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All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f( x )=2 x 2 +2x3In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f(x)=3 x 2 +3x2In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f( x )=3 x 2 +6x+2In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f( x )=2 x 2 +5x+3In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f(x)=4 x 2 6x+2In Problems 33-48, (a) graph each quadratic function by determining whether Us graph opens up or down and by finding its vertex, axis of symmetry, y-intercept,andx-intercepts , if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing. Verify your results using a graphing utility. f(x)=3 x 2 8x+2In Problems 49-54, determine the quadratic function whose graph is given.In Problems 49-54, determine the quadratic function whose graph is given.In Problems 49-54, determine the quadratic function whose graph is given.In Problems 49-54, determine the quadratic function whose graph is given.In Problems 49-54, determine the quadratic function whose graph is given.In Problems 49-54, determine the quadratic function whose graph is given.In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x)=2 x 2 +12xIn Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f( x )=2 x 2 +12xIn Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f( x )=2 x 2 +12x3In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f( x )=4 x 2 8x+3In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x)= x 2 +10x4In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x)=2 x 2 +8x+3In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f( x )=3 x 2 +12x+1In Problems 55-62, determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f( x )=4 x 2 4xIn Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. f( x )= x 2 2x15In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. g(x)= x 2 2x8In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. F( x )=2x5In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. f( x )= 3 2 x2In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. g( x )=2 (x3) 2 +2In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. h( x )=3 ( x+1 ) 2 +4In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. f( x )=2 x 2 +x+1In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. G(x)=3 x 2 +2x+5In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. h(x)= 2 5 x+4In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. f(x)=3x+2In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. H( x )=4 x 2 4x1In Problems 63-74, (a) graph each function, (b) determine the domain and the range of the function, and (c) determine where the function is increasing and where it is decreasing. F( x )=4 x 2 +20x25The graph of the function f( x )=a x 2 +bx+c has vertex at ( 0,2 ) and passes through the point ( 1,8 ) . Find a,b,andc .The graph of the function f(x)=a x 2 +bx+c has vertex at ( 1,4 ) and passes through the point (1,8) . Find a,b,andc .In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f( x )=2x1;g(x)= x 2 4In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f(x)=2x1;g(x)= x 2 9In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f( x )= x 2 +4;g( x )=2x+1In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f( x )= x 2 +9;g( x )=2x+1In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f( x )= x 2 +5x;g(x)= x 2 +3x4In Problems 77-82, for the given functions fandg , (a) Graph fandg on the same Cartesian plane. (b) Solve f( x )=g( x ) . (c) Use the result of part (b) to label the points of intersection of the graphs of fandg . Shade the region for which f( x )g( x ) , that is, the region below f and above g . f( x )= x 2 +7x6;g( x )= x 2 +x6Answer Problems 83 and 84 using the following: A quadratic function of the form f( x )=a x 2 +bx+c with b 2 4ac0 may also be written in the form f( x )=a(x r 1 )(x r 2 ) , where r 1 and r 2 are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are 3 and 1 with a=1;a=2;a=2;a=5 . (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts . What might you conclude?Answer Problems 83 and 84 using the following: A quadratic function of the form f( x )=a x 2 +bx+c with b 2 4ac0 may also be written in the form f( x )=a(x r 1 )(x r 2 ) , where r 1 and r 2 are the x-intercepts of the graph of the quadratic function. (a) Find a quadratic function whose x-intercepts are 5 and 3 with a=1;a=2;a=2;a=5 . (b) How does the value of a affect the intercepts? (c) How does the value of a affect the axis of symmetry? (d) How does the value of a affect the vertex? (e) Compare the x-coordinate of the vertex with the midpoint of the x-intercepts . What might you conclude?Suppose that f(x)= x 2 +4x21 . (a) What is the vertex of f ? (b) What are the x-intercepts of the graph of f ? (c) Solve f(x)=21 for x . What points are on the graph of f ? (d) Use the information obtained in parts (a)(c) to graph f(x)= x 2 +4x21 .Suppose that f( x )= x 2 +2x8 . (a) What is the vertex of f ? (b) What are the x-intercepts of the graph of f ? (c) Solve f( x )=8 for x . What points are on the graph of f ? (d) Use the information obtained in parts (a)(c) to graph f( x )= x 2 +2x8 .Find the point on the line y=x that is closest to the point ( 3,1 ) . [Hint: Express the distance d from the point to the line as a function of x , and then find the minimum value of [d(x)] 2 .]Find the point on the line y=x+1 that is closest to the point ( 4,1 ) .Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is R( p )=4 p 2 +4000p What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?Maximizing Revenue The John Deere company has found that the revenue, in dollars, from sales of riding mowers is a function of the unit price p , in dollars, that it charges. If the revenue R is R( p )= 1 2 p 2 +1900p what unit price p should be charged to maximize revenue? What is the maximum revenue?Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is 6.20 , it cost 6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C (in dollars) to produce x thousand digital music players is given by the function C(x)= x 2 140x+7400 (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?Minimizing Marginal Cost (See Problem 91.) The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given by C( x )=5 x 2 200x+4000 (a) How many cell phones should be manufactured to minimize the marginal cost? (b) What is the minimum marginal cost?Business The monthly revenue R achieved by selling x wristwatches is figured to be R( x )=75x0.2 x 2 . The monthly cost C of selling x wristwatches is C( x )=32x+1750 . (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as P(x)=R( x )C( x ) . What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a)and(c) differ. Explain why a quadratic function is a reasonable model for revenue.Business The daily revenue R achieved by selling x boxes of candy is figured to be R( x )=9.5x0.04 x 2 . The daily cost C of selling x boxes of candy is C( x )=1.25x+250 . (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as P(x)=R(x)C( x ) . What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a)and(c) differ. Explain why a quadratic function is a reasonable model for revenue.Stopping Distance An accepted relationship between stopping distance, d (in feet), and the speed of a car, v (in mph), is d=1.1v+0.06 v 2 on dry, level concrete. (a) How many feet will it take a car traveling 45 mph to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you can be traveling to avoid being involved? (c) What might the term 1.1v represent?Birthrate for Unmarried Women In the United States, the birthrate B for unmarried women (births per 1000 unmarried women) whose age is a is modeled by the function B( a )=0.30 a 2 +16.26a158.90 . (a) What is the age of unmarried women with the highest birthrate? (b) What is the highest birthrate of unmarried women? (c) Evaluate and interpret B( 40 ) . Source: National Vital Statistics System, 2013Let f( x )=a x 2 +bx+c , where a,b,andc are odd integers. If x is an integer, show that f( x ) must be an odd integer. [Hint: x is either an even integer or an odd integer.]Make up a quadratic function that opens down and has only one x-intercept . Compare yours with others in the class. What are the similarities? What are the differences?On one set of coordinate axes, graph the family of parabolas f( x )= x 2 +2x+c=3,c=0andc=1 . Describe the characteristics of a member of this family.On one set of coordinate axes, graph the family of parabolas f( x )= x 2 +bx+1 for b=4,b=0,andb=4 . Describe the general characteristics of this family.State the circumstances that cause the graph of a quadratic function f( x )=a x 2 +bx+c to have no x-intercepts .Why does the graph of a quadratic function open up if a0 and down if a0 ?Can a quadratic function have a range of ( , ) ? Justify your answer.What are the possibilities for the number of times the graphs of two different quadratic functions intersect?Translate the following sentence into a mathematical equation: The total revenue R from selling x hot dogs is 3 times the number of hot dogs sold. (p. A66)Use a graphing utility to find the line of best fit for the following data: (pp. 143-144)Maximizing Revenue The price p (in dollars) and the quantity x sold of a certain product obey the demand equation p= 1 6 x+100 (a) Find a model that expresses the revenue R as a function of x . (Remember, R=xp .) (b) What is the domain of R ? (c) What is the revenue if 200 units are sold? (d) What quantity x maximizes revenue? What is the maximum revenue? (e) What price should the company charge to maximize revenue?Maximizing Revenue The price p (in dollars) and the quantity A sold of a certain