1 a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs of real numbers b introducing a rectangular coordinate sstem (also called a Cartesian coordinate sstem). A rectangular coordinate sstem consists of two perpendicular coordinate lines, called coordinate aes, that intersect at their origins. Usuall, but not alwas, one ais is horizontal with its positive direction to the right, and the other is vertical with its positive direction up. The intersection of the aes is called the origin of the coordinate sstem. It is common to call the horizontal ais the -ais and the vertical ais the -ais, in which case the plane and the aes together are referred to as the -plane (Figure F.). Although labeling the aes with the letters and is common, other letters ma be more appropriate in specific applications. Figure F. shows a uv-plane and a ts-plane the first letter in the name of the plane alwas refers to the horizontal ais and the second to the vertical ais. Origin ais v u s t -plane ais uv-plane ts-plane Figure F. Figure F. -coordinate b P(a, b) a -coordinate Figure F. COORDINATES Ever point P in a coordinate plane can be associated with a unique ordered pair of real numbers b drawing two lines through P, one perpendicular to the -ais and the other perpendicular to the -ais (Figure F.). If the first line intersects the -ais at the point with coordinate a and the second line intersects the -ais at the point with coordinate b, then we associate the ordered pair of real numbers (a, b) with the point P. The number a is called the -coordinate or abscissa of P and the number b is called the -coordinate or ordinate of P. We will sa that P has coordinates (a, b) and write P(a,b)when we want to emphasize that the coordinates of P are (a, b). We can also reverse the above procedure and find the point P associated with the coordinates (a, b) b locating the intersection of the dashed lines in Figure F.. Because of this one-to-one correspondence between coordinates A7
2 A8 Appendi F: Coordinate Planes,Lines,and Linear Functions and points, we will sometimes blur the distinction between points and ordered pairs of numbers b talking about the point (a, b). Recall that the smbol (a, b) also denotes the open interval between a and b; the appropriate interpretation will usuall be clear from the contet. Quadrant II (, +) Quadrant III (, ) Quadrant I (+, +) Quadrant IV (+, ) In a rectangular coordinate sstem the coordinate aes divide the rest of the plane into four regions called quadrants. These are numbered counterclockwise with roman numerals as shown in Figure F.. As indicated in that figure, it is eas to determine the quadrant in which a given point lies from the signs of its coordinates: a point with two positive coordinates (+, +) lies in Quadrant I, a point with a negative -coordinate and a positive -coordinate (, +) lies in Quadrant II, and so forth. Points with a zero -coordinate lie on the -ais and points with a zero -coordinate lie on the -ais. To plot a point P(a,b)means to locate the point with coordinates (a, b) in a coordinate plane. For eample, in Figure F. we have plotted the points Figure F. P(, ), Q(, ), R(, ), and S(, ) Observe how the signs of the coordinates identif the quadrants in which the points lie. Q(, ) R(, ) - S(, ) - - P(, ) Figure F. GRAPHS The correspondence between points in a plane and ordered pairs of real numbers makes it possible to visualize algebraic equations as geometric curves, and, conversel, to represent geometric curves b algebraic equations. To understand how this is done, suppose that we have an -coordinate sstem and an equation involving two variables and, sa =, =, = +, or + = We define a solution of such an equation to be an ordered pair of real numbers (a, b) whose coordinates satisf the equation when we substitute = a and = b. For eample, the ordered pair (, ) is a solution of the equation =, since the equation is satisfied b = and = (verif). However, the ordered pair (, ) is not a solution of this equation, since the equation is not satisfied b = and = (verif). The following definition makes the association between equations in and and curves in the -plane.
