1.2. Lines, Circles, and Parabolas. Cartesian Coordinates in the Plane

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1 . Lines, Circles, and Parabolas 9. Lines, Circles, and Parabolas Positive -ais Negative -ais b Negative -ais Origin a Positive -ais P(a, b) FIGURE.5 Cartesian coordinates in the plane are based on two perpendicular aes intersecting at the origin. (, ) Second quadrant (, ) Third quadrant (, ) First quadrant (, ) (, ) (, ) (, ) (, ) (, ) (, ) Fourth quadrant (, ) FIGURE.6 Points labeled in the coordinate or Cartesian plane. The points on the aes all have coordinate pairs but are usuall labeled with single real numbers, (so (, ) on the -ais is labeled as ). Notice the coordinate sign patterns of the quadrants. This section reviews coordinates, lines, distance, circles, and parabolas in the plane. The notion of increment is also discussed. Cartesian Coordinates in the Plane In the previous section we identified the points on the line with real numbers b assigning them coordinates. Points in the plane can be identified with ordered pairs of real numbers. To begin, we draw two perpendicular coordinate lines that intersect at the -point of each line. These lines are called coordinate aes in the plane. On the horizontal -ais, numbers are denoted b and increase to the right. On the vertical -ais, numbers are denoted b and increase upward (Figure.5). Thus upward and to the right are positive directions, whereas downward and to the left are considered as negative. The origin O, also labeled, of the coordinate sstem is the point in the plane where and are both zero. If P is an point in the plane, it can be located b eactl one ordered pair of real numbers in the following wa. Draw lines through P perpendicular to the two coordinate aes. These lines intersect the aes at points with coordinates a and b (Figure.5). The ordered pair (a, b) is assigned to the point P and is called its coordinate pair. The first number a is the -coordinate (or abscissa) of P; the second number b is the -coordinate (or ordinate) of P. The -coordinate of ever point on the -ais is. The -coordinate of ever point on the -ais is. The origin is the point (, ). Starting with an ordered pair (a, b), we can reverse the process and arrive at a corresponding point P in the plane. Often we identif P with the ordered pair and write P(a, b). We sometimes also refer to the point (a, b) and it will be clear from the contet when (a, b) refers to a point in the plane and not to an open interval on the real line. Several points labeled b their coordinates are shown in Figure.6. This coordinate sstem is called the rectangular coordinate sstem or Cartesian coordinate sstem (after the siteenth centur French mathematician René Descartes). The coordinate aes of this coordinate or Cartesian plane divide the plane into four regions called quadrants, numbered counterclockwise as shown in Figure.6. The graph of an equation or inequalit in the variables and is the set of all points P(, ) in the plane whose coordinates satisf the equation or inequalit. When we plot data in the coordinate plane or graph formulas whose variables have different units of measure, we do not need to use the same scale on the two aes. If we plot time vs. thrust for a rocket motor, for eample, there is no reason to place the mark that shows sec on the time ais the same distance from the origin as the mark that shows lb on the thrust ais. Usuall when we graph functions whose variables do not represent phsical measurements and when we draw figures in the coordinate plane to stud their geometr and trigonometr, we tr to make the scales on the aes identical. A vertical unit of distance then looks the same as a horizontal unit. As on a surveor s map or a scale drawing, line segments that are supposed to have the same length will look as if the do and angles that are supposed to be congruent will look congruent. Computer displas and calculator displas are another matter. The vertical and horizontal scales on machine-generated graphs usuall differ, and there are corresponding distortions in distances, slopes, and angles. Circles ma look like ellipses, rectangles ma look like squares, right angles ma appear to be acute or obtuse, and so on. We discuss these displas and distortions in greater detail in Section.7.

