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mathematics
precalculus

Questions and Answers of Precalculus

(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
Suppose that f(x) = x2 + 4x - 21(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
(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
Suppose that f (x) = x2 + 2x - 8(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
Find the point on the line y = x that is closest to the point (3, 1).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).
Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) isR(p) = - 4p2 + 4000pWhat unit price should be established for the
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 iswhat unit price p should
The marginal cost C (in dollars) of manufacturing x cell phones (in thousands) is given byC(x) = 5x2 + 200x + 4000(a) How many cell phones should be manufactured to minimize the marginal cost?(b)
The monthly revenue R achieved by selling x wristwatches is figured to be R(x) = 75x – 0.2x2.The monthly cost C of selling x wristwatches is C(x) = 32x + 1750.(a) How many wristwatches must the
Business The daily revenue R achieved by selling x boxes of candy is figured to be R(x) = 9.5x - 0.04x2. The daily cost C of selling x boxes of candy is C(x) = 1.25x + 250.(a) How many boxes of candy
Stopping Distance An accepted relationship between stopping distance, d (in feet), and the speed of a car, υ (in mph), is d = 1.1v + 0.06υ2 on dry, level concrete.(a) How many feet will it take a
In the United States, the birthrate B of unmarried women (births per 1000 unmarried women) for women whose age is a is modeled by the function B(a) = -0.27a2 + 14.23a - 120.16.(a) What is the age of
Find a quadratic function whose x-intercepts are -4 and 2 and whose range is [-18, ∞).
Find a quadratic function whose x-intercepts are -1 and 5 and whose range is [∞, 9).
Let f(x) = ax2 + bx + c, where a, b, and c are odd integers. If x is an integer, show that f(x) must be an odd integer.x is either an even integer or an odd integer.
On one set of coordinate axes, graph the family of parabolas f(x) = x2 + 2x + c for c = -3, c = 0, and c = 1. Describe the characteristics of a member of this family.
State the circumstances that cause the graph of a quadratic function f(x) = ax2 + bx + c to have no x-intercepts.
Why does the graph of a quadratic function open up if a> 0 and down if a < 0?
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?
A shot-putter throws a ball at an inclination of 45° to the horizontal. The following data represent the height of the ball h at the instant that it has traveled x feet horizontally.Distance, x
An individual€™s 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 2006. For each age group, let
Use the result obtained in Problem 20 to find the area enclosed by f(x) = -x2 + x + 4, the x-axis, and the lines x= -1 and x = 1.
Use the result obtained in Problem 20 to find the area enclosed by f(x) = x2 + 3x + 5, the x-axis, and the lines x= -4 and x = 4.
A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be 1500 meters.What should the dimensions of the
A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. See the figure. If the perimeter of the window is 20 feet, what dimensions
In problem,(a) Draw a scatter diagram.(b) Select two points from the scatter diagram and find the equation of the line containing the points selected.(c) Graph the line found in part (b) on the
In problem, a linear function is given.(a) Determine the slope and y-intercept of each function.(b) Use the slope and y-intercept to graph the linear function.(c) Determine the average rate of change
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
In problem,(a) Draw a scatter diagram.(b) Select two points from the scatter diagram and find the equation of the line containing the points selected.(c) Graph the line found in part (b) on the
In problem, a linear function is given.(a) Determine the slope and y-intercept of each function.(b) Use the slope and y-intercept to graph the linear function.(c) Determine the average rate of change
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
In problem,(a) Draw a scatter diagram.(b) Select two points from the scatter diagram and find the equation of the line containing the points selected.(c) Graph the line found in part (b) on the
In problem, a linear function is given.(a) Determine the slope and y-intercept of each function.(b) Use the slope and y-intercept to graph the linear function.(c) Determine the average rate of change
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
In problem, a linear function is given.(a) Determine the slope and y-intercept of each function.(b) Use the slope and y-intercept to graph the linear function.(c) Determine the average rate of change
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 bywhere x is the horizontal distance of
True or FalseThe average rate of change of f(x) = 2x + 8 is 8.
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
