A16—A17 3. If 3 and 4 are the legs of a right triangle, the hypotenuse is. A14 4. Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths 11, 60, and 61 is a right triangle. Concepts and Vocabulary 7. If 1x, y2 are the coordinates of a point P in the xy-plane, True or False The distance between two points is some- then x is called the of P and y is the times a negative number.
True or False The point 1 - 1, 42 lies in quadrant IV of the 8. The coordinate axes divide the xy-plane into four sections Cartesian plane.
True or False The midpoint of a line segment is found by 9. Skill Building In Problems 13 and 14, plot each point in the xy-plane. Tell in which quadrant or on what coordinate axis each point lies. Plot the points 12, 02, 12, - 32, 12, 42, 12, 12, and 12, - Describe the set of all points of the form 12, y2, where y is a real number. Plot the points 10, 32, 11, 32, 1 - 2, 32, 15, 32, and 1 - 4, Describe the set of all points of the form 1x, 32, where x is a real number.
In Problems 29—34, plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find its area. In Problems 35—42, find the midpoint of the line segment joining the points P1 and P2. Applications and Extensions If the point 12, 52 is shifted 3 units to the right and 2 units C Median down, what are its new coordinates? If the point 1 - 1, 62 is shifted 2 units to the left and 4 units Midpoint up, what are its new coordinates? Find all points having an x-coordinate of 3 whose distance A B from the point 1 - 2, is Geometry An equilateral triangle is one in which all three a By using the Pythagorean Theorem.
If two vertices of an equilateral b By using the distance formula. How Find all points having a y-coordinate of - 6 whose distance many of these triangles are possible? Find all points on the x-axis that are 6 units from the point 14, - Find all points on the y-axis that are 6 units from the point Geometry Find the midpoint of each diagonal of a square 14, Draw the conclusion that the diagonals. The midpoint of the line segment from P1 to P2 is 1- 1, If of a square intersect at their midpoints.
The midpoint of the line segment from P1 to P2 is 15, - If square. Geometry Verify that the points 0, 0 , a, 0 , and a 23 a a , b are the vertices of an equilateral triangle. Then Geometry The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side see 2 2 the figure. In Problems 55—58, find the length of each side of the triangle determined by the three points P1 , P2 , and P3.
State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. An isosceles triangle is one in which at least two of the sides are of equal length. Find an expression for the distance What is the distance directly from home plate to second base the diagonal of the square? Drafting Error When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as Little League Baseball The layout of a Little League intended and subsequently will form an error triangle.
If playing field is a square, 60 feet on a side. How far is it this error triangle is long and thin, one estimate for the directly from home plate to second base the diagonal of the location of the desired point is the midpoint of the shortest square? The figure shows one such error triangle. Baseball Refer to Problem Overlay a rectangular 2. Use feet as the unit of measurement. See Problem Net Sales The figure illustrates how net sales of Wal-Mart from the center fielder to third base?
Stores, Inc. Use the Little League Baseball Refer to Problem Overlay a midpoint formula to estimate the net sales of Wal-Mart rectangular coordinate system on a Little League baseball Stores, Inc. Wal-Mart Stores, Inc.
The Neon 50 heads east at an average speed of 30 miles per hour, while 0 the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d in miles at the end of t hours. Poverty Threshold Poverty thresholds are determined by Assuming poverty thresholds increase in a straight-line the U.
Census Bureau. A poverty threshold represents the fashion, use the midpoint formula to estimate the poverty minimum annual household income for a family not to be threshold of a family of four with two children under the considered poor. In , the poverty threshold for a family age of 18 in In , the poverty threshold for a family of four Source: U.
Explaining Concepts: Discussion and Writing Write a paragraph that describes a Cartesian plane. Your paragraphs should include the. True 2. The expressions are called the sides of the equation.
Since an equation is a statement, it may be true or false, depending on the value of the variables. Any values of x and y that result in a true statement are said to satisfy the equation.
The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates 1x, y2 satisfy the equation. Graphs play an important role in helping us to visualize the relationships that exist between two variables or quantities. Figure 11 on page 10 shows the relation between the level of risk in a stock portfolio and the average annual rate of return.
Figure 11 To locate some of these points and get an idea of the pattern of the graph , assign some numbers to x and find corresponding values for y. By plotting these points and then connecting them, we obtain the graph of the equation a line , as shown in Figure In Figure 13 we plot these points and connect them with a smooth curve to obtain the graph a parabola. This is referred to as a complete graph.