2 Chapter : Preliminaries B(, 5) (, ) 5, C(5, 6) D(5, ) 5 A(, ) FIGURE.7 Coordinate increments ma be positive, negative, or zero (Eample ). HISTORICAL BIOGRAPHY* René Descartes (596 65) P P (, ) P (, ) (run) (rise) P Q(, ) Q FIGURE.8 Triangles P QP and P Q P are similar, so the ratio of their sides has the same value for an two points on the line. This common value is the line s slope. L Increments and Straight Lines When a particle moves from one point in the plane to another, the net changes in its coordinates are called increments. The are calculated b subtracting the coordinates of the starting point from the coordinates of the ending point. If changes from to, the increment in is = -. EXAMPLE In going from the point As, -d to the point B(, 5) the increments in the - and -coordinates are = - = -, = 5 - s -d = 8. From C(5, 6) to D(5, ) the coordinate increments are = 5-5 =, = - 6 = -5. See Figure.7. Given two points P s, d and in the plane, we call the increments = - and = - the run and the rise, respectivel, between P and P. Two such points alwas determine a unique straight line (usuall called simpl a line) passing through them both. We call the line P P. An nonvertical line in the plane has the propert that the ratio m = rise run = = - - has the same value for ever choice of the two points P s, d and P s, d on the line (Figure.8). This is because the ratios of corresponding sides for similar triangles are equal. DEFINITION The constant Slope is the slope of the nonvertical line P P. m = rise run = = - - The slope tells us the direction (uphill, downhill) and steepness of a line. A line with positive slope rises uphill to the right; one with negative slope falls downhill to the right (Figure.9). The greater the absolute value of the slope, the more rapid the rise or fall. The slope of a vertical line is undefined. Since the run is zero for a vertical line, we cannot evaluate the slope ratio m. The direction and steepness of a line can also be measured with an angle. The angle of inclination of a line that crosses the -ais is the smallest counterclockwise angle from the -ais to the line (Figure.). The inclination of a horizontal line is. The inclination of a vertical line is 9. If f (the Greek letter phi) is the inclination of a line, then f 6 8. To learn more about the historical figures and the development of the major elements and topics of calculus, visit

3 . Lines, Circles, and Parabolas 6 L P (, 5) P (, ) L P (, 6) P (, ) 5 6 The relationship between the slope m of a nonvertical line and the line s angle of inclination f is shown in Figure.: Straight lines have relativel simple equations. All points on the vertical line through the point a on the -ais have -coordinates equal to a. Thus, = a is an equation for the vertical line. Similarl, = b is an equation for the horizontal line meeting the -ais at b. (See Figure..) We can write an equation for a nonvertical straight line L if we know its slope m and the coordinates of one point P s, d on it. If P(, ) is an other point on L, then we can use the two points and P to compute the slope, P m = tan f. m = - - FIGURE.9 The slope of L is m = = 6 - s -d = 8 -. That is, increases 8 units ever time increases units. The slope of L is m = = = -. That is, decreases units ever time increases units. so that - = ms - d or = + ms - d. The equation = + ms - d is the point-slope equation of the line that passes through the point s, d and has slope m. this not this this not this EXAMPLE Write an equation for the line through the point (, ) with slope ->. Solution We substitute =, =, and m = -> into the point-slope equation and obtain FIGURE. Angles of inclination are measured counterclockwise from the -ais. P P L m tan FIGURE. The slope of a nonvertical line is the tangent of its angle of inclination. When =, = 6 so the line intersects the -ais at = 6. EXAMPLE A Line Through Two Points Write an equation for the line through s -, -d and (, ). Solution The line s slope is We can use this slope with either of the two given points in the point-slope equation: With s, d s, d = - + # s - s -dd = = - A - B, or = m = = -5-5 =. With s, d s, d = + # s - d = + - = + = + Same result Either wa, = + is an equation for the line (Figure.).

4 Chapter : Preliminaries 6 5 Along this line, Along this line, (, ) FIGURE. The standard equations for the vertical and horizontal lines through (, ) are = and =. The -coordinate of the point where a nonvertical line intersects the -ais is called the -intercept of the line. Similarl, the -intercept of a nonhorizontal line is the -coordinate of the point where it crosses the -ais (Figure.). A line with slope m and -intercept b passes through the point (, b), so it has equation = b + ms - d, or, more simpl, = m + b. The equation = m + b is called the slope-intercept equation of the line with slope m and -intercept b. (, ) Lines with equations of the form = m have -intercept and so pass through the origin. Equations of lines are called linear equations. The equation A + B = C sa and B not both d is called the general linear equation in and because its graph alwas represents a line and ever line has an equation in this form (including lines with undefined slope). (, ) FIGURE. The line in Eample. EXAMPLE Finding the Slope and -Intercept Find the slope and -intercept of the line =. Solution Solve the equation for to put it in slope-intercept form: = 5 = -8 + = The slope is m = -8>5. The -intercept is b =. b L a Parallel and Perpendicular Lines Lines that are parallel have equal angles of inclination, so the have the same slope (if the are not vertical). Conversel, lines with equal slopes have equal angles of inclination and so are parallel. If two nonvertical lines L and L are perpendicular, their slopes m and m satisf m m = -, so each slope is the negative reciprocal of the other: m =- m, m =- m. FIGURE. Line L has -intercept a and -intercept b. To see this, notice b inspecting similar triangles in Figure.5 that m = -h>a. Hence, m m = sa>hds -h>ad = -. m = a>h, and