3 Appendi F: Coordinate Planes,Lines,and Linear Functions A9 F. definition. The set of all solutions of an equation in and is called the solution set of the equation, and the set of all points in the -plane whose coordinates are members of the solution set is called the graph of the equation. One of the main themes in calculus is to identif the eact shape of a graph. Point plotting is one approach to obtaining a graph, but this method has limitations, as discussed in the following eample. Eample of this method. Use point plotting to sketch the graph of =. Discuss the limitations Solution. The solution set of the equation has infinitel man members, since we can substitute an arbitrar value for into the right side of = and compute the associated to obtain a point (, ) in the solution set. The fact that the solution set has infinitel man members means that we cannot obtain the entire graph of = b point plotting. However, we can obtain an approimation to the graph b plotting some sample members of the solution set and connecting them with a smooth curve, as in Figure F.. The problem with this method is that we cannot be sure how the graph behaves between the plotted points. For eample, the curves in Figure F.7 also pass through the plotted points and hence are legitimate candidates for the graph in the absence of additional information. Moreover, even if we use a graphing calculator or a computer program to generate the graph, as in Figure F.8, we have the same problem because graphing technolog uses point-plotting algorithms to generate graphs. Indeed, in Section. of the tet we see eamples where graphing technolog can be fooled into producing grossl inaccurate graphs. = (, ) 9 9 (, ) (, ) (, ) (, 9) (, ) (, ) (, 9) = Figure F. Figure F.7 In spite of its limitations, point plotting b hand or with the help of graphing technolog can be useful, so here are two more eamples. Figure F.8 [, ] [, ] Scl =, Scl = = Eample Sketch the graph of =. Solution. If <, then is an imaginar number. Thus, we can onl plot points for which, since points in the -plane have real coordinates. Figure F.9 shows the graph obtained b point plotting and a graph obtained with a graphing calculator.
4 A Appendi F: Coordinate Planes,Lines,and Linear Functions = (, ) (, ) (, ) (, ) (,.) (, ) (,.7) (, ) = [, ] [, ] Scl =, Scl = = Figure F.9 Eample Sketch the graph of =. Solution. To calculate coordinates of points on the graph of an equation in and,itis desirable to have epressed in terms of or in terms of. In this case it is easier to epress in terms of, so we rewrite the equation as = Members of the solution set can be obtained from this equation b substituting arbitrar values for in the right side and computing the associated values of (Figure F.). = = (, ) 8 8 (8, ) (, ) (, ) (, ) (, ) (, ) (8, ) Figure F. Most graphing calculators and computer graphing programs require that be epressed in terms of to generate a graph in the -plane. In Section.8 we discuss a method forcircumventing this restriction. Eample Sketch the graph of = /. Solution. Because / is undefined at =, we can onl plot points for which =. This forces a break, called a discontinuit, in the graph at = (Figure F.). INTERCEPTS Points where a graph intersects the coordinate aes are of special interest in man problems. As illustrated in Figure F., intersections of a graph with the -ais have the form (a, ) and intersections with the -ais have the form (,b). The number a is called an -intercept of the graph and the number b a -intercept.
5 Appendi F: Coordinate Planes,Lines,and Linear Functions A = / = / (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, b) -intercept is b Figure F. Eample Find all intercepts of (a) + = (b) = (c) = / (a, ) Solution (a). To find the -intercepts we set = and solve for : -intercept is a = or = Figure F. To find the -intercepts we set = and solve for : = or = As we will see later, the graph of + = is the line shown in Figure F.. Solution (b). To find the -intercepts, set = and solve for : + = = Thus, = is the onl -intercept. To find the -intercepts, set = and solve for : = ( ) = So the -intercepts are = and =. The graph is shown in Figure F.. Figure F. Solution (c). To find the -intercepts, set = : = This equation has no solutions (wh?), so there are no -intercepts. To find -intercepts we would set = and solve for. But, substituting = leads to a division b zero, which is not allowed, so there are no -intercepts either. The graph of the equation is shown in Figure F.. SLOPE To obtain equations of lines we will first need to discuss the concept of slope, which is a numerical measure of the steepness of a line.
6 A Appendi F: Coordinate Planes,Lines,and Linear Functions P (, ) P (, ) (Rise) Consider a particle moving left to right along a nonvertical line from a point P (, ) to a point P (, ). As shown in Figure F., the particle moves units in the - direction as it travels units in the positive -direction. The vertical change is called the rise, and the horizontal change the run. The ratio of the rise over the run can be used to measure the steepness of the line, which leads us to the following definition. Figure F. (Run) F. definition. If P (, ) and P (, ) are points on a nonvertical line, then the slope m of the line is defined b m = rise run = () Observe that this definition does not appl to vertical lines. For such lines we have = (a zero run), which means that the formula for m involves a division b zero. For this reason, the slope of a vertical line is undefined, which is sometimes described informall b stating that a vertical line has infinite slope. P (, ) P (, ) P (, ) P (, ) When calculating the slope of a nonvertical line from Formula (), it does not matter which two points on the line ou use for the calculation, as long as the are distinct. This can be proved using Figure F. and similar triangles to show that m = = Moreover, once ou choose two points to use for the calculation, it does not matter which one ou call P and which one ou call P because reversing the points reverses the sign of both the numerator and denominator of () and hence has no effect on the ratio. Figure F. Eample (a) the points (, ) and (9, 8) (b) the points (, 9) and (, ) (c) the points (, 7) and (, 7). Solution. In each part find the slope of the line through (a) m = 8 9 = = (b) m = 9 = 7 7 = (c) m = ( ) = Eample 7 Figure F. shows the three lines determined b the points in Eample and eplains the significance of their slopes. As illustrated in this eample, the slope of a line can be positive, negative, or zero. A positive slope means that the line is inclined upward to the right, a negative slope means that the line is inclined downward to the right, and a zero slope means that the line is horizontal. An undefined slope means that the line is vertical. Figure F.7 shows various lines through the origin with their slopes.
