units, resulting in the new equation y = 0.5x^2 + 3. Sketching this graph involves plotting points using the new equation and showing a parabola that is narrower and raised compared to the original.
Step-by-step explanation:
To start with, the original graph of y = x^2 is a parabola that opens upwards with its vertex at the origin (0,0). When it is vertically compressed by 0.5, we multiply the y-values by 0.5. This changes the equation to y = 0.5x^2. Next, when the graph is shifted 3 units up, we add 3 to the y-values of the parabola which modifies the equation to y = 0.5x^2 + 3.
The resultant graph has the same shape as the original parabola but is half as tall (due to the compression) and raised so that its vertex is located at (0, 3). To sketch the parabola, you start by plotting the new vertex and then plotting additional points by choosing x-values and calculating the corresponding y-values using the new equation.