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
True or FalseThe slope of a nonvertical line is the average rate of change of the linear function.
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?
If the slope m of the graph of a linear function is _________ , the function is increasing over its domain.
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
For the graph of the linear function H(z) = -4z + 3, the slope is _____ and the y-intercept is ______.
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
For the graph of the linear function f(x) = mx + b, m is the _______ and b is the _____.
The price p (in dollars) and the quantity x sold of a certain product obey the demand equationx = -20p + 500 0 < p ≤ 25(a) Express the revenue R as a function of x.(b) What is the revenue
True or FalseThe graph of the function f(x) = x2 is increasing on the interval (0, ∞).
The price p (in dollars) and the quantity x sold of a certain product obey the demand equationx = -5p + 100 0 ≤ p ≤ 20(a) Express the revenue R as a function of x.(b) What is the revenue if
If f(x) = x2 - 4, find f(-2).
The price p (in dollars) and the quantity x sold of a certain product obey the demand equationP = -1/3 x + 100(a) Find a model that expresses the revenue R as a function of x.(b) What is the domain
The price p (in dollars) and the quantity x sold of a certain product obey the demand equationP = 1/6 x + 100(a) Find a model that expresses the revenue R as a function of x. (Remember, R = xp.)(b)
Find the average rate of change of f (x) = 3x2 - 2, from 2 to 4.
Use a graphing utility to find the line of best fit for the following data: 5 5 3 6. 6 х 12 15 y 10 13 16 19
Find the slope of the line joining the points (2, 5) and (-1, 3).
In problem, write the function whose graph is the graph of y = x3, but is:Horizontally stretched by a factor of 4
In problem:(a) Find the domain of each function.(b) Locate any intercepts.(c) Graph each function.(d) Based on the graph, find the range.(e) Is f continuous on its domain? f(x) = = x + 3 -2x - 3 if x
In problem:(a) Find the domain of each function.(b) Locate any intercepts.(c) Graph each function.(d) Based on the graph, find the range.(e) Is f continuous on its domain? f(x) = x + 3 5 -x + 2 if -2
Use the result obtained in Problem 20 to find the area enclosed by f(x) = 2x2 + 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) = -5x2 + 8, the x-axis, and the lines x= -1 and x = 1.
Calculus: Simpson's Rule The figure shows the graph of y = ax2+ bx + c. Suppose that the points (-h, y0), (0, y1), and (h, y2) are on the graph. It can be shown that the area enclosed by the
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 byV(x) = kx(a - x), 0 ≤ x ≤ 1where k is a
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?Area of an
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, use the given functions f and g.(a) Solve f(x) = 0.(b) Solve g(x) = 0.(c) Solve f(x) = g(x).(d) Solve f(x) > 0.(e) Solve g(x) ≤ 0.(f) Solve f(x) > g(x).(g) Solve f(x) ≥ 1.f(x) =
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, use the given functions f and g.(a) Solve f(x) = 0.(b) Solve g(x) = 0.(c) Solve f(x) = g(x).(d) Solve f(x) > 0.(e) Solve g(x) ≤ 0.(f) Solve f(x) > g(x).(g) Solve f(x) ≥ 1.f(x) =
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, use the given functions f and g.(a) Solve f(x) = 0.(b) Solve g(x) = 0.(c) Solve f(x) = g(x).(d) Solve f(x) > 0.(e) Solve g(x) ≤ 0.(f) Solve f(x) > g(x).(g) Solve f(x) ≥ 1.f(x) =
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, use the given functions f and g.(a) Solve f(x) = 0.(b) Solve g(x) = 0.(c) Solve f(x) = g(x).(d) Solve f(x) > 0.(e) Solve g(x) ≤ 0.(f) Solve f(x) > g(x).(g) Solve f(x) ≥ 1.f(x) =
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, use the given functions f and g.(a) Solve f(x) = 0.(b) Solve g(x) = 0.(c) Solve f(x) = g(x).(d) Solve f(x) > 0.(e) Solve g(x) ≤ 0.(f) Solve f(x) > g(x).(g) Solve f(x) ≥ 1.f(x) =
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
What is the domain of the function f(x) = √x - 3x2?
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
What is the domain of the function f(x) = √x2 - 16?
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.2(2x2 - 3x) > -9
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.6(x2 - 1) > 5x
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.25x2 + 16 < 40x
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.4x2 + 9 < 6x
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, solve each inequality.x2 + 2x + 4 > 0
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, solve each inequality.x2 - x + 1 ≤ 0
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.6x2 < 6 + 5x
In problem,(a) Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.(b) Determine the
In problem, graph the function f by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection).If necessary, write f in the form f(x) = a(x –
In problem, solve each inequality.2x2 < 5x + 3

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