7 Appendi F: Coordinate Planes,Lines,and Linear Functions A P (9, 8) P (, 9) (, 7) (, 7) P (, ) P (, ) m = m = Traveling left to right, a point on the line rises two units for each unit it moves in the positive -direction. Traveling left to right, a point on the line falls three units for each unit it moves in the positive -direction. m = Traveling left to right, a point on the line neither rises nor falls. Figure F. PARALLEL AND PERPENDICULAR LINES The following theorem shows how slopes can be used to tell whether two lines are parallel or perpendicular. Positive m = m = m = m = m = m = m = m = m = Negative slope slope F. theorem. (a) Two nonvertical lines with slopes m and m are parallel if and onl if the have the same slope, that is, m = m (b) Two nonvertical lines with slopes m and m are perpendicular if and onl if the product of their slopes is, that is, m m = This relationship can also be epressed as m = /m or m = /m, which states that nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of one another. Figure F.7 A complete proof of this theorem is a little tedious, but it is not hard to motivate the results informall. Let us start with part (a). Suppose that L and L are nonvertical parallel lines with slopes m and m, respectivel. If the lines are parallel to the -ais, then m = m =, and we are done. If the are not parallel to the -ais, then both lines intersect the -ais; and for simplicit assume that the are oriented as in Figure F.8a. On each line choose the point whose run relative to the point of intersection with the -ais is. On line L the corresponding rise will be m and on L it will be m. However, because the lines are parallel, the shaded triangles in the figure must be congruent (verif), so m = m. Conversel, the condition m = m can be used to show that the shaded triangles are congruent, from which it follows that the lines make the same angle with the -ais and hence are parallel (verif). Now suppose that L and L are nonvertical perpendicular lines with slopes m and m, respectivel; and for simplicit assume that the are oriented as in Figure F.8b. On line L choose the point whose run relative to the point of intersection of the lines is, in which case the corresponding rise will be m ; and on line L choose the point whose rise relative to the point of intersection is, in which case the corresponding run will be /m. Because the lines are perpendicular, the shaded triangles in the figure must be congruent
8 A Appendi F: Coordinate Planes,Lines,and Linear Functions L L L Rise = m Rise = m Rise = Rise = m Run = Run = Run = /m Run = L (a) (b) Figure F.8 (verif), and hence the ratios of corresponding sides of the triangles must be equal. Taking into account that for line L the vertical side of the triangle has length and the horizontal side has length /m (since m is negative), the congruence of the triangles implies that m / = ( /m )/ orm m =. Conversel, the condition m = /m can be used to show that the shaded triangles are congruent, from which it can be deduced that the lines are perpendicular (verif). B(, 7) A(, ) Figure F.9 C(7, ) Eample 8 Use slopes to show that the points A(, ), B(, 7), and C(7, ) are vertices of a right triangle. Solution. We will show that the line through A and B is perpendicular to the line through B and C. The slopes of these lines are m = 7 = and m = 7 7 = Slope of the line through A and B Slope of the line through B and C Since m m =, the line through A and B is perpendicular to the line through B and C; thus, ABC is a right triangle (Figure F.9). (, b) L Figure F. L (a, ) Ever point on L has an -coordinate of a and ever point on L has a -coordinate of b. LINES PARALLEL TO THE COORDINATE AXES We now turn to the problem of finding equations of lines that satisf specified conditions. The simplest cases are lines parallel to the coordinate aes. A line parallel to the -ais intersects the -ais at some point (a, ). This line consists precisel of those points whose -coordinates equal a (Figure F.). Similarl, a line parallel to the -ais intersects the -ais at some point (,b). This line consists precisel of those points whose -coordinates equal b (Figure F.). Thus, we have the following theorem. F. theorem. The vertical line through (a, ) and the horizontal line through (, b) are represented, respectivel, b the equations = a and = b Eample 9 The graph of = is the vertical line through (, ), and the graph of = 7 is the horizontal line through (, 7) (Figure F.).