5 . Lines, Circles, and Parabolas L L C Distance and Circles in the Plane The distance between points in the plane is calculated with a formula that comes from the Pthagorean theorem (Figure.6). Slope m Slope m h A D a B FIGURE.5 ADC is similar to CDB. Hence f is also the upper angle in CDB. From the sides of CDB, we read tan f = a>h. This distance is d ( ) ( ) P(, ) Q(, ) C(, ) FIGURE.6 To calculate the distance between Ps, d and Qs, d, appl the Pthagorean theorem to triangle PCQ. Distance Formula for Points in the Plane The distance between Ps, d and Qs, d is d = s d + s d = s - d + s - d. EXAMPLE 5 Calculating Distance (a) The distance between Ps -, d and Q(, ) is s - s -dd + s - d = sd + sd = = # 5 = 5. a P(, ) (b) The distance from the origin to P(, ) is s - d + s - d = +. B definition, a circle of radius a is the set of all points P(, ) whose distance from some center C(h, k) equals a (Figure.7). From the distance formula, P lies on the circle if and onl if C(h, k) s - hd + s - kd = a, so ( h) ( k) a FIGURE.7 A circle of radius a in the -plane, with center at (h, k). ( - h) + ( - k) = a. () Equation () is the standard equation of a circle with center (h, k) and radius a. The circle of radius a = and centered at the origin is the unit circle with equation + =.

6 Chapter : Preliminaries EXAMPLE 6 (a) The standard equation for the circle of radius centered at (, ) is (b) The circle s - d + s - d = =. s - d + s + 5d = has h =, k = -5, and a =. The center is the point sh, kd = s, -5d and the radius is a =. If an equation for a circle is not in standard form, we can find the circle s center and radius b first converting the equation to standard form. The algebraic technique for doing so is completing the square (see Appendi 9). EXAMPLE 7 Finding a Circle s Center and Radius Find the center and radius of the circle =. Eterior: ( h) ( k) a On: ( h) ( k) a k a (h, k) Interior: ( h) ( k) a h FIGURE.8 The interior and eterior of the circle s - hd + s - kd = a. Solution We convert the equation to standard form b completing the squares in and : = s + d + s - 6 d = a + + a b b + a a -6 b b = s + + d + s d = s + d + s - d = 6 + a b The center is s -, d and the radius is a =. The points (, ) satisfing the inequalit + a -6 b s - hd + s - kd 6 a make up the interior region of the circle with center (h, k) and radius a (Figure.8). The circle s eterior consists of the points (, ) satisfing s - hd + s - kd 7 a. Start with the given equation. Gather terms. Move the constant to the right-hand side. Add the square of half the coefficient of to each side of the equation. Do the same for. The parenthetical epressions on the left-hand side are now perfect squares. Write each quadratic as a squared linear epression. Parabolas The geometric definition and properties of general parabolas are reviewed in Section.. Here we look at parabolas arising as the graphs of equations of the form = a + b + c.

7 . Lines, Circles, and Parabolas 5 (, ) (, ), 9 (, ) (, ) FIGURE.9 The parabola = (Eample 8). EXAMPLE 8 The Parabola = Consider the equation =. Some points whose coordinates satisf this equation are s, d, s, d, a and s -, d. These points (and all others satisfing, 9 b, s -, d, s, d, the equation) make up a smooth curve called a parabola (Figure.9). The graph of an equation of the form = a is a parabola whose ais (ais of smmetr) is the -ais. The parabola s verte (point where the parabola and ais cross) lies at the origin. The parabola opens upward if a 7 and downward if a 6. The larger the value of ƒ a ƒ, the narrower the parabola (Figure.). Generall, the graph of = a + b + c is a shifted and scaled version of the parabola =. We discuss shifting and scaling of graphs in more detail in Section.5. Verte at origin Ais of smmetr 6 FIGURE. Besides determining the direction in which the parabola = a opens, the number a is a scaling factor. The parabola widens as a approaches zero and narrows as ƒ a ƒ becomes large. The Graph of a b c, a The graph of the equation = a + b + c, a Z, is a parabola. The parabola opens upward if a 7 and downward if a 6. The ais is the line =- b a. The verte of the parabola is the point where the ais and parabola intersect. Its -coordinate is = -b>a; its -coordinate is found b substituting = -b>a in the parabola s equation. Notice that if a =, then we have = b + c which is an equation for a line. The ais, given b Equation (), can be found b completing the square or b using a technique we stud in Section.. EXAMPLE 9 Graphing a Parabola Graph the equation = Solution Comparing the equation with = a + b + c we see that a =-, b = -, c =. () Since a 6, the parabola opens downward. From Equation () the ais is the vertical line =- b a =- s -d s ->d = -.

