Final answer:
The slope of a curve at any point is determined by the derivative of the function at that point. For straight lines, slope is the rise over run between any two points on the line, with linear equations taking the form y=mx+b, where m is the slope.
Step-by-step explanation:
The slope of a curve at any point (x, y) is given by the derivative of the function at that point. For a function written as y=f(x), the slope at point (x, y) is found by calculating f'(x), which is the derivative of f with respect to x. In the context of a straight line, the slope is defined as the difference in y-value (the rise) divided by the difference in x-value (the run) of two points on the line.
For a linear equation in the form y=mx+b, the slope is the value of m, which indicates a rise of m units in the y-direction for every increase of 1 unit in the x-direction.
For example, if the function is y = x^2, the derivative is dy/dx = 2x. So, the slope of the curve at any point (x, y) on this function is 2x.
The slope of a curve represents how steep or flat the curve is at a given point. A positive slope indicates that the curve is increasing, a negative slope indicates that the curve is decreasing, and a slope of zero indicates that the curve is flat.
For example, referring to Figure A1 Slope and the Algebra of Straight Lines, a line with an equation y=3x+9 shows a y-intercept at 9 and a slope of 3. This means that for every one-unit increase in x, there's a rise of 3 in y-value. The slope is consistent along the entire length of a straight line. However, this differs for curved lines where the slope can change at different points along the curve.