To calculate the slope of a curve, we need to understand that the slope is not constant; it changes at different points along the curve. This variability means that we must use specific methods to determine the slope at any given point. One effective method is the point method, which involves drawing a tangent line at the point of interest on the curve.

A tangent line is a straight line that touches the curve at exactly one point, representing the slope of the curve at that specific location. To find the slope of the tangent line, we can identify two points on this line. For example, if we have a tangent line that intersects the curve at two points, we can calculate the slope using the formula:

$$\text{slope} = \frac{\text{rise}}{\text{run}}$$

In this context, the rise is the vertical change between the two points, and the run is the horizontal change. For instance, if the rise is 2 units (moving from a y-value of 4 to 6) and the run is also 2 units (moving from an x-value of 3 to 5), we can substitute these values into the slope formula:

$$\text{slope} = \frac{2}{2} = 1$$

This calculation indicates that the slope of the tangent line, and thus the slope of the curve at that specific point, is 1. It is important to note that while the slope at this point is 1, the slope of the curve can vary at other points along the curve.

In summary, the point method allows us to determine the slope of a curve by drawing a tangent line and calculating its slope using the rise over run formula. This approach is essential for understanding how slopes behave in non-linear contexts.