Final answer:
The function with the steepest graph among the options given is y = x², which is a quadratic function. It doesn't have a constant slope, but its graph gets steeper as the value of x increases or decreases far from zero.
Step-by-step explanation:
To determine which function will have the steepest graph, we must consider the slope of the functions. The slope is the rate at which the function increases or decreases as you move along the x-axis.
For the function y = -2, this is a horizontal line with a slope of 0, so it is not steep at all.
The function y = -1/2x is a linear function with a negative slope of -1/2. This line will slope downwards to the right, but it is not very steep.
y = x is a linear function with a slope of 1, which means it will have a consistent, moderate upward slope.
Lastly, y = x² is a quadratic function. It has a varying slope that depends on the value of x, but overall, it has steep sides, especially as the value of x moves away from zero on both sides.
While y = x² does not have a constant slope, at many points, especially far from the origin, it will be much steeper than any of the linear equations provided. Thus, y = x² is the function that will have the steepest graph overall as x increases or decreases drastically.