8 6 Chapter : Preliminaries Verte is, 9 Point smmetric with -intercept (, ) Ais: Intercepts at and Intercept at (, ) FIGURE. The parabola in Eample 9. When = -, we have The verte is s -, 9>d. The -intercepts are where = : =- s -d - s -d + = = = s - ds + d = =, = - We plot some points, sketch the ais, and use the direction of opening to complete the graph in Figure..

9 6 Chapter : Preliminaries EXERCISES. Increments and Distance In Eercises, a particle moves from A to B in the coordinate plane. Find the increments and in the particle s coordinates. Also find the distance from A to B.. As -, d, Bs -, -d. As -, -d, Bs -, d. As -., -d, Bs -8., -d. As, d, Bs,.5d Describe the graphs of the equations in Eercises = 6. + = = Slopes, Lines, and Intercepts Plot the points in Eercises 9 and find the slope (if an) of the line the determine. Also find the common slope (if an) of the lines perpendicular to line AB. 9. As -, d, Bs -, -d. As -, d, Bs, -d. As, d, Bs -, d. As -, d, Bs -, -d In Eercises 6, find an equation for (a) the vertical line and (b) the horizontal line through the given point.. s -, >d. A, -.B 5. A, - B 6. s -p, d In Eercises 7, write an equation for each line described. 7. Passes through s -, d with slope - 8. Passes through s, -d with slope > 9. Passes through (, ) and s -, 5d. Passes through s -8, d and s -, d. Has slope -5> and -intercept 6. Has slope > and -intercept -. Passes through s -, -9d and has slope. Passes through ( >, ), and has no slope 5. Has -intercept and -intercept - 6. Has -intercept -6 and -intercept 7. Passes through s5, -d and is parallel to the line + 5 = 5 8. Passes through A -, B parallel to the line + 5 = 9. Passes through (, ) and is perpendicular to the line 6 - = 5. Passes through (, ) and is perpendicular to the line 8 - = In Eercises, find the line s - and -intercepts and use this information to graph the line.. + =. + = -. - = = - 5. Is there anthing special about the relationship between the lines A + B = C and B - A = C sa Z, B Z d? Give reasons for our answer. 6. Is there anthing special about the relationship between the lines A + B = C and A + B = C sa Z, B Z d? Give reasons for our answer.

10 . Lines, Circles, and Parabolas 7 Increments and Motion 7. A particle starts at As -, d and its coordinates change b increments = 5, = -6. Find its new position. 8. A particle starts at A(6, ) and its coordinates change b increments = -6, =. Find its new position. 9. The coordinates of a particle change b = 5 and = 6 as it moves from A(, ) to Bs, -d. Find and.. A particle started at A(, ), circled the origin once counterclockwise, and returned to A(, ). What were the net changes in its coordinates? Circles In Eercises 6, find an equation for the circle with the given center C(h, k) and radius a. Then sketch the circle in the -plane. Include the circle s center in our sketch. Also, label the circle s - and -intercepts, if an, with their coordinate pairs.. Cs, d, a =. Cs -, d, a =. Cs -, 5d, a =. Cs, d, a = 5. C A -, -B, a = 6. Cs, >d, a = 5 Graph the circles whose equations are given in Eercises 7 5. Label each circle s center and intercepts (if an) with their coordinate pairs = = = s9>d = = = Parabolas Graph the parabolas in Eercises 5 6. Label the verte, ais, and intercepts in each case. 5. = = = = = = = 6. = Inequalities Describe the regions defined b the inequalities and pairs of inequalities in Eercises s - d s - d Ú , , s + d , , Write an inequalit that describes the points that lie inside the circle with center s -, d and radius Write an inequalit that describes the points that lie outside the circle with center s -, d and radius. 7. Write a pair of inequalities that describe the points that lie inside or on the circle with center (, ) and radius, and on or to the right of the vertical line through (, ). 7. Write a pair of inequalities that describe the points that lie outside the circle with center (, ) and radius, and inside the circle that has center (, ) and passes through the origin. Intersecting Lines, Circles, and Parabolas In Eercises 7 8, graph the two equations and find the points in which the graphs intersect. 7. =, + = 7. + =, s - d + = =, = =, = -s - d 77. = -, = =, = s - d =, s - d + = 8. + =, + = Applications 8. Insulation B measuring slopes in the accompaning figure, estimate the temperature change in degrees per inch for (a) the gpsum wallboard; (b) the fiberglass insulation; (c) the wood sheathing. Temperature ( F) Air inside room at 7 F Gpsum wallboard Fiberglass between studs Distance through wall (inches) Sheathing Siding Air outside at F The temperature changes in the wall in Eercises 8 and 8.