product obey the demand equation p= 1 3 x+100 (a) Find a model that expresses the revenue R as a function of x . (b) What is the domain of R ? (c) What is the revenue if 100 units are sold? (d) What quantity x maximizes revenue? What is the maximum revenue? (e) What price should the company charge to maximize revenue?Maximizing Revenue The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x=5p+100 , 0p20 (a) Express the revenue R as a function of x . (b) What is the revenue if 15 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue? (e) What price should the company charge to maximize revenue?Maximizing Revenue The price p (in dollars) and the quantity x sold of a certain product obey the demand equation x=20p+500 , 0p25 (a) Express the revenue R as a function of x . (b) What is the revenue if 20 units are sold? (c) What quantity x maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue? (e) What price should the company charge to earn at least 3000 in revenue?Enclosing a Rectangular Field David has 400 yards of fencing and wishes to enclose a rectangular area. (a) Express the area A of the rectangle as a function of the width w of the rectangle. (b) For what value of w is the area largest? (c) What is the maximum area?Enclosing a Rectangular Field Beth has 3000 feet of fencing available to enclose a rectangular field. (a) Express the area A of the rectangle as a function of x . where x is the length of the rectangle. (b) For what value of x is the area largest? (c) What is the maximum area?Enclosing a Rectangular Field with a Fence A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? (See the figure.)Enclosing a Rectangular Field with a Fence A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?Analyzing the Motion of a Projectile A projectile is fired from a cliff 200 feet above the water at an inclination of 45 to the horizontal, with a muzzle velocity of 50 feet per second. The height h of the projectile above the water is modeled by h( x )= 32 x 2 ( 50 ) 2 +x+200 where x is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Using a graphing utility, graph the function h , 0x200 . (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?Analyzing the Motion of a Projectile A projectile is fired at an inclination of 45 to the horizontal, with a muzzle velocity of 100 feet per second. The height h of the projectile is modeled by h( x )= 32 x 2 ( 100 ) 2 +x where x is the horizontal distance of the projectile from the firing point. (a) At what horizontal distance from the firing point is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the firing point will the projectile strike the ground? (d) Using a graphing utility, graph the function h 0x350 . (e) Use a graphing utility to verify the results obtained in parts (b) and (c). (f) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontally?Suspension Bridge A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.)Architecture A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Choose suitable rectangular coordinate axes and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center.Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90 . See the illustration. (a) What depth will provide maximum cross-sectional area and hence allow the most water to flow? (b) What depths will allow at least 16 square inches of water to flow?Norman Windows A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. See the picture. If the perimeter of the window is 20 feet, what dimensions will admit the most light (maximize the area)? [Hint: Circumference of a circle =2r; area of a circle = r 2 . where r is the radius of the circle.]Constructing a Stadium A track-and-field playing area is in the shape of a rectangle with semicircles at each end. See the figure (top, right).The inside perimeter of the track is to the 1500 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?Architecture A special window has the shape of a rectangle surmounted by an equilateral triangle. See the figure. If the perimeter of the window is 16 feet, what dimensions will admit the most light? [Hint: Area of an equilateral triangle =( 3 4 ) x 2 , where x is the length of a side of the triangle.]Chemical Reactions A self-catalytic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate V is modeled by V( x )=kx( ax ),0xa where k is a positive constant, a is the initial amount of the compound, and x is the variable amount of the compound, for what value of x is the reaction rate a maximum?Calculus: Simpson's Rule The figure shows the graph of y=a x 2 +bx+c . Suppose that the points ( h, y 0 ),( 0, y 1 ), and (h, y 2 ) are on the graph. It can be shown that the area enclosed by the parabola, the x-axis , and the lines x=h and x=h is Area= h 3 (2a h 2 +6c) Show that this area may also be given by Area= h 3 (y0+4y1+y2)Use the