One way to obtain a complete graph of an equation is to plot a sufficient number of points on the graph until a pattern becomes evident.
Then these points are connected with a smooth curve following the suggested pattern. Sometimes knowledge about the equation tells us. Read Section B. In this case, only two points are needed to obtain the graph. Using a Graphing Utility to Graph One purpose of this book is to investigate the properties of equations in order Equations, in Appendix B.
Sometimes we shall graph equations by Figure 14 plotting points. Shortly, we shall investigate various techniques that will enable us to graph an equation without plotting so many points. The x-coordinate of a point at which the graph crosses Graph or touches the x-axis is an x-intercept, and the y-coordinate of a point at which the touches Intercepts graph crosses or touches the y-axis is a y-intercept.
For a graph to be complete, all its x-axis intercepts must be displayed. What are its x-intercepts? What are its y-intercepts? Solution The intercepts of the graph are the points. For x-intercepts, report the x-coordinate of the intercept; for y-intercepts, report the y-coordinate of the intercept. In such cases, a graphing utility can be used. Appendix B, to find out how to locate 2. The equation has two solutions, - 2 and 2. The x-intercepts are - 2 and 2. Another helpful tool for graphing equations involves symmetry, particularly symmetry with respect to the x-axis, the y-axis, and the origin.
Figure 17 illustrates the definition. When a graph is symmetric with respect to the x-axis, notice that the part of the graph above the x-axis is a reflection or mirror image of the part below it, and vice versa.
Figure 17 y Symmetry with respect to the x-axis x, y x, y x, y. Figure 18 Symmetry with respect to the y-axis y —x, y x, y Figure 18 illustrates the definition. When a graph is symmetric with respect to the y-axis, notice that the part of the graph to the right of the y-axis is a reflection of the part to the left of it, and vice versa.
Figure 19 Symmetry with respect to the origin Figure 19 illustrates the definition. Notice that symmetry with respect to the y x, y origin may be viewed in three ways: x, y 1. As a reflection about the y-axis, followed by a reflection about the x-axis 2.
As a projection along a line through the origin so that the distances from the —x, —y x origin are equal —x, —y 3. As half of a complete revolution about the origin. When the graph of an equation is symmetric with respect to a coordinate axis or the origin, the number of points that you need to plot in order to see the pattern is reduced. For example, if the graph of an equation is symmetric with respect to the y-axis, then, once points to the right of the y-axis are plotted, an equal number of points on the graph can be obtained by reflecting them about the y-axis.
Because of this, before we graph an equation, we first want to determine whether it has any symmetry. The following tests are used for this purpose. If an equivalent equation results, the graph of the equation is symmetric with respect to the x-axis. If an equivalent equation results, the graph of the equation is symmetric with respect to the y-axis. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin. Origin: To test for symmetry with respect to the origin, replace x by - x and y by - y.
It is important to know the graphs of these key equations because we use them later. Find any intercepts and check for symmetry first. Solution First, find the intercepts.
The origin 10, 02 is the only intercept. Now test for symmetry. Figure 21 Origin: Replace x by - x and y by - y. See Table 3. Since 11, 12 is on the graph, and the graph is symmetric with respect to the origin, the point 1 -1, is also on the graph. Plot the points from 0, 0 1, 1 Table 3 and use the symmetry. Figure 21 shows the graph. Solution a The lone intercept is 10, The graph is symmetric with respect to the x-axis.
Do you see why? Replace y by - y. Figure 22 shows the graph. We discuss why in Chapter 2. Table 4 Solution Check for intercepts first. We conclude that there is no y-intercept. We conclude that there is no x-intercept. The graph is symmetric with respect to the origin. Because of the symmetry 10 10 10 with respect to the origin, we use only positive values of x. Armed with this information, we can graph the equation. Observe how 1, 1 x —3 2, ——12 the absence of intercepts and the existence of symmetry with respect to the origin were utilized.
A44—A48 2. Concepts and Vocabulary 3. The points, if any, at which a graph crosses or touches the 7. If the graph of an equation is symmetric with respect to the coordinate axes are called. The x-intercepts of the graph of an equation are those is also a point on the graph. True or False To find the y-intercepts of the graph of an 5. True or False The y-coordinate of a point at which the symmetric with respect to the. If the graph of an equation is symmetric with respect to the True or False If a graph is symmetric with respect to the y-axis and - 4 is an x-intercept of this graph, then is x-axis, then it cannot be symmetric with respect to the also an x-intercept.
