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mathematics
linear algebra

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

Match a description to each graph.a.b. c. d. A. increasing more and more rapidly B. decreasing more and more slowly C. increasing more and more slowly D. decreasing more and more rapidly American minimalist painter and sculptor Ellsworth Kelly (b 1923) based many of his works on the shapes of
Harold's concentration often wanders from the game of golf to the mathematics involved in his game. His scorecard frequently contains mathematical doodles and graphs.a. What is a real-world meaning for this graph found on one of his recent scorecards? b. What units might he be using? c. Describe a
Make up a story to go with the graph at right. Be sure to interpret the x- and y-intercepts.
Sketch what you think is a reasonable graph for each relationship described. In each situation, identify the variables and label your axes appropriately. a. The height of a basketball during the last 10 seconds of a game b. The distance it takes to brake a car to a full stop, compared to the car's
Sketch what you think is a reasonable graph for each relationship described. In each situation, identify the variables and label your axes appropriately. In each situation, will the graph be continuous or will it be a collection of discrete points or pieces? Explain why.a. the amount of money you
Describe a relationship of your own and draw a graph to go with it.
Car A and Car B are at the starting line of a race. At the green light, they both accelerate to 60 mi/h in 1 min. The graph at right represents their velocities in relation to time.a. Describe the rate of change for each car. b. After 1 minute, which car will be in the lead? Explain your reasoning.
Some functions can be described as even or odd. An even function has the y-axis as a line of symmetry. If the function f is an even function, then f (- x) = f (x) for all values of x in the domain. Which parent functions that you've seen are even functions? Now graph y = x3, y = 1 / x, and y =
A line of reflection does not have to be the x- or y-axis. Draw the graph of a function and then draw its image when reflected across several different horizontal or vertical lines. Write the equation of each image. Try this with several different functions. In general, if the graph of y = f (x) is
For the graph of the parent function y = x2, you can think of any vertical stretch or shrink as an equivalent horizontal shrink or stretch. For example, the equations y = 4x2 and y = (2x)2 are equivalent, even though one represents a vertical stretch by a factor of 4 and the other represents a
Enter two linear functions into Y1 and Y2 on your calculator. Enter the compositions of the functions as Y3 = Y1 (Y2(x)) and Y4 = Y2(Y1(x)). Graph Y3 and Y4 and look for any relationships between them. (It will help if you turn off the graphs of Y1 and Y2.) Make a conjecture about how the
One way to visualize a composition of functions is to use a web graph. Here's how you evaluate f (g(x)) for any value of x, using a web graph: Choose an x-value. Draw a vertical line from the x-axis to the function g(x). Then draw a horizontal line from that point to the line y = x. Next, draw a
Which of these graphs represent functions? Why or why not?a.b. c.
Consider the function f (x) = 3(x + 1)2 - 4. a. Find f (5). b. Find f (n). c. Find f (x + 2). d. Use your calculator to graph y = f (x) and y = f (x + 2) on the same axes. How do the graphs compare?
Kendall walks toward and away from a motion sensor. Is the graph of his motion a function? Why or why not?
The length of a pendulum in inches, L, is a function of its period, or the length of time it takes to swing back and forth, in seconds, t. The function is defined by the formula L = 9.73t2. a. Find the length of a pendulum if its period is 4 s. b. The Foucault pendulum at the Panthéon in
The number of diagonals of a polygon, d, is a function of the number of sides of the polygon, n, and is given by the formula d = n(n - 3) / 2.a. Find the number of diagonals in a dodecagon (a 12-sided polygon).b. How many sides would a polygon have if it contained 170 diagonals?
Create graphs picturing the water height as each bottle is filled with water at a constant rate.a.b. c.
The five-number summary of this box plot is $2.10, $4.05, $4.95, $6.80, $11.50. The plot summarizes the amounts of money earned in a recycling fund drive by 32 members of the Oakley High School environmental club. Estimate the total amount of money raised. Explain your reasoning.
Given the graph at right, find the intersection of lines
Sketch a graph for a function that has the following characteristics. a. domain: x ≥ 0 range: f (x) ≥ 0 linear and increasing b. domain: - 10 ≤ x ≤ 10 range: - 3 < f (x) ≤ 3 nonlinear and increasing c. domain: x ≥ 0 range: - 2 < f (x) ≤ 10 increasing, then decreasing, then increasing,
You can use rectangle diagrams to represent algebraic expressions. For instance, this diagram demonstrates the equation (x + 5)(2x + 1) = 2x2 + 11x + 5. Fill in the missing values on the edges or in the interior of each rectangle diagram.a. b. c.
