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Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

Identify each equation as true or false. If it is false, correct it to make it true. a. y2 + 11y + 121 = (y + 11)2 b. x2 - 18x + 81 = (x - 9)2 c. 5y2 + 10y + 5 = 5(y + 1)2 d. 4x2 + 24x + 36 = 4(x + 6)2
Use the quadratic formula to solve each equation for y. Graph each curve. a. 25x2 - 4y2 + 100 = 0 b. 4y2 - 10x + 16y + 36 = 0 c. 4x2 + 4y2 + 24x - 8y + 39 = 0 d. 3x2 + 5y2 - 12x + 20y + 8 = 0
Solve each system of equations algebraically, using the substitution or elimination method.a.b. c.
Two seismic monitoring stations recorded the vibrations of an earthquake. The second monitoring station is 50 mi due east of the first. The epicenter was determined to be 30 mi from the first station and 27 mi from the second station. Where could the epicenter of the earthquake be located?
Match each equation to one of the graphs.a. 9x2 + 4y2 - 36 = 0b. x2 - 4y2 - 8x = 0 c. 3x2 - 30x + 5y + 55 = 0 d. x2 + y2 + 2x - 6y - 15 = 0
Use geometry software to construct and explore this sketch.c. Explain why the sum of the distances FP and GP is constant. d. Trace point P as you drag point A. What shape is traced? Why? e. Now move G to a different location within the circle and repeat 9d. Describe how the shape changes. What
Write an equation and graph each transformation of the parent function f (x) = 1/x. a. Translate the graph up 2 units. b. Translate the graph right 3 units. c. Translate the graph down 1 unit and left 4 units. d. Vertically stretch the graph by a scale factor of 2. e. Horizontally stretch the graph
Draw the graph of y = 1/x. a. Label the vertices of the hyperbola. b. The x- and y-axes are the asymptotes for this hyperbola. Draw the box between the two branches of the hyperbola that has the asymptotes as its diagonals. The vertices should lie on the box. c. The dimensions of the box are 2a and
Recall that the general quadratic equation is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Let A = 0, B = 4, C = 0, D = 0, E = 0, and F = - 1. a. Graph this equation. What type of conic section is formed? b. What is the relationship between the inverse variation function, y = 1/x, and the conic sections? c.
Ohm's law states that I = V/R. This law can be used to determine the amount of current I, in amps, flowing in the circuit when a voltage V, in volts, is applied to a resistance R, in ohms. a. If a hairdryer set on high is using a maximum of 8.33 amps on a 120-volt line, what is the resistance in
Using geometry software, draw the curve y = 1/x and plot the foci you found in Exercise 10d. Use measurement tools to verify that y = 1/x satisfies the locus definition of a hyperbola.
A 2 m rod and a 5 m rod are mounted vertically 10 m apart. One end of a 15 m wire is attached to the top of each rod. Suppose the wire is stretched taut and fastened to the ground between the two rods. How far from the base of the 2 m rod is the wire fastened?
Sarah would like to row her boat directly across a river 500 m wide. The current flows 3 km/h and she is able to row 5 km/h.a. At what angle to the riverbank should she point her boat?b. As she starts, how far upstream on the opposite bank should she head?c. Write parametric equations to simulate
Write the general quadratic equations of two concentric circles with center (6, - 4) and radii 5 and 8.
Find exact values of missing side lengths for 19a-d.a.
What are the equations of the asymptotes for each hyperbola? a. y = 2/x + 1 b. y = 3/x -4 c. y = 4/x+ 2 -1 d. y = -2/x+3 - 4
Solve. a. 12 = x - 8/x+3 b. 21 = 3x + 8/x + 5 c. 3 = 2x + 5/4x -7 d. - 4 = -6x + 5/2x + 3
As the rational function y = 1/x is translated, its asymptotes are translated also. Write an equation for the translation of y = 1/x that has the asymptotes described. a. Horizontal asymptote y = 2 and vertical asymptote x = 0 b. Horizontal asymptote y = - 4 and vertical asymptote x = 2 c.
