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Graphs of Polynomial Functions Project

On this site, we will find the equations for polynomial functions given the function’s graph and a point not on the x-axis.

We will identify characteristics of the graph and use them to derive an equation in standard form:
$f(x)=a_nx^n+a_n_-_1x^n^-^1+...a_1x+a_0$
with leading coefficient $a_n$ and degree $n$

Method

Use the end behavior characteristics of the graph to determine basic information about the function.

$f(x)\rightarrow \Box$
as
$x\rightarrow-\infty$ $f(x)\rightarrow \Box$
as
$x\rightarrow\infty$ Degree Leading Coefficient

$\infty$ $\infty$ even positive

$-\infty$ $-\infty$ even negative

– $\infty$ $\infty$ odd positive

$\infty$ – $\infty$ odd negative
Count the $x$ -intercepts, or the zeros of the graph
Count the direction changes and add $1$
The minimum degree is the greater of step $2$ or $3$
Write the $x$ intercepts as factors
Write the function using $a$ as a placeholder for the coefficient
Plug in a point (not on the $x$ axis), and solve for $a$
Rewrite the function