11 8 Chapter : Preliminaries 8. Insulation According to the figure in Eercise 8, which of the materials is the best insulator? the poorest? Eplain. 8. Pressure under water The pressure p eperienced b a diver under water is related to the diver s depth d b an equation of the form p = kd + (k a constant). At the surface, the pressure is atmosphere. The pressure at meters is about.9 atmospheres. Find the pressure at 5 meters. 8. Reflected light A ra of light comes in along the line + = from the second quadrant and reflects off the -ais (see the accompaning figure). The angle of incidence is equal to the angle of reflection. Write an equation for the line along which the departing light travels. Angle of incidence Angle of reflection The path of the light ra in Eercise 8. Angles of incidence and reflection are measured from the perpendicular. 85. Fahrenheit vs. Celsius In the FC-plane, sketch the graph of the equation linking Fahrenheit and Celsius temperatures. On the same graph sketch the line C = F. Is there a temperature at which a Celsius thermometer gives the same numerical reading as a Fahrenheit thermometer? If so, find it. 86. The Mt. Washington Cog Railwa Civil engineers calculate the slope of roadbed as the ratio of the distance it rises or falls to the distance it runs horizontall. The call this ratio the grade of the roadbed, usuall written as a percentage. Along the coast, commercial railroad grades are usuall less than %. In the mountains, the ma go as high as %. Highwa grades are usuall less than 5%. The steepest part of the Mt. Washington Cog Railwa in New Hampshire has an eceptional 7.% grade. Along this part of the track, the seats in the front of the car are ft above those in the rear. About how far apart are the front and rear rows of seats? Theor and Eamples C = 5 sf - d B calculating the lengths of its sides, show that the triangle with vertices at the points A(, ), B(5, 5), and Cs, -d is isosceles but not equilateral. 88. Show that the triangle with vertices A(, ), BA, B, and C(, ) is equilateral. 89. Show that the points As, -d, B(, ), and Cs -, d are vertices of a square, and find the fourth verte. 9. The rectangle shown here has sides parallel to the aes. It is three times as long as it is wide, and its perimeter is 56 units. Find the coordinates of the vertices A, B, and C. A B 9. Three different parallelograms have vertices at s -, d, (, ), and (, ). Sketch them and find the coordinates of the fourth verte of each. 9. A 9 rotation counterclockwise about the origin takes (, ) to (, ), and (, ) to s -, d, as shown in the accompaning figure. Where does it take each of the following points? a. (, ) b. s -, -d c. s, -5d d. (, ) e. (, ) f. (, ) g. What point is taken to (, )? (, ) (, ) (, ) 9. For what value of k is the line + k = perpendicular to the line + =? For what value of k are the lines parallel? 9. Find the line that passes through the point (, ) and through the point of intersection of the two lines + = and - = Midpoint of a line segment Show that the point with coordinates a + (, ) (, ) (, 5) (, ), + b D(9, ) is the midpoint of the line segment joining Ps, d to Qs, d. C

12 . Lines, Circles, and Parabolas The distance from a point to a line We can find the distance from a point Ps, d to a line L: A + B = C b taking the following steps (there is a somewhat faster method in Section.5):. Find an equation for the line M through P perpendicular to L.. Find the coordinates of the point Q in which M and L intersect.. Find the distance from P to Q. Use these steps to find the distance from P to L in each of the following cases. a. Ps, d, L : = + b. Ps, 6d, L : + = c. Psa, bd, L : = - d. Ps, d, L : A + B = C

13 . Functions and Their Graphs 9. Functions and Their Graphs Functions are the major objects we deal with in calculus because the are ke to describing the real world in mathematical terms. This section reviews the ideas of functions, their graphs, and was of representing them. Functions; Domain and Range The temperature at which water boils depends on the elevation above sea level (the boiling point drops as ou ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels from an initial location along a straight line path depends on its speed. In each case, the value of one variable quantit, which we might call, depends on the value of another variable quantit, which we might call. Since the value of is completel determined b the value of, we sa that is a function of. Often the value of is given b a rule or formula that sas how to calculate it from the variable. For instance, the equation A = pr is a rule that calculates the area A of a circle from its radius r. In calculus we ma want to refer to an unspecified function without having an particular formula in mind. A smbolic wa to sa is a function of is b writing = ƒsd s equals ƒ of d In this notation, the smbol ƒ represents the function. The letter, called the independent variable, represents the input value of ƒ, and, the dependent variable, represents the corresponding output value of ƒ at. DEFINITION Function A function from a set D to a set Y is a rule that assigns a unique (single) element ƒsd H Y to each element H D. Input (domain) f Output (range) FIGURE. A diagram showing a function as a kind of machine. f() The set D of all possible input values is called the domain of the function. The set of all values of ƒ() as varies throughout D is called the range of the function. The range ma not include ever element in the set Y. The domain and range of a function can be an sets of objects, but often in calculus the are sets of real numbers. (In Chapters 6 man variables ma be involved.) Think of a function ƒ as a kind of machine that produces an output value ƒ() in its range whenever we feed it an input value from its domain (Figure.). The function