9 Appendi F: Coordinate Planes,Lines,and Linear Functions A = (, 7) = 7 (, ) Figure F. P m LINES DETERMINED BY POINT AND SLOPE There are infinitel man lines that pass through an given point in the plane. However, if we specif the slope of the line in addition to a point on it, then the point and the slope together determine a unique line (Figure F.). Let us now consider how to find an equation of a nonvertical line L that passes through a point P (, ) and has slope m. IfP(,) is an point on L, different from P, then the slope m can be obtained from the points P(,) and P (, ); this gives Figure F. There is a unique line through P with slope m. m = which can be rewritten as = m( ) () With the possible eception of (, ), we have shown that ever point on L satisfies (). But =, = satisfies (), so that all points on L satisf (). We leave it as an eercise to show that ever point satisfing () lies on L. In summar, we have the following theorem. F. theorem. The line passing through P (, ) and having slope m is given b the equation = m( ) () This is called the point-slope form of the line. Eample Find the point-slope form of the line through (, ) with slope. When the equation of a line is written as in (), the slope is the coefficient of and the -intercept is the constant term (Figure F.). Solution. Substituting the values =, =, and m = in () ields the pointslope form + = ( ). = m + b LINES DETERMINED BY SLOPE AND -INTERCEPT A nonvertical line crosses the -ais at some point (,b). point-slope form of its equation, we obtain b = m( ) which we can rewrite as = m + b. To summarize: If we use this point in the Slope -intercept Figure F. F. theorem. The line with -intercept b and slope m is given b the equation This is called the slope-intercept form of the line. = m + b ()
10 A Appendi F: Coordinate Planes,Lines,and Linear Functions Eample equation slope -intercept = + 7 = + = = 8 = m = m = m = m = m = b = 7 b = b = b = 8 b = Eample stated conditions: Find the slope-intercept form of the equation of the line that satisfies the (a) slope is 9; crosses the -ais at (, ) (b) slope is ; passes through the origin (c) passes through (, ); perpendicular to = + (d) passes through (, ) and (, ). Solution (a). = 9. From the given conditions we have m = 9 and b =, so () ields Solution (b). From the given conditions m = and the line passes through (, ), so b =. Thus, it follows from () that = + or =. Solution (c). The given line has slope, so the line to be determined will have slope m =. Substituting this slope and the given point in the point-slope form () and then simplifing ields ( ) = ( ) = + Solution (d). We will first find the point-slope form, then solve for in terms of to obtain the slope-intercept form. From the given points the slope of the line is m = = 9 We can use either of the given points for (, ) in (). We will use (, ). This ields the point-slope form = 9( ) Solving for in terms of ields the slope-intercept form = 9 We leave it for the reader to show that the same equation results if (, ) rather than (, ) is used for (, ) in (). THE GENERAL EQUATION OF A LINE An equation that is epressible in the form A + B + C = () where A, B, and C are constants and A and B are not both zero, is called a first-degree equation in and. For eample, + =
11 Appendi F: Coordinate Planes,Lines,and Linear Functions A7 is a first-degree equation in and since it has form () with A =, B =, C = In fact, all the equations of lines studied in this section are first-degree equations in and. The following theorem states that the first-degree equations in and are precisel the equations whose graphs in the -plane are straight lines. F.7 theorem. Ever first-degree equation in and has a straight line as its graph and, conversel, ever straight line can be represented b a first-degree equation in and. Because of this theorem, () is sometimes called the general equation of a line or a linear equation in and. (, ) Eample Graph the equation + =. Solution. Since this is a linear equation in and, its graph is a straight line. Thus, to sketch the graph we need onl plot an two points on the graph and draw the line through them. It is particularl convenient to plot the points where the line crosses the coordinate aes. These points are (, ) and (, ) (verif), so the graph is the line in Figure F.. Figure F. (, ) + = Eample Find the slope of the line in Eample. Solution. Solving the equation for ields = + which is the slope-intercept form of the line. Thus, the slope is m =. LINEAR FUNCTIONS When and are related b Equation () then we sa that is a linear function of. Linear functions arise in a variet of practial problems. Here