Final answer:
To sketch the graph of an equation, identify the form of the equation, plot the y-intercept, use the slope for a straight line, or use known characteristics for exponential functions. For straight lines, calculate additional points using the slope and for exponential functions, reference points like (0,1).
Step-by-step explanation:
To sketch the graph of an equation, you need to understand the form of the equation and what each term represents. For a straight line in slope-intercept form, expressed as y = mx + b, m represents the slope and b the y-intercept. The slope indicates how steep the line is, and the y-intercept is the point where the line crosses the y-axis. For example, if an equation is given as y = 3x + 9, then the slope (m) would be 3 and the y-intercept (b) would be 9.
Following this information from Figure A1 Slope and the Algebra of Straight Lines, to graph this line, you start at the y-intercept (0,9) on the y-axis. Then, from this point, you use the slope to determine the next point by rising 3 units in the vertical direction for every 1 unit you move to the right along the horizontal axis. Continuing this process, you plot additional points and draw a straight line through them to complete the graph of the line.
For sketching more complex functions like exponential functions, such as y = ex or y = -ex, you will plot the points based on known characteristics of exponential functions. Exponential functions have a consistent base, e, and their graphs will always pass through the point (0,1), since any number to the zero power is 1. If the function is y = -ex, the graph will still pass through (0,1) but will reflect across the x-axis due to the negative sign.