Final answer:
Vertical tangent lines occur when the slope of a curve is undefined or infinite, whereas horizontal tangent lines occur where the slope of the tangent is zero. Both concepts are based on the relationship between the derivatives and the slope of tangent lines on a graph.
Step-by-step explanation:
The discussion of horizontal and vertical tangent lines relates to the concept of slopes and the relationship between different quantities on a graph. In a typical graph with perpendicular axes consisting of a horizontal x-axis and a vertical y-axis, the equation of a straight line is given by y = mx + b, where m represents the slope and b the y-intercept.
A horizontal tangent occurs at points on a curve where the slope of the tangent is zero, which corresponds to a horizontal line. This means the derivative of the curve at that point is 0. Conversely, a vertical tangent line occurs at points where the curve has an undefined or infinite slope, which can be thought of as a result of the derivative of the curve approaching infinity or not existing at that particular point.