is a tpical eample. Cost in dollars C Figure F. C = C = +. Number of parking das Eample A universit parking lot charges $. per da but offers a $. monthl sticker with which the student pas onl $. per da. (a) Find equations for the cost C of parking for das per month under both pament methods, and graph the equations for. (Treat C as a continuous function of, even though onl assumes integer values.) (b) Find the value of for which the graphs intersect, and discuss the significance of this value. Solution (a). The cost in dollars of parking for das at $. per da is C =, and the cost for the $. sticker plus das at $. per da is C = +. (Figure F.). Solution (b). The graphs intersect at the point where = +. which is = /.7.. This value of is not an option for the student, since must be an integer. However, it is the dividing point at which the monthl sticker method
12 A8 Appendi F: Coordinate Planes,Lines,and Linear Functions becomes less epensive than the dail pament method; that is, for it is cheaper to bu the monthl sticker and for it is cheaper to pa the dail rate. DIRECT PROPORTION A variable is said to be directl proportional to a variable if there is a positive constant k, called the constant of proportionalit, such that = k () The graph of this equation is a line through the origin whose slope k is the constant of proportionalit. Thus, linear functions are appropriate in phsical problems where one variable is directl proportional to another. Hooke s law in phsics provides a nice eample of direct proportion. It follows from this law that if a weight of units is suspended from a spring, then the spring will be stretched b an amount that is directl proportional to, that is, = k (Figure F.). The constant k depends on the stiffness of the spring: the stiffer the spring, the smaller the value of k (wh?). is directl proportional to. Eample pounds. Figure F.7 shows an old-fashioned spring scale that is calibrated in (a) Given that the pound scale marks are. in apart, find an equation that epresses the length that the spring is stretched (in inches) in terms of the suspended weight (in pounds). (b) Graph the equation obtained in part (a). Figure F. Length stretched (in) =. 8 Weight (lb) Figure F.7 Solution (a). It follows from Hooke s law that is related to b an equation of the form = k. To find k we rewrite this equation as k = / and use the fact that a weight of = lb stretches the spring =. in. Thus, k = =. =. and hence =. Solution (b). The graph of the equation =. is shown in Figure F.7. Hooke s law, named for the English phsicist Robert Hooke ( 7), applies onl for small displacements that do not stretch the spring to the point of permanentl distorting it.
13 Appendi F: Coordinate Planes,Lines,and Linear Functions A9 EXERCISE SET F. Draw the rectangle, three of whose vertices are (, ), (, ), and (, 7), and find the coordinates of the fourth verte.. Draw the triangle whose vertices are (, ), (, ), and (, ), and find its area. Draw a rectangular coordinate sstem and sketch the set of points whose coordinates (, ) satisf the given conditions.. (a) = (b) = (c) (d) = (e) (f ). (a) = (b) = (c) < (d) and (e) = (f) = Sketch the graph of the equation. (A calculating utilit will be helpful in some of these problems.). =. = + 7. = 8. = =. =. =. =. Find the slope of the line through (a) (, ) and (, ) (b) (, ) and (7, ) (c) (, ) and (, ) (d) (, ) and (, ).. Find the slopes of the sides of the triangle with vertices (, ), (, ), and (, 7).. Use slopes to determine whether the given points lie on the same line. (a) (, ), (, ), and (, ) (b) (, ), (, ), and (, ). Draw the line through (, ) with slope (a) m = (b) m = (c) m =. 7. Draw the line through (, ) with slope (a) m = (b) m = (c) m =. 8. An equilateral triangle has one verte at the origin, another on the -ais, and the third in the first quadrant. Find the slopes of its sides. 9. List the lines in the accompaning figure in the order of increasing slope. I Figure E-9 II. List the lines in the accompaning figure in the order of increasing slope. III IV I Figure E- II. A particle, initiall at (, ), moves along a line of slope m = to a new position (, ). (a) Find if =. (b) Find if =.. A particle, initiall at (7, ), moves along a line of slope m = to a new position (, ). (a) Find if = 9. (b) Find if =.. Let the point (,k) lie on the line of slope m = through (, ); find k.. Given that the point (k, ) is on the line through (, ) and (, ), find k.. Find if the slope of the line through (, ) and (, ) is the negative of the slope of the line through (, ) and (, ).. Find and if the line through (, ) and (, ) has slope, and the line through (, ) and (7, ) has slope. 7. Use slopes to show that (, ), (, ), (, ), and (, ) are vertices of a parallelogram. 8. Use slopes to show that (, ), (, ), and (, 9) are vertices of a right triangle. 9. Graph the equations (a) + = (b) = (c) = (d) = 7.. Graph the equations (a) = (b) = 8 (c) = (d) = +.. Graph the equations (a) = (b) = (c) =.. Graph the equations (a) = (b) = (c) =.. Find the slope and -intercept of (a) = + (b) = (c) + = 8 (d) = (e) a + b =.. Find the slope and -intercept of (a) = + (b) = + (c) + = (d) = (e) a + a = (a = ). Use the graph to find the equation of the line in slopeintercept form. III IV
