Reading Graphs and Function Behavior

Definition The $y$-intercept of the graph of $f$ is the point $(0, f(0))$, which exists only when $0$ is in the domain. By the vertical line test there is at most one. The $x$-intercepts are the points $(a, 0)$ with $f(a) = 0$, and their first coordinates $a$ are the real zeros of $f$. A graph can have no $x$-intercepts, one, or many.

Definition Let $I$ be an interval in the domain of $f$. The function $f$ is increasing on $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in $I$, and decreasing on $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in $I$. Some books call these strictly increasing and strictly decreasing. A few books use increasing for the weaker condition $f(x_1) \le f(x_2)$, which this course calls nondecreasing, as the calculus course does. Increasing and decreasing describe a whole interval, not a single point. An endpoint may be included when the inequality still holds there, so $x^2$ is increasing on $[0, \infty)$. Many books list only open intervals such as $(0, \infty)$, which is also correct. The same condition can be checked on any set of inputs. A function can be decreasing on each of two intervals without being decreasing on their union, so list such intervals separately, and the same goes for increasing.

Definition A function $f$ is constant on an interval $I$ if $f(x_1) = f(x_2)$ for all $x_1$ and $x_2$ in $I$. Its graph over $I$ is a horizontal segment. On an interval with more than one point, a constant function is neither increasing nor decreasing.

Definition A function $f$ is positive at $x$ if $f(x) > 0$ and negative at $x$ if $f(x) < 0$. On the graph, positive values lie above the $x$-axis and negative values lie below it. The sets where $f$ is positive or negative are the solution sets of the inequalities $f(x) > 0$ and $f(x) < 0$. If the graph has no breaks over an interval, $f$ can change sign there only at a zero, so the zeros cut such an interval into pieces of constant sign. A function such as $\frac{1}{x}$ can also change sign at a gap in its domain.

Definition Let $f$ have domain $D$ and let $c$ be in $D$. The value $f(c)$ is the global (or absolute) maximum of $f$ if $f(c) \ge f(x)$ for every $x$ in $D$, and the global minimum if $f(c) \le f(x)$ for every $x$ in $D$. It is a local (or relative) maximum if there is a number $\delta > 0$ such that $f(c) \ge f(x)$ for every $x$ in $D$ with $\lvert x - c \rvert < \delta$, and a local minimum if the same holds with $f(c) \le f(x)$. Maxima and minima together are called extrema. Here, as in the calculus course, a local extremum may sit at an endpoint of the domain, although some books exclude endpoints. Every global extremum is also a local one. A global maximum value is a single number, but it can be attained at several inputs.

Definition The scale of an axis says how many units of input or output one unit of length along the axis stands for, as marked by its tick labels. A viewing window $[x_{\min}, x_{\max}]$ by $[y_{\min}, y_{\max}]$ shows only the points of the graph whose coordinates lie in those intervals. Changing the scale or the window changes how steep or flat the graph looks and which features are visible, but it does not change the function. Heights and slopes must be read from the axis labels, not from distances on the screen or page.

Definition A value read from a graph is an estimate, accurate only up to the precision of the drawing and its scale. An exact value comes from the formula, by evaluating it or by solving an equation algebraically. A reading such as “the zero is about $1.4$” is confirmed or corrected by substituting into the formula, and a number $a$ is an exact zero only if $f(a) = 0$ exactly.