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mathematics
linear algebra

Questions and Answers of Linear Algebra

Graph the equation in the standard window and in the given window. Determine which window better shows the shape of the graph and the x- and y-intercepts. (a) y = 3x2 - 6 [-4, 4, - 4, 4] (b) y = -2x
Find the distance between the pair of points. Give an exact answer and, where appropriate, an approximation to three decimal places. (a) (4, 6) and (5, 9) (b) (- 3, 7) and (2, 11) (c) (-11, -8) and
Percentage of U.S. Population That Is Foreign-Born
The points 1-3, -12 and 19, 42 are the endpoints of the diameter of a circle. Find the length of the radius of the circle.
(a) (-4, 5), (6, 1), and (-8, -5) (b) (-3, 1), (2, -1), and (6, 9) (c) (-4, 3), (0, 5), and (3, -4)
Sprint Cup Series: Tony Stewart in the Top 5
The points 1-3, 42, 12, -12, 15, 22, and 10, 72 are vertices of a quadrilateral. Show that the quadrilateral is a rectangle. Show that the opposite sides of the quadrilateral are the same length and
Find the midpoint of the segment having the given endpoints. (a) (4, -9) and (-12, -3) (b) (7, -2) and (9, 5) (c) (0, 1/2) and (- 2/5, 0)
Use substitution to determine whether the given ordered pairs are solutions of the given equation. (a) (-1, -9), (0, 2); y = 7x - 2 (b) (1/2, 8), (-1, 6); y = -4x + 10 (c) (2/3, 3/4), (1, 3/2) (6x -
Graph the rectangle described in Exercise 82. Then determine the coordinates of the midpoint of each of the four sides. Are the midpoints vertices of a rectangle?
Graph the square with vertices 1-5, -12, 17, -62, 112, 62, and 10, 112. Then determine the midpoint of each of the four sides. Are the midpoints vertices of a square?
The points (√7, -4) and (√2, 3) are endpoints of the diameter of a circle. Determine the center of the circle.
How would you change the window so that the circle is not distorted?(a) (x + 3)2 + (y - 2)2 = 36(b) (x - 4)2 + (y + 5)2 = 49
Find an equation for a circle satisfying the given conditions. (a) Center 12, 32, radius of length 5/3 (b) Center 14, 52, diameter of length 8.2 (c) Center (-1, 4), passes through (3, 7) (d) Center
Draw a graph of a function for which the domain is [-4, 4] and the range is U [3, 5].
Suppose that for some function g, g(x + 3) = 2x + 1. Find g(-1).
Suppose f(x) = | x + 3 | - | x - 4 |. Write f(x) without using absolute-value notation if x is in each of the following intervals. (a) (-∞, - 3) (b) [-3, 4] (c) [4, ∞]
Determine whether the relation is a function. Identify the domain and the range. (a) {(2, 10), (3, 15), (4, 20)} (b) {(3, 1), (5, 1), (7, 1)} (c) {(-7, 3), (-2, 1), (-2, 4), (0, 7)}
Given that g1x2 = 3x2 - 2x + 1, find each of the following. (a) g(0) (b) g(-1) (c) g(3) (d) g(-x) (e) g(1 - t)
Give that f(x) = 5x2 + 4x, find each of the following. (a) f (0) (b) f (- 1) (c) f (3) (d) f (t) (e) f (t - 1)
Given that g(x) = x3, find each of the following. (a) g(2) (b) g(- 2) (c) g (- x) (d) g(3y) (e) g(2 + h)
Given that f (x) = |x| + 3x, find each of the following. (a) f(1) (b) f(- 2) (c) f(- x) (d) f(2y) (e) f(2 - h)
Given that g (x) = x - 4 / x + 3, find each of the following. (a) g(5) (b) g(4) (c) g(- 3) (d) g(- 16.25) (e) g(x + h)
Given that f(x) = x / 2 - x, find each of the following. (a) f(2) (b) f(1) (c) f(- 16) (d) f(- x) (e) f(- 2/3)
Find g(0), g(- 1), g(5), and g(1/2) for g (x) = x / √1 - x2.