Skill Building In Problems 11—16, determine which of the given points are on the graph of the equation. In Problems 17—28, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. In Problems 29—38, plot each point. Then plot the point that is symmetric to it with respect to a the x-axis; b the y-axis; c the origin. In Problems 39—50, the graph of an equation is given.
In Problems 51—54, draw a complete graph so that it has the type of symmetry indicated. Origin y In Problems 55—70, list the intercepts and test for symmetry. In Problems 71—74, draw a quick sketch of each equation.
Given that the point 1, 2 is on the graph of an equation Solar Energy The solar electric generating systems at that is symmetric with respect to the origin, what other point Kramer Junction, California, use parabolic troughs to heat a is on the graph?
This fluid is used If the graph of an equation is symmetric with respect to the to generate steam that drives a power conversion system to y-axis and 6 is an x-intercept of this graph, name another produce electricity. For troughs 7. If the graph of an equation is symmetric with respect to the origin and - 4 is an x-intercept of this graph, name another x-intercept.
If the graph of an equation is symmetric with respect to the x-axis and 2 is a y-intercept, name another y-intercept.
Microphones In studios and on stages, cardioid micro- phones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Source: www. Department of Energy. Draw a graph that contains the points 1 - 2, - 12, 10, 12, noting which graphs are the same.
Are most of the graphs almost straight lines? How same. Explain why, if two of the same. Explain what is meant by a complete graph. Draw a graph of an equation that contains two x-intercepts; Draw a graph that contains the points - 2, 5 , -1, 3 , at one the graph crosses the x-axis, and at the other the and 0, 2 that is symmetric with respect to the y-axis.
Compare your graph with those of other students; comment Make up an equation with the intercepts 12, 02, 14, 02, and on any similarities. Can a graph contain these points and 10, Why Comment on any similarities. Interactive Exercises Ask your instructor if the applets below are of interest to you. Move are the coordinates of point A and the coordinates of point B point A around the Cartesian plane with your mouse.
How related? Origin Symmetry Open the origin symmetry applet. Move related? In this section we study a certain type of equation that contains two variables, called a linear equation, and its graph, a line. Each step contains exactly the same horizontal run and the same vertical rise. The ratio of the rise to the run, called the slope, is a numerical measure of the steepness of the staircase. For example, if the Rise run is increased and the rise remains the same, the staircase becomes less steep.
If the Run run is kept the same, but the rise is increased, the staircase becomes more steep. This important characteristic of a line is best defined using rectangular coordinates. If x1 Z x2 , the slope m of the nonvertical line L containing P and Q is defined by the formula. Figure 27 a on page 20 provides an illustration of the slope of a nonvertical line; Figure 27 b illustrates a vertical line. Figure 27 L. As Figure 27 a illustrates, the slope m of a nonvertical line may be viewed as.
The expression is called the average rate of change of y, with respect to x. Any two distinct points on the line can be used to compute the slope of the line. See Figure 28 for justification. Since any two distinct points can be used to compute the slope of a line, the average rate of change of a line is always the same number. That is, if x increases by 4 units, then y will decrease by 5 units. The average rate of change of y with respect 5 to x is -. Graph all four lines on the same set of coordinate axes.
When the slope of a line is positive, the line slants upward from left to right 1L When the slope of a line is negative, the line slants downward from left to right 1L When the slope is 0, the line is horizontal 1L When the slope is undefined, the line is vertical 1L Figures 30 and 31 on page 21 illustrate that the closer the line is to the vertical position, the greater the magnitude of the slope.
The fact that the slope is means that for every horizontal Figure 32 Run 4 movement run of 4 units to the right there will be a vertical movement rise y of 3 units. Look at Figure If we start at the given point 13, 22 and move 4 units to the right and 3 units up, we reach the point 17, By drawing the line 6 7, 5 through this point and the point 13, 22, we have the graph. If we Figure 33 start at the given point 13, 22 and move 5 units to the right and then 4 units y down, we arrive at the point 18, - By drawing the line through these points, we have the graph.