result obtained in Problem 20 to find the area enclosed by f(x)=5 x 2 +8 , the x-axis , and the lines or x=1 and x=1 .Use the result obtained in Problem 20 to find the area enclosed by f(x)=2 x 2 +8 , the x-axis , and the lines x=2 and x=2 .Use the result obtained in Problem 20 to find the area enclosed by f( x )= x 2 +3x+5 , the x-axis , and the lines x=4 and x=4 .Use the result obtained in Problem 20 to find the area enclosed by f( x )= x 2 +x+4 , the x-axis , and the lines x=1 and x=1 .Life Cycle Hypothesis An individuals income varies with his or her age. The following table shows the median income I of males of different age groups within the United States for 2012. For each age group, let the class midpoint represent the independent variable, x . For the class 65 years and older, we will assume that the class midpoint is 69.5 . (a) Use a graphing utility to draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income. (c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income. (d) Use the function found in part (b) to predict the peak income earned. (e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.Height of a Rail A shot-putter throws a hall at an inclination of 45 to the horizontal. The following data represent the height of the ball h , in feet, at the instant that it has traveled x feet horizontally: (a) Use a graphing utility to draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables. (b) Use a graphing utility to find the quadratic function of best fit that models the relation between distance and height. (c) Use the function found in part (b) to determine how far the ball will travel before it reaches its maximum height. (d) Use the function found in part (b) to find the maximum height of the ball. (e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.Which Model? The following data represent the square footage and rents (dollars per month) for apartments in the La Jolla area of San Diego, California. (a) Using a graphing utility, draw a scatter diagram of the data treating square footage as the independent variable. What type of relation appears to exist between square footage and rent? (b) Based on your response to part (a), find either a linear or a quadratic model that describes the relation between square footage and rent. (c) Use your model to predict the rent for an apartment in San Diego that is 875 square feet.Which Model? An engineer collects the following data showing the speed s of a Toyota Camry and its average miles per gallon, M . (a) Using a graphing utility, draw a scatter diagram of the data, treating speed as the independent variable. What type of relation appears to exist between speed and miles per gallon? (b) Based on your response to part (a), find either a linear model or a quadratic model that describes the relation between speed and miles per gallon. (c) Use your model to predict the miles per gallon for a Camry that is traveling 63 miles per hour.Which Model? The following data represent the birth rate (births per 1000 population) for women whose age is a , in 2012. (a) Using a graphing utility, draw a scatter diagram of the data, treating age as the independent variable. What type of relation appears to exist between age and birth rate? (b) Based on your response to part (a), find either a linear or a quadratic model that describes the relation between age and birthrate, (c) Use your model to predict the birth rate for 35-year- old women.Which Model? A cricket makes a chirping noise by sliding its wings together rapidly. Perhaps you have noticed that the rapidity of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket. (a) Using a graphing utility, draw a scatter diagram of the data, treating temperature as the independent variable. What type of relation appears to exist between temperature and chirps per second? (b) Based on your response to part (a), find either a linear or a quadratic model that best describes the relation between temperature and chirps per second. Use your model to predict the chirps per second if the temperature is 80F .Refer to Example 1 in this section. Notice that if the price charged for the calculators is 0 or 140 , then the revenue is 0 . It is easy to explain why revenue would be 0 if the price charged were 0 , but how can revenue be 0 if the price charged 140 ?Solve the inequality 3x27 .Write (2,7] using inequality notation.