Use the functions f (x) = 3x - 4 and g (x) = x2 + 2 to find these values. a. f (7) b. g (5) c. f (-5) d. g (-3) e. x when f (x)
Miguel works at an appliance store. He gets paid $5.25 an hour and works 8 hours a day. In addition, he earns a 3% commission on all items he sells. Let x represent the total dollar value of the appliances that Miguel sells, and let the function m represent Miguel's daily earnings as a function of
Identify the independent variable for each relation. Is the relation a function? a. the price of a graphing calculator and the sales tax you pay b. the amount of money in your savings account and the time it has been in the account c. the amount your hair has grown since the time of your last
Sketch a reasonable graph for each relation described in Exercise 5. In each situation, identify the variables and label your axes appropriately. In Exercise 5 a. The price of a graphing calculator and the sales tax you pay b. The amount of money in your savings account and the time it has been in
Suppose f (x) = 25 - 0.6x.a. Draw a graph of this function.b. What is f (7)?c. Identify the point (7, f (7)) by marking it on your graph.d. Find the value of x when f (x) = 27.4. Mark this point on your graph.
Sketch a graph for each function. a. y = f (x) has domain all real numbers and range f (x) 0. b. y = g (x) has domain x > 0 and range all real numbers. c. y = h (x) has domain all real numbers and range h (x) = 3.
The Internal Revenue Service has approved ten-year linear depreciation as one method for determining the value of business property. This means that the value declines to zero over a ten-year period, and you can claim a tax exemption in the amount of the value lost each year. Suppose a piece of
Suppose that your basketball team's scores in the first four games of the season were 86 points, 73 points, 76 points, and 90 points.a. What will be your team's mean score if the fifth-game score is 79 points?b. Write a function that gives the mean score in terms of the fifth-game score.c. What
Solve. a. 2(x + 4) = 38 b. 7 + 0.5(x - 3) = 21 c. - 2 + (x + 1) = - 17 d. 4.7 + 2.8(x - 5.1) = 39.7
The three summary points for a data set are M1(3, 11), M2(5, 5), and M3(9, 2). Find the median-median line.
If f (x) = - 2x, find a. f (x + 3) b. - 3 + f (x - 2) c. 5 + f (x + 1)
Consider the line that passes through the points (- 5.2, 3.18) and (1.4, - 4.4), as shown.a. Find an equation of the line. b. Write an equation of the parallel line that is 2 units above this line.
Write an equation of each line. a. the line y = 4.7x translated down 3 units b. the line y = - 2.8x translated right 2 units c. the line y = - x translated up 4 units and left 1.5 units
The graph of y = f (x) is shown at right. Write an equation for each of the graphs below.a.b. c. d.
Jeannette and Keegan collect data about the length of a rope as knots are tied in it. The equation that fits their data is y = 102 - 6.3x, where x represents the number of knots and y represents the length of the rope in centimeters. Mitch had a piece of rope cut from the same source. Unfortunately
Rachel, Pete, and Brian perform Part 2 of the investigation in this lesson. Rachel walks while Pete and Brian hold the motion sensors. They create the unusual graph at right. The horizontal axis has a mark every 1 s, and the vertical axis has a mark every 1 m.a. The lower curve is made from the
Kari's assignment in her computer programming course is to simulate the motion of an airplane by repeatedly translating it across the screen. The coordinate system in the software program is shown at right with the origin, (0, 0), in the upper left corner. In this program, coordinates to the right
Write an equation for each parabola. Each parabola is a translation of the graph of the parent function y = x2.
Solve. a. 3 + (x - 5)2 = 19 b. (x + 3)2 = 49 c. 5 - (x - 1) = - 22 d. - 15 + (x + 6)2 = - 7
This histogram shows the students' scores on a recent quiz in Ms.Noah's class. Sketch what the histogram will look like if Ms. Noaha. Adds five points to everyone's score.b. Subtracts ten points from everyone's score.
Match each recursive formula with the equation of the line that contains the sequence of points, (n, un), generated by the formula. a. u0 = - 8 ...................................... A. y = 3x - 11 un = u(n -1) + 3 where n ≥ 1 .................. B. y = 3x - 8 b. u1 = 3
You need to rent a car for one day. Mertz Rental charges $32 per day plus $0.10 per mile. Saver Rental charges $24 per day plus $0.18 per mile. Luxury Rental charges $51 per day with unlimited mileage.a. Write a cost equation for each rental agency. b. Graph the three equations on the same axes. c.
A car drives at a constant speed along the road pictured at right from point A to point X. Sketch a graph showing the straight line distance between the car and point X as it travels along the road. Mark points A, B, C, D, E, and X on your graph.
