# 1.5 | Analyzing Graphs of Functions

• A function $$f$$ is increasing on an interval if, for any $$x_1$$ and $$x_2$$ in the interval, $$x_1 < x_2$$ implies $$f(x_1) < f(x_2)$$.
• A function $$f$$ is decreasing on an interval if, for any $$x_1$$ and $$x_2$$ in the interval, $$x_1 < x_2$$ implies $$f(x_1) > f(x_2)$$.
• A function $$f$$ is constant on an interval if, for any $$x_1$$ and $$x_2$$ in the interval, $$f(x_1) = f(x_2)$$.
A function value $$f(a)$$ is called a relative maximum of $$f$$ if there exists an interval $$(x_1, x_2)$$ that contains $$a$$ such that
$$x_1 < x < x_2 \text{ implies } f(a) \leq f(x).$$ A function value $$f(a)$$ is called a relative maximum of $$f$$ if there exists an interval $$(x_1, x_2)$$ that contains $$a$$ such that
$$x_1 < x < x_2 \text{ implies } f(a) \geq f(x).$$
The average rate of change between any two points $$(x_1, f(x_1))$$ and $$(x_2, f(x_2))$$ is the slope of the line through the two points and is usually denoted $$f_{avg}$$. The line through the two points is called a secant line, and the slope of the line is denoted $$m_{sec}$$ and can be computed using the slope formula. Therefore,
$$f_{avg} = m_{sec} = \frac{f(x_2) -f(x_1)}{x_2-x_1}$$
A function $$y = f(x)$$ is even if, for each $$x$$ in the domain of $$f$$,
$$f(-x) = f(x).$$
A function $$y = f(x)$$ is odd if, for each $$x$$ in the domain of $$f$$,
$$f(-x) = – f(x).$$