Graphing Polynomials

Topics

Algebra 2

Polynomials

Graph Polynomials

Graphing Polynomials

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Question 1 of 3

Graph

y = \frac{4 x - 10}{x - 3}

$y=(4x-10)/(x-3)$

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Division of Polynomials

$\mathsf{\frac{P}{Divisor}=Quotient+\frac{R}{Divisor}}$

where

$\mathsf{P}$ is the Polynomial and

$\mathsf{R}$ is the Remainder

Long Division

Use long division when a polynomial is divided by a binomial

General Form of a Hyperbola

$y=a/(x-h)+k$

Asymptotes:

$y=k$ and

$x=h$

This polynomial can be written in the General Form of a Hyperbola and then graphed as a hyperbola.

First, divide the fraction using Long Division.

Substitute components into the formula

$\mathsf{P}$ (Polynomial)	$=$	$4x-10$
$\mathsf{Divisor}$	$=$	$x-3$

$=$

Next, solve for the quotient

First term of the quotient:

Divide the first term of the Polynomial by the first term of the Divisor. Place this above the Polynomial

$4xdividex$

$=$

$4$

Multiply $4$ to the divisor. Place this under the Polynomial

$4$

$(x-3)$

$=$

$4x-12$

Subtract $4x-12$ and write the difference one line below

Since

$2$ cannot be divided by the divisor anymore, it is left as a remainder

This also means that the expression at the very top is the Quotient

Combine and substitute the components into the Division of Polynomials formula

$\mathsf{P}$ (Polynomial)	$=$	$4x-10$
$\mathsf{Divisor}$	$=$	$x-3$
$\mathsf{Quotient}$	$=$	$4$
$\mathsf{Remainder}$	$=$	$2$

$\mathsf{\frac{P}{Divisor}}$	$=$	$\mathsf{Quotient+\frac{R}{Divisor}}$	Division of Polynomials

$\frac{4x-10}{x-3}$	$=$	$4+\frac{2}{x-3}$	Substitute

$y$	$=$	$\frac{2}{x-3}+4$

Notice that this expression is in the General Form of a Hyperbola and hence can be graphed as a hyperbola

Identify components of the Hyperbola

$y$	$=$	$a/(x-h)+k$

$y$	$=$	$2/(x-3)+4$

$a$	$=$	$2$
$h$	$=$	$3$
$k$	$=$	$4$

Graph the asymptotes

$x=h$ becomes

$x=3$

$y=k$ becomes

$y=4$

Use sample points to help on graphing the curves

First point using $x=2$ :

$y$	$=$	$2/(x-3)+4$

$y$	$=$	$2/(2-3)+4$	Substitute $x=2$

$y$	$=$	$2/(-1)+4$

$y$	$=$	$-2+4$
$y$	$=$	$2$

Second point using $x=4$ :

$y$	$=$	$2/(x-3)+4$

$y$	$=$	$2/(4-3)+4$	Substitute $x=4$

$y$	$=$	$2/(1)+4$

$y$	$=$	$2+4$
$y$	$=$	$6$

Plot the points $(2,2)$ and $(4,6)$

Finally, graph the curves over the sample points. Make sure that these curves come close but never intersect the asymptotes.

Question 2 of 3