Polynomial and Rational Functions

Explain what the notation $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$means

In Problems 57– 62, find the real zeros of f. If necessary, round to two decimal places.

$f\left(x\right)=x^3+3.2x^2-7.25x-6.3$

The intercepts of the equation $9x^2+4y=36$ are ____________.

Is the expression $4x^3-3.6x^2-\sqrt{2}$ a polynomial? If so, what is its degree?

To graph $y=x^2-4$, you would shift the graph of $y=x^2$_______ a distance of __________ units.

Use graphing utility to approximate (round to two decimal places) the local maximum value and local minimum value of $f\left(x\right)=x^3-2x^2-4x+5$, for $-3<x<3$ .

The x-intercepts of the graph of a function $y=f\left(x\right)$ are the real solutions of the equation $f\left(x\right)=0$ .

If $g\left(5\right)=0$ , what point is on the graph of g ? What is the corresponding x-intercepts of the graph of g ?

The graph of every polynomial function is both ___________ and ____________ .

If $r$ is a real zero of even multiplicity of a function $f$ , then the graph of $f$ _________ (cross\touches) the x-axis at $r$ .

The graph of a power function of the form $f\left(x\right)=x^n$, where $n\ge 2$is an even integer, always contains the points __________, _________ and ___________ .

If $r$ is a solution to the equation $f\left(x\right)=0$, name three additional statements that can be made about $f$ and $r$ assuming $f$ is a polynomial function.

The points at which a graph changes direction (from

increasing to decreasing or decreasing to increasing) are

called _______________.

The graph of the function f(x) =$3x^4-x^3+5x^2-2x-7$ 

will behave like the graph of ________ for large values of |x|.

If $f\left(x\right)=-2x^5+x^3-5x^2+7$, then $\underset{x\to -\infty }{\lim }f\left(x\right)=\_\_\_\_\_\_\_\_$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\_\_\_\_\_\_\_\_$.

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$f\left(x\right)=4x+x^3$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$f\left(x\right)=5x^2+4x^4$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$g\left(x\right)=\frac{1-x^2}{2}$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$h\left(x\right)=3-\frac{1}{2}x$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$f\left(x\right)=1-\frac{1}{x}$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$f\left(x\right)=x(x-1)$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$g\left(x\right)=x^{\frac{3}{2}}-x^2+2$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$h\left(x\right)=\sqrt{x}(\sqrt{x}-1)$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$F\left(x\right)=5x^4-{\mathrm{\pi x}}^3+\frac{1}{2}$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$F\left(x\right)=\frac{x^2-5}{x^3}$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$G\left(x\right)=2{(x-1)}^2(x^2+1)$

In Problems 15–26, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell

why not.

$Q\left(x\right)=-3x^2{(x+2)}^3$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)={\left(x+1\right)}^4$

In Problems 27– 40, use transformations of the graph of $y=x^4$or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)={\left(x-2\right)}^5$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=x^5-3$

In Problems 27– 40, use transformations of the graph of $y=x^4$or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=x^4+2$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=\frac{1}{2}{x}^4$

In Problems 27– 40, use transformations of the graph of $y=x^4$  or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=3x^5$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=-x^5$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.$f\left(x\right)=-x^4$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)={\left(x-1\right)}^5+2$

In Problems 27– 40, use transformations of the graph of $y=x^4$ or $y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)={\left(x+2\right)}^4-3$

In Problems 27– 40, use transformations of the graph of $y=x^4 or y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=2{\left(x+1\right)}^4+1$

In Problems 27– 40, use transformations of the graph of $y=x^4 or y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=\frac{1}{2}{\left(x-1\right)}^5-2$

In Problems 27– 40, use transformations of the graph of $y=x^4 or y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=4-{\left(x-2\right)}^5$

In Problems 27– 40, use transformations of the graph of $y=x^4 or y=x^5$ to graph each function. Verify your results using a graphing utility.

$f\left(x\right)=3-{\left(x+2\right)}^4$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

Zeros:$-1,1,3$; Degree 3

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-2,2,3; degree 3$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-3,0,4, degree 3$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-4,0,2, degree 3$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-4,-1,2,3; degree 4$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-3,-1,2,5; degree 4$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-1,multiplicity 1,3, multiplicity 2, degree 3$

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-2,multiplicity 2,4, multiplicity 1, degree 3$

In Problems 49– 60, for each polynomial function:

(a) List each real zero and its multiplicity. 

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept. 

(c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $\left|x\right|$

$f\left(x\right)=3(x-7){(x+3)}^2$