If a basketball team's present record is 42 wins and 36 losses, how many consecutive games must it win so that its winning record reaches 60%?
Write a rational equation to describe each graph. Some equations will need scale factors.a.b. c. d.
The graph at right shows the concentration of acid in a solution as pure acid is added. The solution began as 55 mL of a 38% acid solution. a. How many milliliters of pure acid were in the original solution? b. Write an equation for f (x). c. Find the amount of pure acid that must be added to
In a container of 2% milk, 2% of the mixture is fat. How much of the liquid in a 1 gal container of 2% milk would need to be emptied and replaced with pure fat so that the container could be labeled as whole (3.25%) milk?
Consider these functions. i. y = 2x - 13/x -5 ii. y = 2x + 11/x+3 a. Rewrite each rational function to show how it is a transformation of y = 1/x. b. Describe the transformations of the graph of y = 1/x that will produce graphs of the equations in 9a. c. Graph each equation on your calculator to
Rewrite each rational expression in factored form.a.b.
Consider the equation y = (x - 1) (x + 4) / (x - 2) (x + 3). a. Describe the features of the graph of this function. b. Describe the end behavior of the graph. c. Sketch the graph.
Solve. Give exact solutions. a. 2/x - 1 + x = 5 b. 2/x - 1 + x = 2
The functional response curve given by the function y = 60x/1 + 0.625x models the number of moose attacked by wolves as the density of moose in an area increases. In this model, x represents the number of moose per 1000 km2, and y represents the number of moose attacked every 100 days. a. How many
A machine drill removes a core from any cylinder. Suppose you want the amount of material left after the core is removed to remain constant. The table below compares the height and radius needed if the volume of the hollow cylinder is to remain constant.a. Plot the data points, (x, h), and draw a
Find the points of intersection, if any, of the circle with center (2, 1) and radius 5 and the line x - 7y + 30 = 0.
A 500 g jar of mixed nuts contains 30% cashews, 20% almonds, and 50% peanuts. a. How many grams of cashews must you add to the mixture to increase the percentage of cashews to 40%? What is the new percentage of almonds and peanuts? b. How many grams of almonds must you add to the original mixture
Solve each quadratic equation. a. 2x2 - 5x - 3 = 0 b. x2 + 4x - 4 = 0 c. x2 + 4x + 1 = 0
Identify the vertical asymptotes for each equation.a.b.
Rewrite each expression in rational form (as the quotient of two polynomials).a.b.
Graph each equation on your calculator, and make a sketch of the graph on your paper. Use a friendly graphing window. Indicate any holes on your sketches.a.b. c. d. What causes a hole to appear in the graph?
Write an equation for each graph.a.b. c.
Graph y = 4/x - 3. a. Describe the end behavior of the graph. b. Describe the behavior of the graph near x = 3. c. Rewrite y = - x + 4/x - 3 in rational function form. d. Factor your answer from 6c. What do the factors tell you about the graph?
Graph each function on your calculator. List all holes and asymptotes, including slant asymptotes. a. y = x - 2 + 1/x b. y = -2x + 3 + 2/x-1 c. y = 7 + 8-4x / x-2
The two graphs below show the same function.a. List all the important facts you can about the graph. b. Find the equation of the slant asymptote in the first graph. c. Give an example of an equation with asymptote x = - 2. d. Name a polynomial with zeros x = - 3 and x = 1. e. Write an equation for
The two graphs below show the same function. Write an equation for this function.
Factor each expression completely and reduce common factors.a.b. c. d.
