The graph of y = x^2 translates upwards when a positive constant is added, resulting in a new equation y = x^2 + k, where k is the number of units the graph shifts up.
The question asks what causes the graph of y = x^2 to translate up. In algebraic terms, a vertical translation of a graph occurs when a constant is added to the function. For the graph of y = x^2 to translate upwards, a positive constant, let's call it k, needs to be added to the equation, making it y = x^2 + k. The value of k determines how much the graph translates vertically. If k is positive, the graph shifts up by k units. The slope of this graph is represented by the derivative of y = x^2, which is 2x, indicating how steep the graph is at any point. However, the slope does not affect the vertical translation.
The graph of y = x^2 can be translated up by adding a constant value to the equation. The constant value represents the amount by which the graph will be shifted vertically. For example, if we have the equation y = x^2 + 3, the graph will be shifted up by 3 units.