14 Chapter : Preliminaries a D domain set f(a) f() Y set containing the range FIGURE. A function from a set D to a set Y assigns a unique element of Y to each element in D. kes on a calculator give an eample of a function as a machine. For instance, the ke on a calculator gives an output value (the square root) whenever ou enter a nonnegative number and press the ke. The output value appearing in the displa is usuall a decimal approimation to the square root of.ifou input a number 6, then the calculator will indicate an error because 6 is not in the domain of the function and cannot be accepted as an input. The ke on a calculator is not the same as the eact mathematical function ƒ defined b ƒsd = because it is limited to decimal outputs and has onl finitel man inputs. A function can also be pictured as an arrow diagram (Figure.). Each arrow associates an element of the domain D to a unique or single element in the set Y. In Figure., the arrows indicate that ƒ(a) is associated with a,ƒ() is associated with, and so on. The domain of a function ma be restricted b contet. For eample, the domain of the area function given b A = pr onl allows the radius r to be positive. When we define a function = ƒsd with a formula and the domain is not stated eplicitl or restricted b contet, the domain is assumed to be the largest set of real -values for which the formula gives real -values, the so-called natural domain. If we want to restrict the domain in some wa, we must sa so. The domain of = is the entire set of real numbers. To restrict the function to, sa, positive values of, we would write =, 7. Changing the domain to which we appl a formula usuall changes the range as well. The range of = is [, q d. The range of =, Ú, is the set of all numbers obtained b squaring numbers greater than or equal to. In set notation, the range is 5 ƒ Ú 6 or 5 ƒ Ú 6 or [, q d. When the range of a function is a set of real numbers, the function is said to be realvalued. The domains and ranges of man real-valued functions of a real variable are intervals or combinations of intervals. The intervals ma be open, closed, or half open, and ma be finite or infinite. EXAMPLE Identifing Domain and Range Verif the domains and ranges of these functions. Function Domain () Range () = = / = = - = - s - q, q d s - q, d s, q d [, q d s - q, ] [-, ] [, q d s - q, d s, q d [, q d [, q d [, ] Solution The formula = gives a real -value for an real number, so the domain is s - q, q d. The range of = is [, q d because the square of an real number is nonnegative and ever nonnegative number is the square of its own square root, = A B for Ú. The formula = > gives a real -value for ever ecept =. We cannot divide an number b zero. The range of = >, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since = >(>). The formula = gives a real -value onl if Ú. The range of = is [, q d because ever nonnegative number is some number s square root (namel, it is the square root of its own square).

15 . Functions and Their Graphs In = -, the quantit - cannot be negative. That is, - Ú, or. The formula gives real -values for all. The range of - is [, q d, the set of all nonnegative numbers. The formula = - gives a real -value for ever in the closed interval from - to. Outside this domain, - is negative and its square root is not a real number. The values of - var from to on the given domain, and the square roots of these values do the same. The range of - is [, ]. Graphs of Functions Another wa to visualize a function is its graph. If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is 5s, ƒsdd ƒ H D6. The graph of the function ƒsd = + is the set of points with coordinates (, ) for which = +. Its graph is sketched in Figure.. The graph of a function ƒ is a useful picture of its behavior. If (, ) is a point on the graph, then = ƒsd is the height of the graph above the point. The height ma be positive or negative, depending on the sign of ƒsd (Figure.5). f() f() f() (, ) FIGURE. The graph of ƒsd = + is the set of points (, ) for which has the value +. EXAMPLE Sketching a Graph Graph the function = over the interval [-, ]. FIGURE.5 If (, ) lies on the graph of f, then the value = ƒsd is the height of the graph above the point (or below if ƒ() is negative). Solution. Make a table of -pairs that satisf the function rule, in this case the equation =.

16 Chapter : Preliminaries. Plot the points (, ) whose coordinates appear in the table. Use fractions when the are convenient computationall.. Draw a smooth curve through the plotted points. Label the curve with its equation. (, ) (, ), 9 (, ) (, ) Computers and graphing calculators graph functions in much this wa b stringing together plotted points and the same question arises. How do we know that the graph of = doesn t look like one of these curves??? p 5 Time (das) FIGURE.6 Graph of a fruit fl population versus time (Eample ). t To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? The answer lies in calculus, as we will see in Chapter. There we will use the derivative to find a curve s shape between plotted points. Meanwhile we will have to settle for plotting points and connecting them as best we can. EXAMPLE Evaluating a Function from Its Graph The graph of a fruit fl population p is shown in Figure.6. (a) Find the populations after and 5 das. (b) What is the (approimate) range of the population function over the time interval t 5? Solution (a) We see from Figure.6 that the point (, ) lies on the graph, so the value of the population p at is psd =. Likewise, p(5) is about. (b) The range of the population function over t 5 is approimatel [, 5]. We also observe that the population appears to get closer and closer to the value p = 5 as time advances.

