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This section shows and explains graphical examples of function curvature.

Another common aspect of the graph of a function, is the graphs curvature. In general, curvature (like local extrema) is difficult to determine without tools from calculus and as such, it is a major area of study in calculus. However, for this class we will restrict ourselves to a description of what different types of curvature ‘look like’ on a graph so that we can identify curvature types visually.

One typically refers to the curvature as a combination of whether the function is increasing or decreasing, and whether it is concave up or concave down. We will describe each of these terms/features by themselves and then give graphical representations of their combinations which may be more helpful to understanding.

Increasing:
A function is increasing on an interval $(a,b)$ if, for every single pair of values $x$ and $w$ such that $a < x < w < b$, we have $f(x) \leq f(w)$.
In ‘human speak’ a function is increasing on an interval $(a,b)$, if larger numbers in that interval always get sent to larger values by $f$.
Decreasing:
A function is decreasing on an interval $(a,b)$ if, for every single pair of values $x$ and $w$ such that $a < x < w < b$, we have $f(x) \geq f(w)$.
In ‘human speak’ a function is decreasing on an interval $(a,b)$, if larger numbers in that interval always get sent to smaller values by $f$.
Concave Up:
A function is concave up on an interval $(a,b)$ if, for every single pair of values $x$ and $w$ such that $a < x < w < b$, the inequality $\frac {f(a) - f(x)}{a-x} \leq \frac {f(w)-f(b)}{w-b}$ is true.
In ‘human speak’ a function is concave up if the line connecting $x$ to $w$ is always above the graph of $f$ for every value between $x$ and $w$. Alternatively, one can say an interval $(a,b)$ is concave up if the graph ‘bends upward’ as you graph it from left to right.
Concave Down:
A function is concave down on an interval $(a,b)$ if, for every single pair of values $x$ and $w$ such that $a < x < w < b$, the inequality $\frac {f(a) - f(x)}{a-x} \geq \frac {f(w)-f(b)}{w-b}$ is true.
In ‘human speak’ a function is concave down if the line connecting $x$ to $w$ is always below the graph of $f$ for every value between $x$ and $w$. Alternatively, one can say an interval $(a,b)$ is concave down if the graph ‘bends downward’ as you graph it from left to right.

The following are graphical representations of various combinations of increasing/decreasing and concave up/down;

Increasing and Concave Up:
Consider the graph for $f(x)$ Increasing and Concave down:
Consider the graph for $f(x)$ Decreasing and Concave Up:
Consider the graph for $f(x)$ Decreasing and Concave Down:
Consider the graph for $f(x)$ Combination of each of the above in the same graph:
Consider the graph of the polynomial $f(x)$ Key:

• Decreasing and Concave Up
• Increasing and Concave Down
• Decreasing and Concave Down
• Increasing and Concave Up.

Thus we have all four combinations in one polynomial, which is not unusual for higher degree polynomials.