Final answer:
To transform the graph of x² to 2x²-20x+43, you assess the vertical stretch by the factor 2, the horizontal shift due to -20x, and the vertical shift up by 43 units. Calculate the new vertex, rescale the axes, and sketch the new graph with these transformations.
Step-by-step explanation:
To transform the graph of x² into 2x²-20x+43, you would follow several steps. First, you would compare the new equation to the standard quadratic form ax² + bx + c. Here, a is 2, b is -20, and c is +43. Assessing the changes, you recognize that the graph will be stretched vertically by a factor of 2, will shift horizontally due to the -20x term, and move vertically up by 43 units due to the constant term +43.
Next, to visualize these transformations, you would plot the original function and then apply the transformations step by step. This involves identifying the vertex of the parabola, which can be found using the vertex formula h = -b/(2a) and k = c - b²/(4a). The vertex of x² is at (0,0), while the vertex of the transformed function is going to be shifted. You can then rescale your axes if necessary, labeling them with f(x) and x. Finally, sketch the new graph considering the stretch and shifts you've calculated.