17 . Functions and Their Graphs Representing a Function Numericall We have seen how a function ma be represented algebraicall b a formula (the area function) and visuall b a graph (Eamples and ). Another wa to represent a function is numericall, through a table of values. Numerical representations are often used b engineers and applied scientists. From an appropriate table of values, a graph of the function can be obtained using the method illustrated in Eample, possibl with the aid of a computer. The graph of onl the tabled points is called a scatterplot. EXAMPLE A Function Defined b a Table of Values Musical notes are pressure waves in the air that can be recorded. The data in Table. give recorded pressure displacement versus time in seconds of a musical note produced b a tuning fork. The table provides a representation of the pressure function over time. If we first make a scatterplot and then connect the data points (t, p) from the table, we obtain the graph shown in Figure.7. TABLE. Tuning fork data Time Pressure Time Pressure p (pressure) Data t (sec) FIGURE.7 A smooth curve through the plotted points gives a graph of the pressure function represented b Table.. The Vertical Line Test Not ever curve ou draw is the graph of a function. A function ƒ can have onl one value ƒ() for each in its domain, so no vertical line can intersect the graph of a function more than once. Thus, a circle cannot be the graph of a function since some vertical lines intersect the circle twice (Figure.8a). If a is in the domain of a function ƒ, then the vertical line = a will intersect the graph of ƒ in the single point (a, ƒ(a)). The circle in Figure.8a, however, does contain the graphs of two functions of ; the upper semicircle defined b the function ƒsd = - and the lower semicircle defined b the function gsd = - - (Figures.8b and.8c).

18 Chapter : Preliminaries (a) (b) (c) FIGURE.8 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of a function ƒsd = -. (c) The lower semicircle is the graph of a function gsd = - -. f() FIGURE. To graph the function = ƒsd shown here, we appl different formulas to different parts of its domain (Eample 5). FIGURE.9 The absolute value function has domain s - q, q d and range [, q d. Piecewise-Defined Functions Sometimes a function is described b using different formulas on different parts of its domain. One eample is the absolute value function whose graph is given in Figure.9. Here are some other eamples. EXAMPLE 5 The function Graphing Piecewise-Defined Functions is defined on the entire real line but has values given b different formulas depending on the position of. The values of ƒ are given b: = - when 6, = when, and = when 7. The function, however, is just one function whose domain is the entire set of real numbers (Figure.). EXAMPLE 6 ƒ ƒ = e, Ú -, 6, ƒsd = The Greatest Integer Function -, 6,, 7 The function whose value at an number is the greatest integer less than or equal to is called the greatest integer function or the integer floor function. It is denoted :;, or, in some books, [] or [[]] or int. Figure. shows the graph. Observe that :.; =, :.9; =, :; =, : -.; = -, :; =, :.; =, : -.; = - : -; = -.

19 . Functions and Their Graphs 5 FIGURE. The graph of the greatest integer function = :; lies on or below the line =, so it provides an integer floor for (Eample 6). f() (, ) (, ) FIGURE. The graph of the least integer function = <= lies on or above the line =, so it provides an integer ceiling for (Eample 7). FIGURE. The segment on the left contains (, ) but not (, ). The segment on the right contains both of its endpoints (Eample 8). EXAMPLE 7 The Least Integer Function The function whose value at an number is the smallest integer greater than or equal to is called the least integer function or the integer ceiling function. It is denoted <=. Figure. shows the graph. For positive values of, this function might represent, for eample, the cost of parking hours in a parking lot which charges $ for each hour or part of an hour. EXAMPLE 8 Writing Formulas for Piecewise-Defined Functions Write a formula for the function = ƒsd whose graph consists of the two line segments in Figure.. Solution We find formulas for the segments from (, ) to (, ), and from (, ) to (, ) and piece them together in the manner of Eample 5. Segment from (, ) to (, ) The line through (, ) and (, ) has slope m = s - d>s - d = and -intercept b =. Its slope-intercept equation is =. The segment from (, ) to (, ) that includes the point (, ) but not the point (, ) is the graph of the function = restricted to the half-open interval 6, namel, Segment from (, ) to (, ) The line through (, ) and (, ) has slope m = s - d>s - d = and passes through the point (, ). The corresponding pointslope equation for the line is The segment from (, ) to (, ) that includes both endpoints is the graph of = - restricted to the closed interval, namel, Piecewise formula =, 6. = + ( - ), or = -. = -,. Combining the formulas for the two pieces of the graph, we obtain ƒsd = e, 6 -,.