This approach —2 8, —2 brings us to the point 1 - 2, 62, which is also on the graph shown in Figure No matter what y-coordinate is used, the corresponding x-coordinate always equals 3. Figure 34 y 4. As suggested by Example 4, we have the following result:. To overcome this, most graphing utilities have special commands for drawing vertical lines. Consult your manual to determine the correct methodology for your graphing utility. See L. Figure For any other point 1x, y2 on L, we have x, y.
Solution Because all the y-values are equal on a horizontal line, the slope of a horizontal line Figure 37 is 0. As suggested by Example 6, we have the following result:. Graph the line. In the solution to Example 7, we could have used the other point, 1 - 4, 52, instead of the point 12, The equation that results, although it looks different, is equivalent to the equation that we obtained in the example. Try it for yourself. In this event, we know both the slope m of the line and a point 10, b2 on the line; then we use the point—slope form, equation 2 , to obtain the following equation:.
Graph the equation. Solution To obtain the slope and y-intercept, write the equation in slope—intercept form by solving for y. Graph the line using 4 2 1 2 the fact that the y-intercept is 2 and the slope is -. Then, starting at the point 0, 2 1 2 10, 22, go to the right 2 units and then down 1 unit to the point 12, If B Z 0 in 4 , then we can solve the equation for y and write the equation A in slope—intercept form as we did in Example 8. Another approach to graphing the equation 4 would be to find its intercepts.
Remember, the intercepts of the graph of an equation are the points where the graph crosses or touches a coordinate axis. The x-intercept is 4 and the point 14, 02 is on the graph of the equation. Every line has an equation that is equivalent to an equation written in general form.
Because the equation of every line can be written in general form, any equation equivalent to equation 4 is called a linear equation.
There we have drawn two y parallel lines and have constructed two right triangles by drawing sides parallel to the coordinate axes. The right triangles are similar. Two angles are equal. Because the triangles are similar, the ratios of corresponding sides are equal. Rise Rise. If two nonvertical lines are parallel, then their slopes are equal and they have different y-intercepts.
If two nonvertical lines have equal slopes and they have different y-intercepts, then they are parallel. Figure 45 The slope is - 2. Since the line that we seek also has slope - 2 and contains the y point 12, , use the point—slope form to obtain its equation. If two nonvertical lines are perpendicular, then the product of their slopes is If two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.
Figure 47 Proof Let m1 and m2 denote the slopes of the two lines. There is no loss in gener- y ality that is, neither the angle nor the slopes are affected if we situate the lines so that they meet at the origin. Suppose that the lines are perpendicular. Then triangle OAB is a right triangle. If the lines are perpendicular, the product of their slopes is - 1.
Graph the two lines. Solution First write the equation of the given line in slope—intercept form to find its slope. Any line perpendicular to this line will have slope 3. Figure 48 shows the graphs. Otherwise, the angle between the two lines will appear distorted.
A discussion of square screens is given in Section B. The slope of a vertical line is ; the slope of a 7. Two nonvertical lines have slopes m1 and m2 , respectively. The lines are parallel if and the 2.
A horizontal line is given by an equation of the form , where b is the. True or False Vertical lines have an undefined slope. True or False Perpendicular lines have slopes that are 6. Skill Building In Problems 11—14, a find the slope of the line and b interpret the slope. In Problems 15—22, plot each pair of points and determine the slope of the line containing them. In Problems 23—30, graph the line containing the point P and having slope m.
In Problems 31—36, the slope and a point on a line are given. Use this information to locate three additional points on the line. Answers may vary. See Example 3. Slope 4; point 11, 22 Slope 2; point 1 - 2, 32 Slope - ; point 12, 3 2 Slope ; point 1 - 3, 22 Slope - 2; point 1 - 2, - 32 Slope - 1; point 14, 12 4 3 In Problems 37—44, find an equation of the line L. In Problems 45—70, find an equation for the line with the given properties. Express your answer using either the general form or the slope—intercept form of the equation of a line, whichever you prefer.
Containing the points 11, 32 and 1 - 1, 22 Containing the points 1 - 3, 42 and 12, 52 Slope undefined; containing the point 12, 42 Slope undefined; containing the point 13, Horizontal; containing the point 1 - 3, 22 Vertical; containing the point 14, - In Problems 71—90, find the slope and y-intercept of each line.
In Problems 91—, a find the intercepts of the graph of each equation and b graph the equation. Find an equation of the x-axis. Find an equation of the y-axis. In Problems —, the equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither.
