Writing to Learn Determine whether has a zero outside the

In Exercises 1 and 2, write an equation for the linear function ƒ satisfying the given conditions. Graph \(y=f(x)\). \f(-3)=-2\) and \(f(4)=-9\)

In Exercises 1 and 2, write an equation for the linear function ƒ satisfying the given conditions. Graph \(y=f(x)\). \(f(-3)=6\) and \(f(1)=-2\)

In Exercises 3 and 4, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch the graph by hand and support your answer with a grapher. \(h(x)=3(x-2)^{2}+4\)

In Exercises 3 and 4, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch the graph by hand and support your answer with a grapher. \(g(x)=-(x+3)^{2}+1\)

In Exercises 5–8, find the vertex and axis of the graph of the function. Support your answer graphically. \(f(x)=-2(x+3)^{2}+5\)

In Exercises 5–8, find the vertex and axis of the graph of the function. Support your answer graphically. \(g(x)=4(x-5)^{2}-7\)

In Exercises 5–8, find the vertex and axis of the graph of the function. Support your answer graphically. \(f(x)=-2 x^{2}-16 x-31\)

In Exercises 5–8, find the vertex and axis of the graph of the function. Support your answer graphically. \(g(x)=3 x^{2}-6 x+2\)

In Exercises 9 and 10, write an equation for the quadratic function whose graph contains the given vertex and point. Vertex \((-2,-3)\), point \((1,2)\)

In Exercises 9 and 10, write an equation for the quadratic function whose graph contains the given vertex and point. Vertex \((-1,1)\), point \((3,-2)\)

In Exercises 11 and 12, write an equation for the quadratic function with graph shown, given one of the labeled points is the vertex of the parabola.

In Exercises 11 and 12, write an equation for the quadratic function with graph shown, given one of the labeled points is the vertex of the parabola.

In Exercises 13–16, graph the function in a viewing window that shows all of its extrema and x-intercepts. \(f(x)=x^{2}+3 x-40\)

In Exercises 13–16, graph the function in a viewing window that shows all of its extrema and x-intercepts. \(f(x)=-8 x^{2}+16 x-19\)

In Exercises 13–16, graph the function in a viewing window that shows all of its extrema and x-intercepts. \(f(x)=x^{3}+x^{2}+x+5\)

In Exercises 13–16, graph the function in a viewing window that shows all of its extrema and x-intercepts. \(f(x)=x^{3}-x^{2}-20 x-2\)

In Exercises 17 and 18, write the statement as a power function equation. Let k be the constant of variation. The surface area S of a sphere varies directly as the square of the radius r.

In Exercises 17 and 18, write the statement as a power function equation. Let k be the constant of variation. The force of gravity F acting on an object is inversely proportional to the square of the distance d from the object to the center of the earth.

In Exercises 19 and 20, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. F = kx ,where F is the force it takes to stretch a spring x units from its unstressed length and k is the spring’s force constant.

In Exercises 19 and 20, write a sentence that expresses the relationship in the formula, using the language of variation or proportion. \(A=\pi \cdot r^{2}\), where A and r are the area and radius of a circle and is the usual mathematical constant.

In Exercises 21–24, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=4 x^{1 / 3}\)

In Exercises 21–24, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=-2 x^{3 / 4}\)

In Exercises 21–24, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=-2 x^{-3}\)

In Exercises 21–24, state the values of the constants k and a for the function \(f(x)=k \cdot x^{a}\). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether ƒ is even, odd, or undefined for \(x<0\). Describe the rest of the curve if any. Graph the function to see whether it matches the description. \(f(x)=(2 / 3) x^{-4}\)

In Exercises 25–28, divide by \(f(x) \text { by } d(x)\), and write a summary statement in polynomial form. \(f(x)=2 x^{3}-7 x^{2}+4 x-5 ; d(x)=x-3\)

In Exercises 25–28, divide by \(f(x) \text { by } d(x)\), and write a summary statement in polynomial form. \(f(x)=x^{4}+3 x^{3}+x^{2}-3 x+3 ; d(x)=x+2\)

In Exercises 25–28, divide by \(f(x) \text { by } d(x)\), and write a summary statement in polynomial form. \(f(x)=2 x^{4}-3 x^{3}+9 x^{2}-14 x+7 ; d(x)=x^{2}+4\)

In Exercises 25–28, divide by \(f(x) \text { by } d(x)\), and write a summary statement in polynomial form. \(f(x)=3 x^{4}-5 x^{3}-2 x^{2}+3 x-6 ; d(x)=3 x+1\)

In Exercises 29 and 30, use the Remainder Theorem to find the remainder when f (x) is divided by \(x-k\). Check by using synthetic division. \(f(x)=3 x^{3}-2 x^{2}+x-5 ; k=-2\)