How long should a traffic light stay yellow before turning red? One study suggests that for a car approaching a 40 ft wide intersection under normal driving conditions, the length of time, y, that a light should stay yellow, in seconds, is given by the equation y = 1 + v/25+ 50/v, where v is the
The graph at right is the image of y = 1/x after a transformation.a. Write an equation for each asymptote.b. What translations are involved in transforming y = 1/x to its image?c. The point (4, - 1) is on the image. What is the vertical scale factor in the transformation?d. Write an equation of the
A block with mass 10 kg is sliding down a 35° incline, acted on by a gravity force vector of 98 N (newtons).a. Sketch this gravity vector in two components, one parallel to the incline, vi, and one perpendicular to the incline, vn.b. Find the magnitude of each component.
If you invest $1000 at 6.5% interest for 5 years, how much interest do you earn in each of these scenarios? a. The interest is compounded annually. b. The interest is compounded monthly. c. The interest is compounded weekly. d. The interest is compounded daily.
What is the least common denominator for each pair of rational expressions?a.b. c. d.
Add or subtract as indicated.a.b. c. d.
Multiply or divide as indicated. Reduce any common factors to simplify.a.b. c. d.
Rewrite as a single rational expression.a.b.
Graph y = x + 1/x2 - 7x - 8 - x/2(x - 8) on your calculator. a. List all asymptotes, holes, and intercepts based on your calculator's graph. b. Rewrite the right side of the equation as a single rational expression. c. Use your answer from 6b to verify your observations in 6a. Explain.
Consider the equation y = x - 3/x2 - 4. a. Without graphing, identify the zeros and asymptotes of the graph of the equation. Explain your methods. b. Verify your answers by graphing the function.
Consider the equation y = x + 1 + 1/x-1. a. Without graphing, name the asymptotes of the function. b. Rewrite the equation as a single rational function. c. Sketch a graph of the function without using your calculator. d. Confirm your work by graphing the function with your calculator.
You saw in Lesson 9.7 that a transformation of the parent function f (x) = 1/x is a rotated hyperbola. Are there other kinds of rational functions that are also rotated hyperbolas? Recall that any conic section can be written in the general quadratic form, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. a.
Sketch the graph of each equation. Label foci when appropriate.a. x - 3/-2 = (y - 2)2b. x2 + (y - 2)2 = 16c.d.
On her way to school, Ellen drives at a steady speed for the first 2 mi. After glancing at her watch, she drives 20 mi/h faster during the remaining 3.5 mi. How fast does she drive during the two portions of this trip if the total time of her trip is 10 min?
Rewrite each expression as a single rational expression in factored form.a.b. c.
Solve this system of equations algebraically, then confirm your answer graphically.
Write an equation of the image of the absolute-value function, y = |x|, after performing each of the following transformations in order. Sketch a graph of your final equation. a. Stretch vertically by a factor of 2. b. Then translate right 4 units. c. Then translate down 3 units.
Earth's orbit is an ellipse with the Sun at one of the foci. Perihelion is the point at which Earth is closest to the Sun, and aphelion is the point at which it is farthest from the Sun. The distances from perihelion to the Sun and from the Sun to aphelion are in an approximate ratio of 59:61. If
Perform the following matrix computations. If a computation is not possible, explain why.a.b. c. d. [B][A] e. 3[C] - [A]
Identify each sequence as arithmetic, geometric, or neither. State the next three terms, and then write a recursive formula to generate each sequence. a. 9,12,15,18, . . . b. 1, 1, 2, 3, 5, 8, 13, . . . c. - 3, 6, - 12, 24, . . .
Evan is standing at one corner of the football field when he sees his dog, Spot, start to run diagonally across the field as shown. Evan knows that Spot can run to the opposite corner in 15 s. The dimensions of the football field are 100 yd by 52 yd.a. What is Spot's rate in yards per second?b.
