Chapter 3

Describe a translation of the graph that will result in a function with (a) four distinct real zeros, (b) two real zeros, each of multiplicity \(2,(\mathrm{c})\) two real zeros and two imaginary zeros, and (d) four imaginary zeros.

Determine whether the statement is true or false. Justify your answer. The graphs of \(f(x)=-4 x^{2}-10 x+7\) and \(g(x)=12 x^{2}+30 x+1\) have the same axis of symmetry.

Find a polynomial function that has the given zeros. (There are many correct answers.) \(3,2+\sqrt{7}, 2-\sqrt{7}\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=x^{4}-4 x^{3}+16 x-16\) Upper bound: \(x=5\) Lower bound: \(x=-3\)

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x-\frac{2}{x+1}$$

Describe the graph of the function and identify the vertex. $$f(x)=x^{2}-7 x-8$$

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(-2,\) multiplicity: 2 Zero: \(-1,\) multiplicity: 1 Degree: 3

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=2 x^{4}-8 x+3\) Upper bound: \(x=3\) Lower bound: \(x=-4\)

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=2 x-\frac{8}{x}$$

Describe the graph of the function and identify the vertex. $$f(x)=-x^{2}+x+6$$

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(3,\) multiplicity: 1 Zero: \(2,\) multiplicity: 3 Degree: 4

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x+2-\frac{1}{x+1}$$

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=(x-z)^{2}\)

Describe the graph of the function and identify the vertex. $$f(x)=6 x^{2}+5 x-6$$

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(-4,\) multiplicity: 2 Zero: \(3,\) multiplicity: 2 Degree: 4

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{3}{2} x^{2}-\frac{23}{2} x+6=\frac{1}{2}\left(2 x^{3}-3 x^{2}-23 x+12\right)$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=2 x-1+\frac{1}{x-2}$$

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=x^{2}-z\)

Describe the graph of the function and identify the vertex. $$f(x)=4 x^{2}+2 x-12$$

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x+1+\frac{2}{x-1}$$

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=z(x-3)^{2}\)

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(-1,\) multiplicity: 2 Zero: \(-2,\) multiplicity: 1 Degree: 3 Rises to the left, Falls to the right

Find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x+2+\frac{2}{x+2}$$

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=z x^{2}+4\)

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(1,\) multiplicity: 2 Zero: \(4,\) multiplicity: 2 Degree: Falls to the left, Falls to the right

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-1$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x+3-\frac{2}{2 x-1}$$

Find the value of \(b\) such that the function has the given maximum or minimum value. \(f(x)=-x^{2}+b x-75 ;\) Maximum value: 25

Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers.) Third-degree polynomial with two real zeros and a negative leading coefficient.

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-2$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=x-1-\frac{2}{2 x-3}$$

Find the value of \(b\) such that the function has the given maximum or minimum value. \(f(x)=-x^{2}+b x-16 ;\) Maximum value: 48

Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers.) Fourth-degree polynomial with three real zeros and a positive leading coefficient.

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-x$$

A 1000-liter tank contains 50 liters of a \(25 \%\) brine solution. You add \(x\) liters of a \(75 \%\) brine solution to the tank. (a) Show that the concentration \(C\) (the ratio of brine to the total solution) of the final mixture is given by $$C=\frac{3 x+50}{4(x+50)}$$ (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the function. As the tank is filled, what happens to the rate at which the concentration of brine increases? What percent does the concentration of brine appear to approach?

Find the value of \(b\) such that the function has the given maximum or minimum value. \(f(x)=x^{2}+b x+26 ;\) Minimum value: 10

Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers.) Fifth-degree polynomial with three real zeros and a positive leading coefficient.

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-2 x$$

A rectangular region of length \(x\) and width \(y\) has an area of 500 square meters. (a) Write the width \(y\) as a function of \(x .\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch a graph of the function and determine the width of the rectangle when \(x=30\) meters.

Find the value of \(b\) such that the function has the given maximum or minimum value. \(f(x)=x^{2}+b x-25 ;\) Minimum value: -50

Sketch the graph of a polynomial function that satisfies the given conditions. If not possible, explain your reasoning. (There are many correct answers.) Fourth-degree polynomial with two real zeros and a negative leading coefficient

A page that is \(x\) inches wide and \(y\) inches high contains 30 square inches of print (see figure). The margins at the top and bottom are 2 inches deep and the margins on each side are 1 inch wide. (a) Show that the total area \(A\) of the page is given by $$A=\frac{2 x(2 x+11)}{x-2}$$ (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size such that the minimum amount of paper will be used. Verify your answer numerically using the table feature of the graphing utility.

Let \(x\) and \(y\) be two positive real numbers whose sum is \(S .\) Show that the maximum product of \(x\) and \(y\) occurs when \(x\) and \(y\) are both equal to \(S / 2\)

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=x^{3}-9 x\)

Assume that the function \(f(x)=a x^{2}+b x+c\) \(a \neq 0,\) has two real xeros. Show that the \(x\) -coordinate of the vertex of the graph is the average of the zeros of \(f\) (Hint: Use the Quadratic Formula.)

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(g(x)=x^{4}-4 x^{2}\)

The ordering and transportation cost \(C\) (in thousands of dollars) for the components used in manufacturing a product is given by $$C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad x \geq 1$$ where \(x\) is the order size (in hundreds). Use a graphing utility to graph the cost function. From the graph, estimate the order size that minimizes cost.