In Exercises 29 and 30, use the Remainder Theorem to find the remainder when f (x) is divided by \(x-k\). Check by using synthetic division. \(f(x)=-x^{2}+4 x-5 ; k=3\)

In Exercises 31 and 32, use the Factor Theorem to determine whether the first polynomial is a factor of the second polynomial. \(x-2 ; x^{3}-4 x^{2}+8 x-8\)

In Exercises 31 and 32, use the Factor Theorem to determine whether the first polynomial is a factor of the second polynomial. \(x+3 ; x^{3}+2 x^{2}-4 x-2\)

In Exercises 33 and 34, use synthetic division to prove that the number k is an upper bound for the real zeros of the function ƒ. \(k=5 ; f(x)=x^{3}-5 x^{2}+3 x+4\)

In Exercises 33 and 34, use synthetic division to prove that the number k is an upper bound for the real zeros of the function ƒ. \(k=4 ; f(x)=4 x^{4}-16 x^{3}+8 x^{2}+16 x-12\)

In Exercises 35 and 36, use synthetic division to prove that the number k is a lower bound for the real zeros of the function ƒ. \(k=-3 ; f(x)=4 x^{4}+4 x^{3}-15 x^{2}-17 x-2\)

In Exercises 35 and 36, use synthetic division to prove that the number k is a lower bound for the real zeros of the function ƒ. \(k=-3 ; f(x)=2 x^{3}+6 x^{2}+x-6\)

In Exercises 37 and 38, use the Rational Zeros Theorem to write a list of all potential rational zeros. Then determine which ones, if any, are zeros. \(f(x)=2 x^{4}-x^{3}-4 x^{2}-x-6\)

In Exercises 37 and 38, use the Rational Zeros Theorem to write a list of all potential rational zeros. Then determine which ones, if any, are zeros. \(f(x)=6 x^{3}-20 x^{2}+11 x+7\)

In Exercises 39–42, perform the indicated operation, and write the result in the form \(a+b i\). \((1+i)^{3}\)

In Exercises 39–42, perform the indicated operation, and write the result in the form \(a+b i\). \((1+2 i)^{2}(1-2 i)^{2}\)

In Exercises 39–42, perform the indicated operation, and write the result in the form \(a+b i\). \(i^{29}\)

In Exercises 39–42, perform the indicated operation, and write the result in the form \(a+b i\). \(\sqrt{-16}\)

In Exercises 43 and 44, solve the equation. \(x^{2}-6 x+13=0\)

In Exercises 43 and 44, solve the equation. \(x^{2}-2 x+4=0\)

In Exercises 45–48, match the polynomial function with its graph. Explain your choice. \(f(x)=(x-2)^{2}\)

In Exercises 45–48, match the polynomial function with its graph. Explain your choice. \(f(x)=(x-2)^{3}\)

In Exercises 45–48, match the polynomial function with its graph. Explain your choice. \(f(x)=(x-2)^{4}\)

In Exercises 45–48, match the polynomial function with its graph. Explain your choice. \(f(x)=(x-2)^{5}\)

In Exercises 49–52, find all of the real zeros of the function, finding exact values whenever possible. Identify each zero as rational or irrational. State the number of nonreal complex zeros. \(f(x)=x^{4}-10 x^{3}+23 x^{2}\)

In Exercises 49–52, find all of the real zeros of the function, finding exact values whenever possible. Identify each zero as rational or irrational. State the number of nonreal complex zeros. \(k(t)=t^{4}-7 t^{2}+12\)

In Exercises 49–52, find all of the real zeros of the function, finding exact values whenever possible. Identify each zero as rational or irrational. State the number of nonreal complex zeros. \(h(x)=x^{3}-2 x^{2}-8 x+5\)

In Exercises 49–52, find all of the real zeros of the function, finding exact values whenever possible. Identify each zero as rational or irrational. State the number of nonreal complex zeros. \(k(x)=x^{4}-x^{3}-14 x^{2}+24 x+5\)

In Exercises 53–56, find all of the zeros and write a linear factorization of the function. \(f(x)=2 x^{3}-9 x^{2}+2 x+30\)

In Exercises 53–56, find all of the zeros and write a linear factorization of the function. \(f(x)=5 x^{3}-24 x^{2}+x+12\)

In Exercises 53–56, find all of the zeros and write a linear factorization of the function. \(f(x)=6 x^{4}+11 x^{3}-16 x^{2}-11 x+10\)

In Exercises 53–56, find all of the zeros and write a linear factorization of the function. \(f(x)=x^{4}-8 x^{3}+27 x^{2}-50 x+50\), given that \(1+2 i\) is a zero.