Find h(0), h(2), and h(- x) for h(x) = x + √x2 - 1.
Use a graphing calculator and the Table feature set in ASK mode. (a) Given that g(x) = 0.06x3 - 5.2x2 - 0.8x, find g(-2.1), g(5.08) and g(10.003). (b) Given that h(x) = 3x4 - 10x3 + 5x2 - x + 6.
Graph the function. (a) f(x) = 1/2 x + 3 (b) f(x) = √x - 1 (c) f(x) = - x2 + 4
A graph of a function is shown. Using the graph, find the indicated function values; that is, given the inputs, find the outputs.(a) h(1), h(3), and h(4)(b) t(- 4), t(0), and t(3) (c) s(- 4), s(- 2),
Determine whether the graph is that of a function. An open circle indicates that the point does not belong to the graph.(a)(b) (c)
Determine the domain and the range of the function.(a)(b) (c)
Graph the function with a graphing calculator. Then visually estimate the domain and the range. (a) f(x) = | x | (b) f(x) = | x | - 2 (c) f(x) = 3x - 2
In 2008, it took $21.57 to equal the value of $1 in 1913. In 1985, it took only $10.87 to equal the value of $1 in 1913. The amount it takes to equal the value of $1 in 1913 can be estimated by the
The elevation E, in meters, above sea level at which the boiling point of water is t degrees Celsius is given by the functionE(t) = 1000(100 - t) + 580(100 - t) 2.At what elevation is the boiling
Use substitution to determine whether the given ordered pairs are solutions of the given equation. (a) (-3, -2), (2, -3); y2 - x2 = -5 (b) (0, -7), (8, 11); y = 0.5x + 7 (c) (4/5, -2), (11/5, 1/10);
Graph the equation. (a) y = (x - 1)2 (b) y = 1/3 x - 6 (c) -2x - 5y = 10
Find the domain of the function. Do not use a graphing calculator.(a) f(x) = 4√2x + 5 + 3(b) f(x) = √x + 1 / x
Answer the following questions for each table. (a) Is the change in the inputs x the same? (b) Is the change in the outputs y the same? (c) Is the function linear? 1. x
Determine the slope, if it exists, of the graph of the given linear equation. (a) y = 1.3x - 5 (b) y = - 2/5x + 7 (c) x = -2
The fireworks industry in the United States had a revenue of $945 million in 2009. This figure had increased from approximately $425 million in 1998. (Source: American Pyrotechnics Association) Find
The population of Flint, Michigan, decreased from 124,943 in 2000 to 102,434 in 2010. Find the average rate of change in the population of Flint, Michigan, over the 10-year period.
The population of St. Louis, Missouri, decreased from 348,194 in 2000 to 319,294 in 2010. Find the average rate of change in the population of St. Louis, Missouri, over the 10-year period.
To cut costs, many corporations have been selling their private jets. The number of used jets for sale worldwide has increased from 1022 in 1999 to 3014 in 2009. Find the average rate of change in
In 2009, electric bike sales in China totaled 21.0 million. Sales are estimated to rise to 25.0 million by 2012. Find the average rate of change in sales of electric bikes in China from 2009 to 2012.
