Question

vocabulary and core concept check

1. writing explain what is meant by the end behavior of a polynomial function.
2. which one doesnt belong? which function does not belong with the other three?

explain your reasoning.
$f(x)=7x^5 + 3x^2 - 2x$
$g(x)=3x^3 - 2x^8 + \frac{5}{4}$
$h(x)= -3x^4 + 5x^{-1} - 3x^2$
$k(x)=\sqrt{3}x + 8x^4 + 2x + 1$
monitoring progress and modeling with mathematics
in exercises 3–8, decide whether the function is a polynomial function. if so, write it in standard form and state its degree, type, and leading coefficient. (see example 1.)

1. $f(x)= -3x + 5x^5 - 6x^2 + 2$
2. $p(x)= \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1$
3. $f(x)=9x^4 + 8x^3 - 6x^{-2} + 2x$
4. $g(x)=\sqrt{3} - 12x + 13x^2$
5. $h(x)= \frac{5}{3}x^2 - \sqrt{7}x^4 + 8x^5 - \frac{1}{2} + x$
6. $h(x)=3x^4 + 2x - \frac{5}{x} + 9x^3 - 7$

error analysis in exercises 9 and 10, describe and correct the error in analyzing the function.

1. $f(x)=8x^3 - 7x^3 - 9x - 3x^2 + 11$

$f$ is a polynomial function.
the degree is 3 and $f$ is a cubic function.
the leading coefficient is 8.

1. $f(x)=2x^4 + 4x - 9\sqrt{x} + 3x^2 - 8$

$f$ is a polynomial function.
the degree is 4 and $f$ is a quartic function.
the leading coefficient is 2.
in exercises 11–16, evaluate the function for the given value of $x$. (see example 2.)

1. $h(x)= -3x^4 + 2x^5 - 12x - 6; x = -2$
2. $f(x)=7x^4 - 10x^2 + 14x - 26; x = -7$
3. $g(x)=x^6 - 64x^3 + x^2 - 7x - 51; x = 8$
4. $g(x)= -x^3 + 3x^2 + 5x + 1; x = -12$
5. $p(x)=2x^3 + 4x^2 + 6x + 7; x = \frac{1}{2}$
6. $h(x)=5x^3 - 3x^2 + 2x + 4; x = -\frac{1}{3}$

in exercises 17–20, describe the end behavior of the graph of the function. (see example 3.)

1. $h(x)= -5x^4 + 7x^3 - 6x^2 + 9x + 2$
2. $g(x)=7x^7 + 12x^3 - 6x^2 - 2x - 18$
3. $f(x)= -2x^4 + 12x^8 + 17 + 15x^2$
4. $f(x)=11 - 18x^2 - 5x^5 - 12x^4 - 2x$

in exercises 21 and 22, describe the degree and leading coefficient of the polynomial function using the graph.

1. graph of a polynomial function
2. graph of a polynomial function
3. using structure determine whether the function is a polynomial function. if so, write it in standard form and state its degree, type, and leading coefficient.

$f(x)=5\sqrt{2}x^3 + \frac{3}{4}x^4 - 9x^4 + \sqrt{2}x^2 + 4x - 1 - x^{-5}x^8 - 4$

1. writing let $f(x)=13$. state the degree, type, and leading coefficient. describe the end behavior of the function. explain your reasoning.

Explanation:

Answer:

1. \( h(x)=-3x^{4}+5x^{-1}-3x^{2} \)
2. Polynomial; \( 5x^{3}-3x^{2}-3x + 2 \); degree \( 3 \); cubic; leading coefficient \( 5 \)
3. Polynomial; \( \frac{1}{2}x^{5}+6x^{4}-4x^{3}+3x - 1 \); degree \( 5 \); quintic; leading coefficient \( \frac{1}{2} \)
4. Polynomial; \( 13x^{2}-12x+\sqrt{3} \); degree \( 2 \); quadratic; leading coefficient \( 13 \)
5. Polynomial; \( \frac{5}{3}x^{4}+8x^{3}-\sqrt{7}x^{2}+x-\frac{1}{2} \); degree \( 4 \); quartic; leading coefficient \( \frac{5}{3} \)
6. \( -3(-2)^{4}+2(-2)^{3}-12(-2)-6=-46 \)
7. \( 7(-7)^{4}-10(-7)^{2}+14(-7)-26 = 16774 \)
8. \( 8^{6}-64(8)^{4}+8^{2}-7(8)-51=-563 \)
9. \( -(-12)^{4}+3(-12)^{2}+5(-12)+1=-20027 \)
10. \( 2(\frac{1}{2})^{3}+4(\frac{1}{2})^{2}+6(\frac{1}{2})+7=\frac{23}{2} \)
11. \( 5(-\frac{1}{3})^{3}-3(-\frac{1}{3})^{2}+2(-\frac{1}{3})+4=\frac{89}{27} \)
12. As \( x\to\infty \), \( h(x)\to-\infty \); as \( x\to-\infty \), \( h(x)\to-\infty \)
13. As \( x\to\infty \), \( g(x)\to\infty \); as \( x\to-\infty \), \( g(x)\to-\infty \)
14. As \( x\to\infty \), \( f(x)\to-\infty \); as \( x\to-\infty \), \( f(x)\to-\infty \)
15. As \( x\to\infty \), \( f(x)\to-\infty \); as \( x\to-\infty \), \( f(x)\to-\infty \)