D'Andre surveyed a randomly chosen group of 15 teachers at his school and asked them how many students were enrolled in their third-period classes. Here is the data set he collected. {27, 29, 18, 34, 42, 38, 34, 33, 25, 28, 45, 35, 32, 19, 36} a. List the mean, median, and mode. b. Make a box plot
Rewrite each equation in standard form, and identify the type of curve. a. 25x2 - 4y2 + 100 = 0 b. 4y2 - 10x + 16y +36 = 0 c. 4x2 + 4y2 + 24x - 8y + 39 = 0 d. 3x2 + 5y2 - 12x + 20 y + 8 = 0
Consider this ellipse.a. Write the equation for the graph shown in standard form.b. Write the parametric equations for this graph.c. Name the coordinates of the center and foci.d. Write the general quadratic form of the equation for this graph.
The towers of a parabolic suspension bridge are 400 m apart and reach 50 m above the suspended roadway. The cable is 4 m above the roadway at the halfway point. Write an equation that models the shape of the cable. Assume the origin, (0, 0), is located at the halfway point of the roadway.
Solve algebraically. Round answers to the nearest hundredth. a. 4 + 5x =18 b. 12(0.5)2x = 30 c. log3 15 = log x/log x d. log6 100 = x e. 2 logx = 2.5 f. log5 53 = x g. 4 logx = log 16 h. log(5 + x) - log 5 = 2 i. xlog 5x = 12
The chart below shows average fuel efficiency of new U.S. passenger cars.a. Find the median-median line for the data. b. What is the root mean square error for the median-median line model? c. What is the real-world meaning of the root mean square error in 22b? d. Is the median-median line a good
The bases on a baseball diamond form a square that is 90 ft on each side. Deanna has a 12 ft lead and can run the remaining 78 ft from first to second base at 28 ft/s. The catcher releases the ball from home plate toward second base 1.5 s after Deanna starts to steal the base, and the ball travels
Use P = 1 + 3i, Q = - 2 + i, and R = 3 - 5i to evaluate each expression. Give answers in the form a + bi. a. P + Q - R b. PQ c. Q2 d. P ÷ Q
The equation (x/a)2 + (y/b)2 = 1 is the standard form of an ellipse. What shape will the equations (x/a)3 + (y/b)3 = 1, (x/a)4 + (y/b)4 = 1, or generally (x/a)n + (y/b)n = 1 create? These equations are called Lamé curves, or superellipses, when n > 2. Investigate several Lamé curve
How can you find the horizontal asymptote of a rational function without graphing or using a table? Use a calculator to explore these functions, and look for patterns. Some equations may not have horizontal asymptotes. Make a conjecture about how to determine the equation of a horizontal asymptote
You have seen that for an ellipse, the sum of the distances from any point to the two foci is constant. For a hyperbola, the difference of the distances from any point to the two foci is constant. What shape is created if the product of the distances is constant? What if the ratio of the distances
In Lesson 9.5, you explored the number of points of intersection of two conic sections. You saw that, for example, a hyperbola and an ellipse can intersect 0, 1, 2, 3, or 4 times. However, points of intersection on a coordinate plane only include the real solutions to a system of equations. If you
Consider the hyperbola graphed at right.a. Write the equations of the asymptotes for this hyperbola.b. Write the general quadratic equation for this hyperbola.c. Write an equation that will give the vertical distance, d, between the asymptote with positive slope and a point on the upper portion of
Write the general quadratic equation x2 + y2 + 8x - 2y - 8 = 0 in standard form. Identify the shape described by the equation and describe its features.
Write the general quadratic equation y2 - 8y - 4x + 28 = 0 in standard form. Determine the vertex, focus, and directrix of the parabola defined by this equation. Sketch a graph.
Pure gold is too soft to be used for jewelry, so gold is always mixed with other metals. 18-karat gold is 75% gold and 25% other metals. How much pure gold must be mixed with 5 oz of 18-karat gold to make a 22-karat (91.7%) gold mixture?
Write an equation of each rational function described as a translation of the graph of y = 1/x. a. The rational function has asymptotes x = - 2 and y = 1. b. The rational function has asymptotes x = 0 and y = - 4.