(a) f( x )0 (b) f( x )0(a) g( x )0 (b) g( x )0(a) g( x )f( x ) (b) f( x )g( x )(a) f( x )g( x ) (b) f( x )g( x )x 2 3x100x 2 +3x100x 2 4x0x 2 +8x0x 2 90x 2 10x 2 +x12x 2 +7x122 x 2 5x+36 x 2 6+5xx 2 x+10x 2 +2x+404 x 2 +96x25 x 2 +1640x6( x 2 1 )5x2( 2 x 2 3x )9What is the domain of the function f( x )= x 2 16 ?What is the domain of the function f( x )= x3 x 2 ?In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 1 g( x )=3x+3In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 +3 g( x )=3x+3In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 +1 g( x )=4x+1In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 +4 g( x )=x2In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 4 g( x )= x 2 +4In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 2x+1 g( x )= x 2 +1In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 x2 g( x )= x 2 +x2In Problems 25-32, use the given functions f and g . (a) Solvef( x )=0 . (b) Solveg( x )=0 . (c) Solvef( x )=g( x ) . (d) Solvef( x )0 . (e) Solveg( x )0 . (f) Solvef( x )g( x ) . (g) Solvef( x )1 . f( x )= x 2 x+1 g( x )= x 2 +x+6Physics A ball is thrown vertically upward with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s( t )=80t16 t 2 . (a) At what time t will the ball strike the ground? (b) For what time t is the ball more than 96 feet above the ground?Physics A ball is thrown vertically upward with an initial velocity of 96 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s( t )=96t16 t 2 . (a) At what time t will the ball strike the ground? (b) For what time t is the ball more than 128 feet above the ground?Revenue Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is p dollars, the revenue R (in dollars) is R(p)=4 p 2 +4000p (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed 800,000 ?Revenue The John Deere company has found that the revenue from sales of heavy-duty tractors is a function of the unit price p , in dollars, that it charges. The revenue R , in dollars, is given by R(p)= 1 2 p 2 +1900p (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed 1,200,000 ?Artillery A projectile Fired from the point ( 0,0 ) at an angle to the positive x-axis has a trajectory given by y=cx( 1+ c 2 )( g 2 ) ( x v ) 2 where x= horizontal distance in meters y= height in meters v= initial muzzle velocity in meters per second (m/sec) g= acceleration due to gravity =9.81 meters per second squared (m/sec2) c0 is a constant determined by the angle of elevation. A howitzer Fires an artillery round with a muzzle velocity of 897 m/sec. (a) If the round must clear a hill 200 meters high at a distance of 2000 meters in front of the howitzer, what c values are permitted in the trajectory equation? (b) If the goal in part (a) is to hit a target on the ground 75 kilometers away, is it possible to do so? If so, for what values of c ? If not, what is the maximum distance the round will travel? Source: www. answers, comRunaway Car Using Hooke's Law, we can show that the work done in compressing a spring a distance of x feet from its at-rest position is W= 1 2 k x 2 , where k is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is given by W = w 2g v 2 , Where w= weight of the object (in Ibs), g= acceleration due to gravity( 32.2ft/ sec 2 ), and v= object’s velocity (in ft/sec). A parking garage has a spring shock absorber at the end of a ramp to stop runaway cars. The spring has a stiffness constant k=9450lb/ft and must be able to stop a 4000-lb car traveling at 25 mph. What is the least compression required of the spring? Express your answer using feet to the nearest tenth. [Hint: Solve W W ~ ,x0 ]. Source: www.scifonims.comShow that the inequality ( x4 ) 2 0 has exactly one solution.Show that the inequality ( x2 ) 2 0 has one real number that is not a solution.Explain why the inequality x 2 +x+10 has all real numbers as the solution set.Explain why the inequality x 2 x+10 has the empty set as the solution set.Explain the circumstances under which the x-intercepts of the graph of a quadratic function are included in the solution set of a quadratic inequality.1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE1CT2CT3CT4CT5CT6CT7CT8CT9CT10CT11CT1CR2CR3CR4CR5CR6CR7CR8CR9CR10CR11CR12CR13CR14CR15CR16CR17CR18CR19CR20CR21CR22CR23CR24CRThe intercepts of the equation 9 x 2 +4y=36 are ______. (pp.18-19)Is the expression 4 x 3 3.6 x 2 2 a polynomial? If so, what is its degree? (pp. A22-A23)To graph y= x 2 4 , you would shift the graph of y= x 2 ______ a distance of ______ units. (pp. 106-114)Use a graphing utility to approximate (rounded to two decimal places) the local maximum value and local minimum value of f( x )= x 3 -2 x 2 -4x+5 , for 3x3 . (pp. 87-88)True or False The x-intercepts of the graph of a function y=f( x ) are the real solutions of the equation f( x )=0 . (pp. 73-75)If g( 5 )=0 , what point is on the graph of g ? What is the corresponding x-intercept of the graph of g ? (pp. 73-75)The graph of every polynomial function is both _______ and _______.If r is a real zero of even multiplicity of a polynomial function f , then the graph of f _______ (crosses/touches) the x-axis at r .The graphs of power functions of the form f(x)= x n , where n is an even integer, always contain the points ________, _______, and ______.If r