In 2010, the traffic fatality rate in the United States was the lowest since the federal government began keeping records in 1966. There were 43,510 highway deaths in 2005. This number had decreased
The U.S. annual per-capita consumption of broccoli was 3.1 lb in 1990. By 2008, this amount had risen to 5.5 lb. Find the average rate of change in the consumption of broccoli per capita from 1990 to
Bank revenue from overdraft fees for checking accounts, ATMs, and debit cards is increasing in the United States. In 2003, account overdraft revenue was $27.1 billion. This amount is estimated to
Find the slope and the y-intercept of the line with the given equation. (a) y = 3 / 5 x - 7 (b) f(x) = -2x + 3 (c) x = - 2/5
Find the slope of the line containing the given points.(a)(b) (c)
Graph the equation using the slope and the y-intercept. (a) y = - 1 / 2 x - 3 (b) y = 3 / 2 x + 1 (c) f(x) = 3x - 1
Whales can withstand extreme atmospheric pressure changes because their bodies are flexible. Their rib cages and lungs can collapse safely under pressure. Sperm whales can hunt for squid at depths of
The stopping distance (at some fixed speed) of regular tires on glare ice is a function of the air temperature F, in degrees Fahrenheit. This function is estimated by D(F) = 2F + 115, where D1F2 is
Suppose that while driving a car, you suddenly see a deer standing in the road. Your brain registers the information and sends a signal to your foot to hit the brake. The car travels a distance D, in
A contractor buys a new truck for $23,000. The truck is purchased on January 1 and is expected to last 5 years, at the end of which time its trade-in, or salvage, value will be $4500. If the company
If f(x) = x2 - 3x, find each of the following. (a) f(1/2) (b) f(5) (c) f(-5)
A treadmill is 5 ft long and is set at an 8% grade. How high is the end of the treadmill?
Find the slope of the line containing the given points. (a) (a, a2) and (a + h, (a + h2) (b) (r, s + t) and (r, s)
Suppose that f is a linear function. Determine whether the statement is true or false. (a) f(c - d) = f(c) - f(d) (b) f(kx) = kf(x)
Let f(x) = mx + b. Find a formula for f(x) given each of the following. (a) f(x + 2) = f(x) + 2 (b) f(3x) = 3f(x)
Find the slope and the y-intercept of the graph of the linear equation. Then write the equation of the line in slope-intercept form.(a)(b) (c)
Find a linear function h given h(1) = 4 and h( -2) = 13. Then find h(2). (a) (0, -3) (b) (- 1/4, 7) (c) (2/11, - 1) (d) (0.03, 0)
Find a linear function h given h(1) = 4 and h(-2) = 13. Then find h(2).
Find a linear function g given g(- 1 / 4) = - 6 and g(2) = 3. Then find g(-3).
Find a linear function f given f(5) = 1 and f(-5) = -3. Then find f(0).
Find a linear function h given h(-3) = 3 and h(0) = 2. Then find h(-6).
Determine whether the pair of lines is parallel, perpendicular, or neither. (a) y = 26 / 3 x - 11, y = - 3 / 26 x - 11 (b) y = - 3x + 1, y = - 1 / 3 + 1 (c) y = 2 / 5 x - 4, y = - 2 / 5 x + 4
Write a slope-intercept equation for a line passing through the given point that is parallel to the given line. Then write a second equation for a line passing through the given point that is
Determine whether the statement is true or false. The lines y = 2x - 3 and y = 5 are perpendicular.
Determine whether the statement is true or false. The lines y = 2 / 5 x + 4 and y = 2 / 5 x - 4 are parallel.
Determine whether the statement is true or false. The intersection of the lines y = 2 and x = - 3 / 4 is (- 3 / 4, 2).
Determine the intervals on which the function is (a) increasing, (b) decreasing, and (c) constant.1.2.
Using the graph, determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing.(a) f(x) = x2 + 5x - 3(b) f(x) = x2 - 2x + 3
Graph the function. Estimate the intervals on which the function is increasing or decreasing and any relative maxima or minima. (a) f(x) = x2 (b) f(x) = 4 - x2 (c) f(x) = 5 - | x |
Graph the function using the given viewing window. Find the intervals on which the function is increasing or decreasing and find any relative maxima or minima. Change the viewing window if it seems
A wholesale garden center estimates that it will sell N units of a basket of three potted amaryllises after spending a dollar on advertising, where N(a) = - a2 + 300a + 6, ‰¤ a
The temperature of a patient during an illness is given by the function T(t) = -0.1t2 + 1.2t + 98.6, 0 ≤ t ≤ 12, Where T is the temperature, in degrees Fahrenheit, at time t, in days, after the
Use a graphing calculator to find the intervals on which the function is increasing or decreasing. Consider the entire set of real numbers if no domain is given. (a) f(x) = 8x / x2 + 1 (b) f(x) = -4
Rick's lumberyard has 480 yd of fencing with which to enclose a rectangular area. If the enclosed area is x yards long, express its area as a function of its length.