Use geometry software to construct a segment whose length represents the starting term of a sequence. Then use transformations, such as translations and dilations, to create segments whose lengths represent additional terms in the sequence. For example, the segments at right represent the first ten
Use geometry software to investigate the form y = ax + b of a linear function. a. On the same coordinate plane, graph the lines y = 0.5x + 4, y = x + 4, y = 2x + 4, y = 5x + 4, y = - 3x + 4, and y = - 0.25x + 4. Describe the graphs of the family of lines y = ax + 4 as a takes on different
Each parabola described is congruent to the graph of y = x2. Write an equation for each parabola and sketch its graph. a. The parabola is translated down 5 units. b. The parabola is translated up 3 units. c. The parabola is translated right 3 units. d. The parabola is translated left 4 units.
If f (x) = x2, then the graph of each equation below is a parabola. Describe the location of the parabola relative to the graph of f (x) = x2. a. y = f (x) - 3 b. y = f (x) + 4 c. y = f (x - 2) d. y = f (x + 4)
Describe what happens to the graph of y = x2 in the following situations. a. x is replaced with (x - 3). b. x is replaced with (x + 3). c. y is replaced with (y - 2). d. y is replaced with (y + 2).
Solve. a. x2 = 4 b. x2 + 3 = 19 c. (x - 2)2 = 25
Write an equation for each parabola at right.
The red parabola below is the image of the graph of y = x2 after a translation right 5 units and down 3 units.a. Write an equation for the red parabola. b. Where is the vertex of the red parabola? c. What are the coordinates of the other four points if they are 1 or 2 horizontal units from the
Given the graph of y = f (x) at right, draw a graph of each of these related functions.a. y = f (x + 2) b. y = f (x - 1) - 3
This table of values compares the number of teams in a pee wee teeball league and the number of games required for each team to play every other team twice (once at home and once away from home).a. Continue the table out to 10 teams. b. Plot each point and describe the graph produced. c. Write an
Each graph at right is a transformation of the graph of the parent function y = ˆšx Write an equation for each graph.
Write the equation of a parabola that is congruent to the graph of y = - (x + 3)2 + 4, but translated right 5 units and down 2 units.
Police measure the lengths of skid marks to determine the initial speed of a vehicle before the brakes were applied. Many variables, such as the type of road surface and weather conditions, play an important role in determining the speed. The formula used to determine the initial speed is S =
Identify each relation that is also a function. For each relation that is not a function, explain why not.a. independent variable: city dependent variable: area code. b. independent variable: any pair of whole numbers dependent variable: their greatest common factor. c. independent variable: any
Solve for x. Solving square root equations often results in extraneous solutions, or answers that don't work in the original equation, so be sure to check your work. a. 3 + √x - 4 = 20 b. √x + 7 = - 3 c. 4 - (x - 2)2 = - 21 d. 5 - √-(x + 4) = 2
Find the equation of the parabola with vertex (- 6, 4), a vertical line of symmetry, and containing the point (- 5, 5).
The graph of the line ℓ1 is shown at right. a. Write the equation of the line ℓ1. b. The line ℓ2 is the image of the line ℓ1 translated right 8 units. Sketch the line ℓ2 and write its equation in a way that shows the horizontal translation. c. The line ℓ2 also can be thought of as the
Consider this data set: {37, 40, 36, 37, 37, 49, 39, 47, 40, 38, 35, 46, 43, 40, 47, 49, 70, 65, 50, 73} a. Give the five-number summary. b. Display the data in a box plot. c. Find the interquartile range. d. Identify any outliers, based on the interquartile range.
Describe what happens to the graph of y = in the following situations. a. x is replaced with (x - 3). b. x is replaced with (x + 3). c. y is replaced with (y - 2). d. y is replaced with (y + 2).
Each curve at right is a transformation of the graph of the parent function y = ˆšx. Write an equation for each curve.
Given the graph of y = f (x) below, draw a graph of each of these related functions.a. y = f (- x) b. y = - f (x) c. y = - f (- x)
Consider the parent function f (x) = √x a. Name three pairs of integer coordinates that are on the graph of y = f(x + 4) - 2. b. Write y = f (x + 4) - 2 using a radical, or square root symbol, and graph it. c. Write y = - f (x - 2) + 3 using a radical, and graph it.
Consider the parabola at right:a. Graph the parabola on your calculator. What two functions did you use? b. Combine both functions from 6a using notation to create a single relation. Square both sides of the relation. What is the resulting equation?
Refer to the two parabolas shown.a. Explain why neither graph represents a function. b. Write a single equation for each parabola using notation. c. Square both sides of each equation in 7b. What is the resulting equation of each parabola?
As Jake and Arthur travel together from Detroit to Chicago, each makes a graph relating time and distance. Jake, who lives in Detroit and keeps his watch on Detroit time, graphs his distance from Detroit. Arthur, who lives in Chicago and keeps his watch on Chicago time (1 hour earlier than
Write the equation of each parabola. Each parabola is a transformation of the graph of the parent function y = x2.
Each graph is a transformation of the graph of one of the parent functions you've studied. Write an equation for each graph.
Sketch a graph of each of these equations.a.b. c.