is a solution to the equation f(x)=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 ________.12AYUIf f( x )=2 x 5 + x 3 5 x 2 +7 , then lim x f( x )= _____ and lim x f( x )= _____.Explain what the notation lim x f( x )= means.In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. f(x)=4x+ x 3In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. f( x )=5 x 2 +4 x 4In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. g(x)= 1 x 2 2In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. h(x)=3 1 2 xIn Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. f(x)=1 1 xIn Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. f(x)=x(x1)In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. g( x )= x 3/2 x 2 +2In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. h(x)= x ( x 1 )In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. F(x)=5 x 4 x 3 + 1 2In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. F(x)= x 2 5 x 3In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. G( x )=2 ( xl ) 2 ( x 2 +1)In Problems 17-28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not. Write each polynomial in standard form. Then identify the leading term and the constant term. G( x )=3 x 2 (x+2) 3In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= ( x+1 ) 428AYUIn Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= x 5 3In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= x 4 +2In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= 1 2 x 4In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )=3 x 5In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= x 5In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= x 4In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= ( x1 ) 5 +2In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= ( x+2 ) 4 3In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )=2 ( x+1 ) 4 +1In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )= 1 2 ( x1 ) 5 2In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )=4 ( x2 ) 5In Problems 29-42, use transformations of the graph of y= x 4 or y= x 5 to graph each function. f( x )=3 ( x+2 ) 4In Problems 43-50, 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 43-50, 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 43-50, 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 43-50, 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 43-50, 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 43-50, 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 43-50, 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 43-50, 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 57-68, 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 | x | . f(x)=3( x7 ) ( x+3 ) 2In Problems 57-68, 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 | x | . f( x )=4( x+4 ) ( x+3 ) 3In Problems 57-68, 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 | x | . f( x )=4( x 2 +1 ) ( x2 ) 3In Problems 57-68, 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 | x | . f( x )=2( x3 ) ( x 2 +4 ) 3In Problems 57-68, 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 | x | . f( x )=2 ( x+ 1 2 ) 2 ( x+4 ) 3In Problems 57-68, 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 | x | . f( x )= ( x 1 3 ) 2 ( x1 ) 3In Problems 57-68, 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 | x | . f( x )= ( x5 ) 3 ( x+4 ) 2In Problems 57-68, 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 | x | . f( x )= ( x+ 3 ) 2 ( x2 ) 4In Problems 57-68, 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 | x | . f( x )=3( x 2 +8 ) ( x 2 +9 ) 2In Problems 57-68, 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 | x | . f( x )=2 ( x 2 +3 ) 3In Problems 57-68, 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 | x | . f( x )=2 x 2 ( x 2 2 )In Problems 57-68, 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 | x | . f( x )=4x( x 2 3 )61AYU62AYU63AYU64AYU65AYUIn Problems 73-76, construct a polynomial function that might have the given graph. (More than one answer may be possible.)67AYU68AYUIn Problems 77-80, write a polynomial function whose graph is shown (use the smallest degree possible).In Problems 77-80, write a polynomial function whose graph is shown (use the smallest degree possible).In Problems 77-80, write a polynomial function whose graph is shown (use the smallest degree possible).In Problems 77-80, write a polynomial function whose graph is shown (use the smallest degree possible).73AYUIn Problems 81-98, analyze each polynomial function by following Steps 1 through 8 on page 193. f( x )=x ( x+2 ) 275AYU76AYUIn Problems 81-98, analyze each polynomial function by following Steps 1 through 8 on page 193. f( x )=2( x+2 ) ( x2 ) 3In Problems 81-98, analyze each polynomial function by following Steps 1 through 8 on page 193. f( x )= 1 2 ( x+4 ) ( x1 ) 379AYU