A seamstress is designing a triangular flag so that the length of the base of the triangle, in inches, is 7 less than twice the height h. Express the area of the flag as a function of the height.
A hot-air balloon rises straight up from the ground at a rate of 120 ft / min. The balloon is tracked from a rangefinder on the ground at point P, which is 400 ft from the release point Q of the
An airplane is flying at an altitude of 3700 ft. The slanted distance directly to the airport is d feet. Express the horizontal distance h as a function of d.
A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 40 m. Each vertex of the rhombus is a midpoint of a side of the rectangle. Express the area of the rhombus as a
A carpet installer uses 46 ft of linen tape to bind the edges of a rectangular hall runner. If the runner is w feet wide, express its area as a function of the width.
A device used in golf to estimate the distance d, in yards, to a hole measures the size s, in inches, that the 7-ft pin appears to be in a viewfinder. Express the distance d as a function of s.
A gas tank has ends that are hemispheres of radius r feet. The cylindrical midsection is 6 ft long. Express the volume of the tank as a function of r.
A car dealership has 30 ft of dividers with which to enclose a rectangular play space in a corner of a customer lounge. The sides against the wall require no partition. Suppose the play space is x
A rancher has 360 yd of fencing with which to enclose two adjacent rectangular corrals, one for horses and one for cattle. A river forms one side of the corrals. Suppose the width of each corral is x
Designs Unlimited plans to produce a one-component vertical file by bending the long side of an 8-in. by 14-in. sheet of plastic along two lines to form a U shape.(a) Express the volume of the file
From a 12-cm by 12-cm piece of cardboard, square corners are cut out so that the sides can be folded up to make a box.(a) Express the volume of the box as a function of the side x, in centimeters, of
A rectangle that is x feet wide is inscribed in a circle of radius 8 ft.(a) Express the area of the rectangle as a function of x.(b) Find the domain of the function.(c) Graph the function with a
A rectangular box with volume 320 ft3 is built with a square base and top. The cost is $1.50 / ft2 for the bottom, $2.50 / ft2 for the sides, and $1 / ft2 for the top. Let x = the length of the base,
For each piecewise function, find the specified function values.(a)g(-4), g(0), g(1), and g(3) (b) f(-5), f(-2), f(0), and f(2) (c) h(-4), f(-2), f(4), and f(6)
Make a hand-drawn graph of each of the following. Check your results using a graphing calculator.(a)(b) (c)
Determine the domain and the range of the piecewise function. Then write an equation for the function.(a)(b) (c)
Given f(x) = 5x2 - 7, find each of the following. (a) f(-3) (b) f(3) (c) f(a) (d) f(-a)
Given f 1x2 = 4x3 - 5x, find each of the following. (a) f(2) (b) f(-2) (c) f(a) (d) f(-a)
Write an equation of the line perpendicular to the graph of the line 8x - y = 10 and containing the point (-1, 1).
Find the slope and the y-intercept of the line with equation 2x - 9y + 1 = 0.
Using a graphing calculator, estimate the interval on which the function is increasing or decreasing and any relative maxima or minima. (a) f(x) = x4 + 4x3 - 36x2 - 160x + 400 (b) f(x) = 3.22x5 -
A parking garage charges $2 for up to (but not including) 1 hr of parking, $4 for up to 2 hr of parking, $6 for up to 3 hr of parking, and so on. Let C(t) = the cost of parking for t hours. (a) Graph

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