20 6 Chapter : Preliminaries EXERCISES. Functions In Eercises 6, find the domain and range of each function.. ƒsd = +. ƒsd = -. Fstd =. Fstd = t + t 5. gszd = - z 6. gszd = - z In Eercises 7 and 8, which of the graphs are graphs of functions of, and which are not? Give reasons for our answers. 7. a. b.. Epress the side length of a square as a function of the length d of the square s diagonal. Then epress the area as a function of the diagonal length.. Epress the edge length of a cube as a function of the cube s diagonal length d. Then epress the surface area and volume of the cube as a function of the diagonal length.. A point P in the first quadrant lies on the graph of the function ƒsd =. Epress the coordinates of P as functions of the slope of the line joining P to the origin. Functions and Graphs Find the domain and graph the functions in Eercises a. b. ƒ ƒ ƒ ƒ ƒ 5. ƒsd = 5-6. ƒsd = gsd = ƒ 8. gsd = - 9. Fstd = t>ƒ t. Gstd = >ƒ t. Graph the following equations and eplain wh the are not graphs of functions of. a. = b. =. Graph the following equations and eplain wh the are not graphs of functions of. a. ƒ ƒ + ƒ ƒ = b. ƒ + ƒ = Piecewise-Defined Functions Graph the functions in Eercises Consider the function = s>d -. a. Can be negative? b. Can =? c. Can be greater than? d. What is the domain of the function?. Consider the function = -. a. Can be negative? b. Can be greater than? c. What is the domain of the function? ƒsd = e, -, 6 gsd = e -, -, 6 Fsd = e -,, 7 Gsd = e >, 6, 7. Find a formula for each function graphed. a. b. (, ) Finding Formulas for Functions. Epress the area and perimeter of an equilateral triangle as a function of the triangle s side length. t

21 . Functions and Their Graphs 7 8. a. b. (, ) 5 (, ) Theor and Eamples 7. Abo with an open top is to be constructed from a rectangular piece of cardboard with dimensions in. b in. b cutting out equal squares of side at each corner and then folding up the sides as in the figure. Epress the volume V of the bo as a function of. 9. a. b. (, ) (, ). a. b. (T, ) A (, ) (, ) (, ) 8. The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hpotenuse is units long. a. Epress the -coordinate of P in terms of. (You might start b writing an equation for the line AB.) b. Epress the area of the rectangle in terms of. B T T A T T T T t P(,?) A T T. a. Graph the functions ƒsd = > and gsd = + s>d together to identif the values of for which 7 +. b. Confirm our findings in part (a) algebraicall.. a. Graph the functions ƒsd = >s - d and gsd = >s + d together to identif the values of for which b. Confirm our findings in part (a) algebraicall. The Greatest and Least Integer Functions. For what values of is a. :; =? b. <= =?. What real numbers satisf the equation :; = <=? 5. Does < -= = -:; for all real?give reasons for our answer. 6. Graph the function ƒsd = e :;, Ú <=, 6 Wh is ƒ() called the integer part of? 9. A cone problem Begin with a circular piece of paper with a in. radius as shown in part (a). Cut out a sector with an arc length of. Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in part (b). (a) in. a. Eplain wh the circumference of the base of the cone is 8p -. b. Epress the radius r as a function of. c. Epress the height h as a function of. d. Epress the volume V of the cone as a function of. h (b) r in.

22 8 Chapter : Preliminaries. Industrial costs Daton Power and Light, Inc., has a power plant on the Miami River where the river is 8 ft wide. To la a new cable from the plant to a location in the cit mi downstream on the opposite side costs $8 per foot across the river and $ per foot along the land. 8 ft P Power plant Q mi (Not to scale) Daton a. Suppose that the cable goes from the plant to a point Q on the opposite side that is ft from the point P directl opposite the plant. Write a function C() that gives the cost of laing the cable in terms of the distance. b. Generate a table of values to determine if the least epensive location for point Q is less than ft or greater than ft from point P.. For a curve to be smmetric about the -ais, the point (, ) must lie on the curve if and onl if the point s, -d lies on the curve. Eplain wh a curve that is smmetric about the -ais is not the graph of a function, unless the function is =.. A magic trick You ma have heard of a magic trick that goes like this: Take an number. Add 5. Double the result. Subtract 6. Divide b. Subtract. Now tell me our answer, and I ll tell ou what ou started with. Pick a number and tr it. You can see what is going on if ou let be our original number and follow the steps to make a formula ƒ() for the number ou end up with.