In Problems —, write an equation of each line. Geometry Use slopes to show that the triangle whose a Write a linear equation that relates the monthly charge vertices are 1 -2, 52, 11, 32, and 1 - 1, 02 is a right triangle. C, in dollars, to the number x of kilowatt-hours used in a Geometry Use slopes to show that the quadrilateral month, 0 … x … Geometry Use slopes to show that the quadrilateral whose vertices are 1 - 1, 02, 12, 32, 11, - 22, and 14, 12 is a e Interpret the slope of the line.
Source: Commonwealth Edison Company, January, Geometry Use slopes and the distance formula to show Write a C, in dollars, to the number x of kilowatt-hours used in a linear equation that relates the cost C, in dollars, of renting month, 0 … x … What is the cost b Graph this equation.
Cost Equation The fixed costs of operating a business are e Interpret the slope of the line. Fixed costs include rent, fixed salaries, and costs of leasing Measuring Temperature The relationship between Celsius machinery. Variable costs is linear. Write a linear equation that relates the daily cost C, in Measuring Temperature The Kelvin K scale for measuring dollars, of manufacturing the jeans to the number x of jeans temperature is obtained by adding to the Celsius manufactured.
What is the cost of manufacturing pairs temperature. Cost of Driving a Car The annual fixed costs for owning a Problem Access Ramp A wooden access ramp is being built to Write a linear equation that relates the cost C and the reach a platform that sits 30 inches above the floor. The number x of miles driven annually. Electricity Rates in Illinois Commonwealth Edison ramp above the floor to the horizontal distance x from Company supplies electricity to residential customers for a the platform.
Will this ramp meet the requirements? In , Is this result reasonable? Show that the line containing the points 1a, b2 and 1b, a2, Source: www. Also show that the On one set of coordinate axes, graph the of money it spends on advertising.
Prove that if two nonvertical lines have slopes whose product on advertising to the number x of boxes the company is -1 then the lines are perpendicular. Figure 47 and use the converse of the Pythagorean Theorem. Which of the following equations might have the graph More than one answer is possible. Investigate the origin of this symbolism. Write a brief essay on your findings. Which of the following equations might have the graph shown?
Carpentry Carpenters use the term pitch to describe the The figure shows the graph of two parallel lines. Which of the steepness of staircases and roofs. How does pitch relate to following pairs of equations might have such a graph? Are there any lines that have no intercepts? Assume that the x-intercept is The figure shows the graph of two perpendicular lines. Which of the following pairs of equations might have such a If two distinct lines have the same slope, but different graph?
Which form of the equation of a line do you prefer to use? Have reasons. What Went Wrong? Is he correct? If not, what went wrong? Interactive Exercises Ask your instructor if the applet below is of interest to you.
Slope Open the slope applet. Move point B around the Cartesian plane with your mouse. What is the slope of the line?
What happens to the value of the slope as the x-coordinate approaches 1? What can be said about a line whose slope is negative? What can be said about a line whose slope is 0?
What can be said about the steepness of a line with positive slope as its slope increases? Move B to the point whose coordinates are 15, Move B to the point whose coordinates are 1 - 1, A29—A30 p.
Consider, for example, the following geometric statement that defines a circle. The fixed distance r is called the radius, and the fixed point 1h, k2 is called the center of the circle.
Figure 49 shows the graph of a circle. To find the equation, let 1x, y2 represent Figure Then the y x, y distance between the points 1x, y2 and 1h, k2 must always equal r. That is, by the r distance formula. Notice that the graph of the unit circle is symmetric with respect to the x-axis, the y-axis, and the origin.
To graph the equation, compare the given equation to the standard form of the equation of a circle. The comparison yields information about the circle. The circle has center 1 -3, 22 and a —7, 2 1, 2 —3, 2 radius of 4 units.
To graph this circle, first plot the center 1 - 3, Since the radius is —10 —5 2 x 4, we can locate four points on the circle by plotting points 4 units to the left, to the —3, —2 right, up, and down from the center. These four points can then be used as guides to obtain the graph. Solution This is the equation discussed and graphed in Example 2. The x-intercepts are - 3 - 2 23 L - 6. Look back at Figure 51 to verify the approximate locations of the intercepts. If an equation of a circle is in the general form, we use the method of completing the square to put the equation in standard form so that we can identify its center and radius.