In Exercises 57–60, write the function as a product of linear and irreducible quadratic factors all with real coefficients. \(f(x)=x^{3}-x^{2}-x-2\)

In Exercises 57–60, write the function as a product of linear and irreducible quadratic factors all with real coefficients. \(f(x)=9 x^{3}-3 x^{2}-13 x-1\)

In Exercises 57–60, write the function as a product of linear and irreducible quadratic factors all with real coefficients. \(f(x)=2 x^{4}-9 x^{3}+23 x^{2}-31 x+15\)

In Exercises 57–60, write the function as a product of linear and irreducible quadratic factors all with real coefficients. \(f(x)=3 x^{4}-7 x^{3}-3 x^{2}+17 x+10\)

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 3; zeros: \(\sqrt{5},-\sqrt{5}, 3\)

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 2; -3 only real zero

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 4; zeros: 3, \(-2,1 / 3,-1 / 2\)

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 3; zeros: \(1+i, 2\)

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 4; zeros: -2 (multiplicity 2), 4(multiplicity 2)

In Exercises 61–66, write a polynomial function with real coefficients whose zeros and their multiplicities include those listed. Degree 3; zeros: \(2-i,-1 \text {, and } f(2)=6\)

In Exercises 67 and 68, describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function \(f(x)=1 / x\). Identify the horizontal and vertical asymptotes. \(f(x)=\frac{-x+7}{x-5}\)

In Exercises 67 and 68, describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function \(f(x)=1 / x\). Identify the horizontal and vertical asymptotes. \(f(x)=\frac{3 x+5}{x+2}\)

In Exercises 69–72, find the asymptotes and intercepts of the function, and graph it. \(f(x)=\frac{x^{2}+x+1}{x^{2}-1}\)

In Exercises 69–72, find the asymptotes and intercepts of the function, and graph it. \(f(x)=\frac{2 x^{2}+7}{x^{2}+x-6}\)

In Exercises 69–72, find the asymptotes and intercepts of the function, and graph it. \(f(x)=\frac{x^{2}-4 x+5}{x+3}\)

In Exercises 69–72, find the asymptotes and intercepts of the function, and graph it. \(g(x)=\frac{x^{2}-3 x-7}{x+3}\)

In Exercises 73–74, find the intercepts, analyze, and graph the given rational function. \(f(x)=\frac{x^{3}+x^{2}-2 x+5}{x+2}\)

In Exercises 73–74, find the intercepts, analyze, and graph the given rational function. \(f(x)=\frac{-x^{4}+x^{2}+1}{x-1}\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(2 x+\frac{12}{x}=11\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(\frac{x}{x+2}+\frac{5}{x-3}=\frac{25}{x^{2}-x-6}\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(2 x^{3}+3 x^{2}-17 x-30<0\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(3 x^{4}+x^{3}-36 x^{2}+36 x+16 \geq 0\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(\frac{x+3}{x^{2}-4} \geq 0\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(\frac{x^{2}-7}{x^{2}-x-6}<1\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \((2 x-1)^{2}|x+3| \leq 0\)

In Exercises 75–82, solve the equation or inequality algebraically, and support graphically. \(\frac{(x-1)|x-4|}{\sqrt{x+3}}>0\)

Writing to Learn Determine whether \(f(x)=x^{5}-10 x^{4}-3 x^{3}+28 x^{2}+20 x-2\) has a zero outside the viewing window. Explain. (See graph.)

Launching a Rock Larry uses a slingshot to launch a rock straight up from a point 6 ft above level ground with an initial velocity of 170 ft/sec. (a) Find an equation that models the height of the rock t seconds after it is launched and graph the equation. (See Example 8 in Section 2.1.) (b) What is the maximum height of the rock? When will it reach that height? (c) When will the rock hit the ground?

Volume of a Box Villareal Paper Co. has contracted to manufacture a box with no top that is to be made by removing squares of width x from the corners of a 30-in. by 70-in. Piece of cardboard. (a) Find an equation that models the volume of the box. (b) Determine x so that the box has a volume of \(5800 \text { in. }^{3}\).

Architectural Engineering DeShanna, an engineer at J. P. Cook, Inc., completes structural specifications for a 255-ft-long steel beam anchored between two pilings 50 ft above ground, as shown in the figure. She knows that when a 250-lb object is placed d feet from the west piling, the beam bends s feet where \(s=\left(8.5 \times 10^{-7}\right) d^{2}(255-d)\). (a) Graph the function s. (b) What are the dimensions of a viewing window that shows a graph for the values that make sense in this problem situation? (c) What is the greatest amount of vertical deflection s, and where does it occur? (d) Writing to Learn Give a possible scenario explaining why the solution to part (c) does not occur at the halfway point.