Given the graph of y = f (x), draw graphs of these related functions.a. b. c.
A chemistry class gathered these data on the conductivity of a base solution as acid is added to it. Graph the data and use transformations to find a model to fit the data.
A panel of judges rate 20 science fair exhibits as shown. The judges decide that the top rating should be 100, so they add 6 points to each rating.a. What are the mean and the standard deviation of the ratings before adding 6 points? b. What are the mean and the standard deviation of the ratings
This table shows the percentage of households with computers in the United States in various years.a. Make a scatter plot of these data. b. Find the median-median line. c. Use the median-median line to predict the percentage of households with computers in 2002. d. Is a linear model a good model
Describe what happens to the graph of y = f (x) in these situations. a. x is replaced with x/3. b. x is replaced with - x. c. x is replaced with 3x. d. y is replaced withy/2. e. y is replaced with - y. f. y is replaced with 2y.
Solve each equation for y. a. y + 3 = 2(x - 5)2 b. y + 5 / 2 = |x + 1 / 3/| c. y + 7 /-2 = √x - 6 /-3
Choose a few different values for a. What can you conclude about y = a | x | and y = | ax |? Are they the same function?
The graph at right shows how to solve the equation | x - 4 | = 3 graphically. The equations y = | x - 4 | and y = 3 are graphed on the same coordinate axes.a. What is the x-coordinate of each point of intersection?
You can use a single radio receiver to find the distance to a transmitter by measuring the strength of the signal. Suppose these approximate distances are measured with a receiver while you drive along a straight road. Find a model that fits the data. Where do you think the transmitter might be
Assume that you know the vertex of a parabola is (5, - 4). a. If the parabola is stretched vertically by a factor of 2 in relation to the graph of y = x2, what are the coordinates of the point 1 unit to the right of the vertex? b. If the parabola is stretched horizontally by a factor of 3 in
Given the parent function y = x2, describe the transformations represented by the function y - 2 / 3 = (x + 7 / 4)2. Sketch a graph of the transformed parabola
A parabola has vertex (4.7, 5) and passes through the point (2.8, 9). a. What is the equation of the axis of symmetry for this parabola? b. What is the equation of this parabola? c. Is this the only parabola passing through this vertex and point? Explain. Sketch a graph to support your answer.
Each equation represents a single transformation. Copy and complete this table.
Refer to Exercise 13 in Lesson 4.6. The original data is shown at right. Instead of adding the same number to each score, one of the judges suggests that perhaps they should multiply the original scores by a factor that makes the highest score equal 100. They decide to try this method.a. By what
Find the next three terms in this sequence: 16, 40, 100, 250, . . .
Solve. Give answers to the nearest 0.01.a.b. c. d.
This table shows the distances needed to stop a car on dry pavement in a minimum length of time for various speeds. Reaction time is assumed to be 0.75 s.a. Construct a scatter plot of these data. b. Find the equation of a parabola that fits the points and graph it. c. Find the residuals for this
Consider the linear function y = 3x + 1. a. Write the equation of the image of the graph of y = 3x + 1 after a reflection across the x-axis. Graph both lines on the same axes. b. Write the equation of the image of the graph of y = 3x + 1 after a reflection across the y-axis. Graph both lines on the
Use f(x) = √1 - x2 to graph each of the transformations below. a. g(x) = - f (x b. h(x) = - 2f (x) c. (x) = - 3 + 2f (x)
Each curve is a transformation of the graph of y = ˆš1 - x2. Write an equation for each curve.a.b. c. d. e. f.
Write an equation and draw a graph for each transformation of the unit circle. Use the form u = ± √1 - x2. a. Replace y with (y - 2). b. Replace x with (x + 3). c. Replace y with y / 2. d. Replace x with x / 2.
To create the ellipse at right, the x-coordinate of each point on a unit circle has been multiplied by a factor of 3.a. Write the equation of this ellipse. b. What expression did you substitute for x in the parent equation? c. If y = f (x) is the function for the top half of a unit circle, then
Given the unit circle at right, write the equation that generates each transformation. Use the form x2 + y2 = 1.a. Each y-value is half the original y-value. b. Each x-value is half the original x-value. c. Each y-value is half the original y-value, and each x-value is twice the original x-value.
Consider the ellipse at right.a. Write two functions that you could use to graph this ellipse.b. Use to write one equation that combines the two equations in 8a. c. Write another equation for the ellipse by squaring both sides of the equation in 8b.
Follow these steps to explore a relationship between linear, quadratic, square root, absolute-value, and semicircle functions.Use friendly windows of an appropriate size.a. Graph these equations simultaneously on your calculator. The first four functions intersect in the same two points. What are
Given the functions f(x) = 3 + √x + 5 and g(x) = 2 + (x - 1)2, find these values. a. f(4) b. f (g(4)) c. g(- 1) d. g( f (- 1))

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