Remember that any number added on the left side of the equation must also be added on the right. This equation is the standard form of the equation of a circle with radius 1 and 1 x center , To graph the equation use the center 1 - 2, 32 and the radius 1.
To graph this Figure 53 equation, solve for y. Answers are given at the end of these exercises. Use the Square Root Method to solve the equation subtract the number. True or False Every equation of the form 5. True or False The center of the circle has a circle as its graph. For a circle, the is the distance from the center to any is 3, - 2.
Skill Building In Problems 7—10, find the center and radius of each circle. Write the standard form of the equation. In Problems 11—20, write the standard form of the equation and the general form of the equation of each circle of radius r and center 1h, k2.
Graph each circle. In Problems 21—34, a find the center 1h, k2 and radius r of each circle; b graph each circle; c find the intercepts, if any. Center at the origin and containing the point 1 - 2, 32 Center 11, 02 and containing the point 1 - 3, Center 12, 32 and tangent to the x-axis Center 1 - 3, 12 and tangent to the y-axis. With endpoints of a diameter at 11, 42 and 1 - 3, 22 With endpoints of a diameter at 14, 32 and 10, In Problems 43—46, match each graph with the correct equation.
Find the area of the square in the figure. A weather satel- lite circles 0. Find the equation for the orbit of the satellite on this map. Find the area of the blue shaded region in the figure, assum- ing the quadrilateral inside the circle is a square.
See the figure. Find an equation for the wheel if the center of the wheel is on the y-axis. Source: inventors. It has a maximum height of b b meters and a diameter of meters, with one full rotation c The tangent line is perpendicular to the line containing taking approximately 30 minutes.
Find an equation for the the center of the circle and the point of tangency. The Greek Method The Greek method for finding the Source: Wikipedia equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular see Problem Refer to Problem Find the center of the circle.
If a circle of radius 2 is made to roll along the x-axis, what is an equation for the path of the center of the circle? If the circumference of a circle is 6p, what is its radius? Explain how the center and radius of a circle can be used to graph the circle. Why is this incorrect? Suppose that you have a rectangular field that requires watering. Your watering system consists of an arm of variable length that rotates so that the watering pattern is a circle.
Decide where to position the arm and what length it should be so that the entire field is watered most efficiently. When does it become desirable to use more than one arm? Write equations for the circle s swept out by the watering arm s. Square field Rectangular field, one arm Rectangular field, two arms. Place the cursor on the center of the circle and hold b Move B to a point in the Cartesian plane directly above the mouse button.
Drag the center around the Cartesian the center such that the radius of the circle is 5. What is the equa- d Move B to a point in the Cartesian plane such that the tion of the circle?
What is the e Find the coordinates of two points with integer coordi- equation of the circle? What is the a circle of radius 5 with center equal to that found in equation of the circle?
What is the f Use the concept of symmetry about the center, vertical equation of the circle? Place the cursor on point B, press and hold the equal to that found in part a.
Drag B around the Cartesian plane. Equations of Lines and Circles Vertical line p. Objectives Section You should be able to. Examples Review Exercises 1. Review Exercises In Problems 1—6, find the following for each pair of points: 8. List the intercepts of the graph below.
In Problems 9—16, list the intercepts and test for symmetry with respect to the x-axis, the y-axis, and the origin. In Problems 17—20, find the standard form of the equation of the circle whose center and radius are given. In Problems 21—26, find the center and radius of each circle. Find the intercepts, if any, of each circle.
In Problems 27—36, find an equation of the line having the given characteristics. Vertical; containing the point 1 - 3, 42 Containing the points 13, - 42 and 12, In Problems 37—40, find the slope and y-intercept of each line. Graph the line, labeling any intercepts. In Problems 41—44, find the intercepts and graph each line. Graph the line with slope 3 Chapter Project Each problem should a By using the converse of the Pythagorean Theorem involve a different concept. Be sure that your directions are b By using the slopes of the lines joining the vertices clearly stated.
The endpoints of the diameter of a circle are 1 - 3, 22 and Describe each of the following graphs in the xy-plane. Give 15, - Find the center and radius of the circle. Write the justification. Write the slope—intercept form of the line with slope -2 1. Find the distance from P1 to P2.
Find the midpoint of the line segment joining P1 and P2. Write the general form of the circle with center 14, and radius 5. Graph this circle. Also find a line perpendicular to it 5. Treating year as the independent variable and the winning value as the dependent variable, find linear equations relating these variables separately for men and women using the data for the years and Compare the Internet-based Project equations and comment on any similarities or differences.
