 Features of Function Graphs MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts The features of a function graph can show us many aspects of the relationship represented by the function. Let's take a look at the more popular graphical features. Intercepts are the locations (points) where the graph crosses (or touches) either the x-axis or y-axis.
 • To find the y-intercept, look where x = 0. Remember: the y-intercept will have an x-coordinate of 0. y = -2x + 2 y = -2(0) + 2;    y = 2     y-intercept: (0,2) (You can also read the y-intercept, b, from the function equation if it is in y = mx + b form.) • To find the x-intercept, look where y = 0. Remember: the x-intercept will have a y-coordinate of 0. y = -2x + 2 0 = -2x + 2;     2x = 2;     x = 1     x-intercept: (1,0)    y-values positive or y-values negative • The positive regions of a function are those intervals where the function is above the x-axis. It is where the y-values are positive (not zero). • The negative regions of a function are those intervals where the function is below the x-axis. It is where the y-values are negative (not zero). • y-values that are on the x-axis are neither positive nor negative. The x-axis is where y = 0.
 Some functions are positive over their entire domain (All y-values above the x-axis.) positive: -∞ < x < +∞ or "all Reals", or (-∞,+∞) Some functions are negative over their entire domain. (All y-values below the x-axis.) negative: -∞ < x < +∞ or "all Reals", or (-∞,+∞) Some functions have both positive and negative regions. (y-values above and below x-axis) positive: x > 0 or (0,+∞) negative: x < 0 or (-∞,0) (do not include zero) Secret to Finding the Intervals! The secret to correctly stating the intervals where a function is positive or negative is to remember that the intervals ALWAYS pertain to the locations of the x-values. Think of reading the graph from left to right along the x-axis. Do NOT read numbers off the y-axis for the intervals. Stay on the x-axis!  When looking for sections of a graph that are increasing or decreasing, be sure to look at (or "read") the graph from left to right.
 • Increasing: A function is increasing, if as x increases (reading from left to right), y also increases . In plain English, as you look at the graph, from left to right, the graph goes up-hill. The graph has a positive slope. Example: The function (graph) at the right is increasing from the point (-5,-3) to the point (-2,1), which is described as increasing when -5 < x < -2 . It also increases from the point (1,1) to the point (3,4), described as increasing when 1 < x < 3.  Decreasing: A function is decreasing, if as x increases (reading from left to right), y decreases. In plain English, as you look at the graph, from left to right, the graph goes down-hill. The graph has a negative slope.

Example: The graph shown above is decreasing from the point (3,4) to the point (5,-5), described as decreasing when 3 < x < 5. Constant: A function is constant, if as x increases (reading from left to right), y stays the same. In plain English, as you look at the graph, from left to right, the graph goes flat (horizontal). The graph has a slope of zero.

Example: The graph shown above is constant from the point (-2,1) to the point (1,1), described as constant when -2 < x < 1. The y-values of all points in this interval are "one".  Intervals of increasing, decreasing or constant ALWAYS pertain to x-values. Do NOT read numbers off the y-axis. Stay on the x-axis for these intervals! NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".