BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

15th Edition

ISBN: 9781608408405

Author: HOUGHTON MIFFLIN HARCOURT

Publisher: Cengage Learning

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Question

Chapter 4.1, Problem 29E

To determine

To sketch: the graph of the given polynomial function.

Expert Solution & Answer

Explanation of Solution

Given information:

A polynomial function is given as

$h(x)=x4−2x3+3x$

Concept used:

A polynomial function is of the form

$f(x)=anxn+an−1xn−1+...+a1x+a0$

Terms of a polynomial function should be arranged in descending order according to its degree to express it in a standard form and degree of each term should be a positive integer or whole number. The coefficients should be real numbers.

Leading coefficient of a polynomial function is the coefficient of the leading term.

Degree of the polynomial is the degree of leading term or the height degree in the polynomial function.

For the polynomial $f(x)=anxn+an−1xn−1+...+a1x+a0$ the leading coefficient is $an$ and the degree is $n$ .

The end behavior can describe the graph of a polynomial function as $x$ approaches $+∞$ or as $x$ approaches $−∞$ .

The end behavior of a polynomial function can be determined by the leading coefficient and the degree of the polynomial.

If degree is even and leading coefficient is negative.

$f(x)→−∞$ as $x→−∞$ and $f(x)→−∞$ as $x→+∞$ .

If degree is odd and leading coefficient is negative.

$f(x)→∞$ as $x→−∞$ and $f(x)→−∞$ as $x→+∞$ .

If degree is odd and leading coefficient is positive.

$f(x)→−∞$ as $x→−∞$ and $f(x)→∞$ as $x→+∞$ .

If degree is even and leading coefficient is positive.

$f(x)→∞$ as $x→−∞$ and $f(x)→∞$ as $x→+∞$ .

Calculation:

Consider the given polynomial function.

$h(x)=x4−2x3+3x$

Now, tabulate some points as shown below:

$x$	$h(x)=x4−2x3+3x$	$(x,h(x))$
$0$	$h(0)=(0)4−2(0)3+3(0)=0$	$(0,0)$
$1$	$h(1)=(1)4−2(1)3+3(1)=2$	$(1,2)$
$−1$	$h(−1)=(1)4−2(−1)3+3(1)=6$	$(−1,6)$
$2$	$h(2)=(2)4−2(2)3+3(2)=6$	$(2,6)$
$−2$	$h(−2)=(−2)4−2(−2)3+3(−2)=26$	$(−2,26)$

Graph:

Plot the points and draw a smooth curve to sketch the graph.

BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015, Chapter 4.1, Problem 29E

Interpretation:

Degree is even and leading coefficient is positive.

$h(x)→∞$ as $x→−∞$ and $h(x)→∞$ as $x→+∞$ .

Chapter 4.1, Problem 28E

Chapter 4.1, Problem 30E

Chapter 4 Solutions

BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

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