Storage Container A liquid storage container on a truck is in the shape of a cylinder with hemispheres on each end as shown in the figure. The cylinder and hemispheres have the same radius. The total length of the container is 140 ft. (a) Determine the volume V of the container as a function of the radius x. (b) Graph the function \(y=V(x)\). (c) What is the radius of the container with the largest possible volume? What is the volume?

Pell Grants The maximum loan permitted under the federal student-aid program is given in Table 2.24 for several years. Let x = 0 represent 1990, x = 1 represent 1991, and so forth. (a) Find a quadratic regression model for the Pell Grant amounts, and graph it together with a scatter plot of the data. (b) Find a quartic regression model for the Pell Grant amounts, and graph it together with a scatter plot of the data. (c) Use each regression equation to predict the amount of a Pell Grant in 2006. (d) Writing to Learn Determine the end behavior of the two regression models. What does the end behavior say about future Pell Grant amounts?

National Institutes of Health Spending Table 2.25 shows the spending at the National Institutes of Health for several years. Let x = 0 represent 1990, x = 1 represent 1991, and so forth. (a) Find a linear regression model, and graph it together with a scatter plot of the data. (b) Find a quadratic regression model, and graph it together with a scatter plot of the data. (c) Use the linear and quadratic regression models to estimate when the amount of spending will exceed $30 billion.

Breaking Even Midtown Sporting Goods has determined that it needs to sell its soccer shinguards for $5.25 a pair in order to be competitive. It costs $4.32 to produce each pair of shinguards, and the weekly overhead cost is $4000. (a) Express the average cost that includes the overhead of producing one shinguard as a function of the number x of shinguards produced each week. (b) Solve algebraically to find the number of shinguards that must be sold each week to make $8000 in profit. Support your work graphically.

Deer Population The number of deer P at any time t(in years) in a federal game reserve is given by \(P(t)=\frac{800+640 t}{20+0.8 t}\). (a) Find the number of deer when t is 15, 70, and 100. (b) Find the horizontal asymptote of the graph of \(y=P(t)\). (c) According to the model, what is the largest possible deer population?

Resistors The total electrical resistance R of two resistors connected in parallel with resistances \(R_{1} \text { and } R_{2}\) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). The total resistance is 1.2 ohms. Let \(x=R_{1}\). (a) Express the second resistance \(R_{2}\) as a function of x. (b) Find \(R_{2}\) if \(x_{1}\) is 3 ohms.

Acid Mixture Suppose that x ounces of distilled water are added to 50 oz of pure acid. (a) Express the concentration of the new mixture as a function of x. (b) Use a graph to determine how much distilled water should be added to the pure acid to produce a new solution that is less than 60% acid. (c) Solve (b) algebraically.

Industrial Design Johnson Cannery will pack peaches in 1-L cylindrical cans. Let x be the radius of the base of the can in centimeters. (a) Express the surface area S of the can as a function of x. (b) Find the radius and height of the can if the surface area is \(900 \mathrm{cm}^{2}\). (c) What dimensions are possible for the can if the surface area is to be less than \(900 \mathrm{cm}^{2}\)?

Industrial Design Gilman Construction is hired to build a rectangular tank with a square base and no top. The tank is to hold \(1000 \mathrm{ft}^{3}\) of water. Let x be a length of the base. (a) Express the outside surface area S of the tank as a function of x. (b) Find the length, width, and height of the tank if the outside surface area is \(600 \mathrm{ft}^{2}\). (c) What dimensions are possible for the tank if the outside surface area is to be less than \(600 \mathrm{ft}^{2}\)?

If you collected motion data using a \(\mathrm{CBL}^{\mathrm{TM}} \text { or } \mathrm{CBR}^{\mathrm{TM}}\), a plot of height versus time or distance versus time should be shown on your grapher or computer screen. Either plot will work for this project. If you do not have access to a CBL or CBR, enter the data from the table provided into your grapher or computer. Create a scatter plot for the data.

Find values for a, h, and k so that the equation \(y=a(x-h)^{2}+k\) fits one of the bounces contained in the data plot. Approximate the vertex 1h, k2 from your data plot and solve for the value of an algebraically.

Change the values of a, h, and k in the model found above and observe how the graph of the function is affected on your grapher or computer. Generalize how each of these changes affects the graph.

Expand the equation you found in #2 above so that it is in the standard quadratic form: \(y=a x^{2}+ bx+c\).

Use your grapher or computer to select the data from the bounce you modeled above and then use quadratic regression to find a model for this data set. (See your grapher’s guidebook for instructions on how to do this.) How does this model compare with the standard quadratic form found in #4?

Complete the square to transform the regression model to the vertex form of a quadratic, and compare it to the original vertex model found in #2. (Round the values of a, b, and c to the nearest 0.001 before completing the square if desired.)