Predicting Olympic Performance Measurements of human 2. Interpret the slopes in your equations from part 1. Do the performance over time sometimes follow a strong linear y-intercepts have a reasonable interpretation? Why or why relationship for reasonably short periods. In the Summer not? Olympic Games returned to Greece, the home of both the 3. Use your equations to predict the winning time in the ancient Olympics and the first modern Olympics.
The following Olympics. Compare your predictions to the actual results data represent the winning times in hours for men and women 2. How well in the Olympic marathon.
Repeat parts 1 to 3 using the data for the years and 6. Pick your favorite Winter Olympics event and find the How do your results compare? Would your equations be useful in predicting the winning Winter Olympics prior to Repeat parts 1 to 3 using marathon times in the Summer Olympics? Why or your selected event and years and compare to the actual why not? Functions and Their Graphs Outline 2. Choosing a Cellular Telephone Plan Most consumers choose a cellular telephone provider first, and then select an appropriate plan from that provider.
The choice as to the type of plan selected depends upon your use of the phone. For example, is text messaging important? How many minutes do you plan to use the phone? Do you desire a data plan to browse the Web? The mathematics learned in this chapter can help you decide the plan best-suited for your particular needs.
So far, our discussion has focused on techniques for graphing equations containing two variables. In this chapter, we look at a special type of equation involving two variables called a function.
This chapter deals with what a function is, how to graph functions, properties of functions, and how functions are used in applications. For him, a function simply meant any positive integral power of a variable x. Gottfried Wilhelm Leibniz — , who always emphasized the geometric side of mathematics, used the word function to denote any quantity associated with a curve, such as the coordinates of a point on the curve. Leonhard Euler — employed the word to mean any equation or formula involving variables and constants.
His idea of a function is similar to the one most often seen in courses that precede calculus. Later, the use of functions in investigating heat flow equations led to a very broad definition, due to Lejeune Dirichlet — , which describes a function as a correspondence between two sets. It is his definition that we use here. Engine size is linked to gas mileage. When the value of one variable is related to the value of a second variable, we have a relation.
A relation is a correspondence Figure 1 between two sets. If x and y are two elements in these sets and if a relation exists y between x and y, then we say that x corresponds to y or that y depends on x, and we 5 write x : y. There are a number of ways to express relations between two sets. It says that 1, 2 if we take some number x, multiply it by 3, and then subtract 1 we obtain the corresponding value of y.
Not only can a relation be expressed through an equation or graph, but we can also express a relation through a technique called mapping.
Ordered pairs can be used to represent x : y as 1x, y2. In this relation, Alaska corresponds to 1, Arizona corresponds to 8, and so on. Using ordered pairs, this relation would be expressed as. One of the most important concepts in algebra is the function. A function is a special type of relation. See Figure 3. Notice that Colleen has two Dan — telephone numbers. Figure 4 is a relation that shows a correspon- Colleen — dence between animals and life expectancy.
Neutrosophy means the study of ideas and notions that are not true, nor false, but in between i. Each field has a neutrosophic part, i. Thus, there were born the neutrosophic logic, neutrosophic set, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus, etc. There exist many types of indeterminacies — that is why neutrosophy can be developed in many different ways. The Larson Calculus program has a long history of innovation in the calculus market.
Two primary objectives guided the authors in writing this book: to develop precise, readable materials for students that clearly define and demonstrate concepts and rules of calculus and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and saves the instructor time. These changes have been incorporated without sacrificing the mathematical soundness that has been paramount to the success of this text and series.
With the Ninth Edition, the author continues to revolutionize the way students learn material by incorporating more real-world applications, on-going review and innovative technology.
The HYBRID demonstrates Larson's commitment to revolutionizing the way instructors teach and students learn material by moving all the end-of-section exercises from the text online as well as incorporating more real-world applications, ongoing review, and innovative technology. The book emphasizes integrated and engaging applications that show students the real-world relevance of topics and concepts. Applied problems drawn from government sources, industry, current events, and other disciplines provide well-rounded examples and appeal to students' diverse interests.
The Ninth Edition builds upon its applications emphasis through updated exercises and relevant examples. Pedagogical features--from algebra review to study tips--continue to provide extra guidance and practice.
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