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The graph of a polynomial function provides a visual representation of how the function's value changes as the input value changes. The shape of this graph can vary greatly depending on the degree and coefficients of the function.
The most fundamental way to draw a graph is by determining several point pairs that satisfy the function, then connecting them with a smooth curve.
Steps:
Graph the function .
We choose several values and calculate :
Plot the points and connect them:
Graph the function .
Table of values:
Plot the points and connect with a smooth curve (parabola):
Graph the function .
Table of values:
Plot the points and connect with a smooth curve:
Graphs of polynomial functions are always smooth (no sharp corners) and continuous (no jumps or breaks). Their general shape is heavily influenced by the degree of the polynomial.
In general, the graph of a polynomial function of degree can intersect the -axis at most and has at most turning points (peaks or valleys).
One important characteristic of polynomial graphs is their end behavior, which describes the direction of the graph as approaches positive infinity () or negative infinity ().
The end behavior is determined solely by the leading term :
There are four possible combinations:
Even, (Positive):
As , (rises right )
As , (rises left )
Examples: ,
Even, (Negative):
As , (falls right )
Odd, (Positive):
As , (rises right )
Odd, (Negative):
As , (falls right )
Knowing the end behavior is very helpful for identifying the graph of a polynomial function without having to plot it in detail.
Application Example:
Match the following functions with their likely end behavior:
By analyzing the leading term, we can predict the general shape of the graph at its ends.
As , (falls left )
Examples: ,
As , (falls left )
Examples: , ,
As , (rises left )
Examples: ,
