Chapter 3

What type of model best represents data that follow a parabolic pattern?

Fill in the blank(s). For the rational function \(f(x)=N(x) / D(x),\) if the degree of \(N(x)\) is exactly one more than the degree of \(D(x),\) then the graph of \(f\) has a _______ (or oblique) ______.

Fill in the blank. Functions of the form \(f(x)=N(x) / D(x),\) where \(N(x)\) and \(D(x)\) are polynomials and \(D(x)\) is not the zero polynomial, are called _____.

The _________ of __________ states that if \(f(x)\) is a polynomial of degree \(n(n>0),\) then \(f\) has at least one zero in the complex number system.

Two forms of the Division Algorithm are shown below. Identify and label each part. \(f(x)=d(x) q(x)+r(x) \quad \frac{f(x)}{d(x)}=q(x)+\frac{r(x)}{d(x)}\)

A polynomial function with degree \(n\) and leading coefficient \(a_{n}\) is a function of the form \(f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}, a_{n} \neq 0,\) where \(n\) is _____ and \(a_{n}, a_{n-1}, \ldots, a_{2}, a_{1}, a_{0}\) are _____ members.

Fill in the blank(s). The graph of a polynomial function is _______ , so it has no breaks, holes, or gaps.

Which coefficient of determination indicates a better model for a set of data, \(r^{2}=0.0365\) or \(r^{2}=0.9688 ?\)

Fill in the blank(s). The graph of \(f(x)=1 / x\) has a _______ asymptote at \(x=0\).

Fill in the blank. If \(f(x) \rightarrow \pm \infty\) as \(x \rightarrow a\) from the left (or right), then \(x=a\) is a _____ of the graph of \(f .\)

A quadratic factor that cannot be factored as a product of linear factors containing real numbers is said to be _________ over the _________.

Fill in the blank(s). The rational expression \(p(x) / q(x)\) is called _______ when the degree of the numerator is greater than or equal to that of the denominator.

A _____ function is a second-degree polynomial function, and its graph is called a _____.

Fill in the blank(s). A polynomial function of degree n has at most _______ real zeros and at most _______ relative extrema.

Does the graph of \(f(x)=\frac{x^{3}-1}{x^{2}+2}\) have a slant asymptote?

What feature of the graph of \(y=\frac{9}{x-3}\) can you find by solving \(x-3=0 ?\)

How many linear factors does a polynomial function \(f\) of degree \(n\) have, where \(n>0 ?\)

Is the quadratic function \(f(x)=(x-2)^{2}+3\) written in standard form? Identify the vertex of the graph of \(f\)

Fill in the blank(s). When \(x=a\) is a zero of a polynomial function \(f,\) the following statements are true. (a) \(x=a\) is a ____________ of the polynomial equation \(f(x)=0\). (b) ___________ is a factor of the polynomial \(f(x)\) (c) The point____________ is an \(x\) -intercept of the graph of \(f\).

Using long division, you find that \(f(x)=\frac{x^{2}+1}{x+1}=x-1+\frac{2}{x+1} .\) What is the slant asymptote of the graph of \(f ?\)

Is \(y=\frac{2}{3}\) a horizontal asymptote of the function \(f(x)=\frac{2 x}{3 x^{2}-5} ?\)

Three of the zeros of a fourth-degree polynomial function \(f\) are \(-1,3,\) and \(2 i .\) What is the other zero of \(f ?\)

The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called _______ of _______ .

Does the graph of the quadratic function \(f(x)=-3 x^{2}+5 x+2\) have a relative minimum value at its vertex?

If a zero of a polynomial function \(f\) is of even multiplicity, then the graph of \(f\)____________ the \(x\) -axis, and if the zero is of odd multiplicity, then the graph of \(f\) _____________the \(x\) -axis.

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{1}{x-1}$$

Sketch the graph of the function \(g\) and describe how the graph is related to the graph of \(f(x)=1 / x\) $$g(x)=\frac{-1}{x}+2$$

Match the function with its exact number of zeros. $$f(x)=-2 x^{4}+32$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

Fill in the blank(s). A real number \(c\) is a(n) __________ bound for the real zeros of \(f\) when no zeros are greater than \(c,\) and is a(n) __________ bound when no real zeros of \(f\) are less than \(c .\)

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{5 x}{x-1}$$

Sketch the graph of the function \(g\) and describe how the graph is related to the graph of \(f(x)=1 / x\) $$g(x)=\frac{1}{x-6}$$

Match the function with its exact number of zeros. $$f(x)=x^{5}-x^{3}$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

How many negative real zeros are possible for a polynomial function \(f,\) given that \(f(-x)\) has 5 variations in sign?

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{3 x}{|x-1|}$$

Sketch the graph of the function \(g\) and describe how the graph is related to the graph of \(f(x)=1 / x\) $$g(x)=\frac{1}{x-3}-1$$

Match the function with its exact number of zeros. $$f(x)=x^{3}+3 x^{2}+2 x$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

You divide the polynomial \(f(x)\) by \((x-4)\) and obtain a remainder of \(7 .\) What is \(f(4) ?\)

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{3}{|x-1|}$$

Sketch the graph of the function \(g\) and describe how the graph is related to the graph of \(f(x)=1 / x\) $$g(x)=\frac{-1}{x+2}-4$$

Match the function with its exact number of zeros. $$f(x)=x-14$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. (0,2.1),(1,2.4),(2,2.5),(3,2.8),(4,2.9),(5,3.0) (6,3.0),(7,3.2),(8,3.4),(9,3.5),(10,3.6)

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{3 x^{2}}{x^{2}-1}$$

Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x)+1$$

Confirm that the function has the indicated zeros. $$f(x)=x^{2}+5 ;-\sqrt{5} i, \sqrt{5} i$$

Use long division to divide and use the result to factor the dividend completely. $$\left(x^{2}+5 x+6\right) \div(x+3)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-x^{2}\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. $$\begin{aligned} &(-2,11.0),(-1,10.7),(0,10.4),(1,10.3),(2,10.1)\\\ &(3,9.9),(4,9.6),(5,9.4),(6,9.4),(7,9.2),(8,9.0) \end{aligned}$$

(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 0.5 & \\ \hline 0.9 & \\ \hline 0.99 & \\ \hline 0.999 & \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.5 & \\ \hline 1.1 & \\ \hline 1.01 & \\ \hline 1.001 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 5 & \\ \hline 10 & \\ \hline 100 & \\ \hline 1000 & \\ \hline \end{array}$$ $$f(x)=\frac{4 x}{x^{2}-1}$$

Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x-1)$$

Confirm that the function has the indicated zeros. $$f(x)=x^{3}+9 x ; 0,-3 i, 3 i$$