Graph y = 2x - 14/x - 5. Write equations for the horizontal and vertical asymptotes.
How can you modify the equation y = 2x - 14/x - 5 so that the graph of the new equation is the same as the original graph except for a hole at x = - 3? Verify your new equation by graphing it on your calculator.
After 300 s, the paddle wheel in the investigation has rotated 300°. Draw a reference triangle and find the frog's height at this time.
Suppose sin θ ≈ - 0.7314 and 180° ≤ θ ≤ 270°. a. Locate the point where the terminal side of intersects the unit circle. b. Find θ and cos θ. c. What other angle has the same cosine value? Use domain 0° ≤ α ≤ 360°
Find an angle in standard position θ, for a plane flying on these bearings. Use domain - 180° ≤ θ ≤ 180°. a. 105° b. 325° c. 180° d. 42°
Find sinθ and cosθ for each angle in standard position described. a. The terminal side of angle θ passes through the point (2, 3). b. The terminal side of angle θ passes through the point (-2, 3).
Find each angle with the given trigonometric value. Use domain 0 ≤ θ ≤ 360°. a. cos θ = -√3/2 b. cos θ - √2/2 c. sin θ = -3/5 d. sin θ = 1
Make a table of the values of sinθ, cosθ, tanθ, and sinθ / cosθ, using values of at intervals of 30° over the domain 0°≤ θ ≤ 360°. What do you notice? Use the definitions of trigonometric ratios to explain your conjecture.
For the past several hundred years, astronomers have kept track of the number of sunspots. This table shows the average number of sunspots each year from 1972 to 1999.a. Make a scatter plot of the data and describe any patterns that you notice. b. Estimate the length of a cycle. c. Predict the next
Annie is standing on a canyon floor 20 m from the base of a cliff. Looking through her binoculars, she sees the remains of ancient cliff dwellings in the cliff face. Annie holds her binoculars at eye level, 1.5 m above the ground. a. Write an equation that relates the angle at which she holds the
Convert to the specified units using ratios. For example, to convert 0.17 meter to inches:a. 0.500 day to seconds b. 3.0 mi/h to ft/s (There are 5280 feet per mile.)
Find the circumference and area of the circle with equation 2x2 + 2y2 - 2x + 7y - 38 = 0.
Rewrite each expression as a single rational expression in factored form.a.b. c.
Use your calculator to find each value, approximated to four decimal places. Then draw diagrams in a unit circle to show the meaning of the value. Name the reference angle. a. sin (-175°) b. cos 147° c. sin 280° d. cos 310° e. sin (-47°)
Write an equation for a rational function, f (x), that has vertical asymptotes x = - 4 and x = 1, horizontal asymptote y = 2, and zeros x = - 2 and x = 5. Check your answer by graphing the equation on your calculator.
Create a sine or cosine graph, and trace to find the value of each expression. a. sin 120° b. sin (- 120°) c. cos (- 150°) d. cos 150°
Which of the following functions are periodic? For each periodic function, identify the period.
Identify an angle that is coterminal with the given angle. Use domain 0° ≤ θ ≤ 360°. a. - 25° b. - 430° c. 435° d. 1195°
For each Quadrant, I - IV, shown at right, identify whether the values of cosÎ¸ and sinÎ¸ are positive or negative.
Carefully sketch a graph of the function y = sin x over the domain - 360° x 360°. Identify all values of x in this interval for which sin x = 0.
Carefully sketch a graph of the function y = cos x over the domain - 360° x 360°. Identify all values of x in this interval for which cos x = 0.
Convert between radians and degrees. Give exact answers. (Remember that degree measures are always labeled °, and radians generally are not labeled.) a. 80o b. 570 o c. -4π/4 d. 11π/9 e. -3π/4 f. 3π g. -900 o h. 5π/6

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