14 A Appendi F: Coordinate Planes,Lines,and Linear Functions.. (a) Figure E- (a) Figure E- 7 8 Find the slope-intercept form of the line satisfing the given conditions. 7. Slope =, -intercept =. 8. m =, b =. 9. The line is parallel to = and its -intercept is 7.. The line is parallel to + = and passes through (, ).. The line is perpendicular to = + 9 and its -intercept is.. The line is perpendicular to = 7 and passes through (, ).. The line passes through (, ) and (, 7).. The line passes through (, ) and (, ).. The -intercept is and the -intercept is.. The -intercept is b and the -intercept is a. 7. The line is perpendicular to the -ais and passes through (, ). 8. The line is parallel to = and passes through (, 8). 9. In each part, classif the lines as parallel, perpendicular, or neither. (a) = 7 and = + 9 (b) = and = 7 (c) + = and + 7 = (d) A + B + C = and B A + D = (e) = ( ) and 7 = ( ). In each part, classif the lines as parallel, perpendicular, or neither. (a) = + and = (b) = ( ) and = ( + 7) (c) = and + 9 = (b) (b) (d) A + B + C = and A + B + D = (e) = and =. For what value of k will the line + k = (a) have slope (b) have -intercept (c) pass through the point (, ) (d) be parallel to the line = (e) be perpendicular to the line + =?. Sketch the graph of = and eplain how this graph is related to the graphs of = and =.. Sketch the graph of ( )( + ) = and eplain how it is related to the graphs of = and + =.. Graph F = 9 C + in a CF-coordinate sstem.. Graph u = v in a uv-coordinate sstem.. Graph Y = X + inayx-coordinate sstem. 7. A point moves in the -plane in such a wa that at an time t its coordinates are given b = t + and = t. B epressing in terms of, show that the point moves along a straight line. 8. A point moves in the -plane in such a wa that at an time t its coordinates are given b = + t and = t. B epressing in terms of, show that the point moves along a straight-line path and specif the values of for which the equation is valid. 9. Find the area of the triangle formed b the coordinate aes and the line through (, ) and (, ).. Draw the graph of 9 =.. A spring with a natural length of in stretches to a length of in when a -lb object is suspended from it. (a) Use Hooke s law to find an equation that epresses the amount b which the spring is stretched (in inches) in terms of the suspended weight (in pounds). (b) Graph the equation obtained in part (a). (c) Find the length of the spring when a -lb object is suspended from it. (d) What is the largest weight that can be suspended from the spring if the spring cannot be stretched to more than twice its natural length?. The spring in a heav-dut shock absorber has a natural length of ft and is compressed. ft b a load of ton. An additional load of tons compresses the spring an additional ft. (a) Assuming that Hooke s law applies to compression as well as etension, find an equation that epresses the length that the spring is compressed from its natural length (in feet) in terms of the load (in tons). (b) Graph the equation obtained in part (a). (c) Find the amount that the spring is compressed from its natural length b a load of tons. (d) Find the maimum load that can be applied if safet regulations prohibit compressing the spring to less than half its natural length.
15 Appendi F: Coordinate Planes,Lines,and Linear Functions A Confirm that a linear function is appropriate for the relationship between and. Find a linear equation relating and, and verif that the data points lie on the graph of our equation Table E-. Table E-. There are two common sstems for measuring temperature, Celsius and Fahrenheit. Water freezes at Celsius ( C) and Fahrenheit ( F); it boils at C and F. (a) Assuming that the Celsius temperature T C and the Fahrenheit temperature T F are related b a linear equation, find the equation. (b) What is the slope of the line relating T F and T C if T F is plotted on the horizontal ais? (c) At what temperature is the Fahrenheit reading equal to the Celsius reading? (d) Normal bod temperature is 98. F. What is it in C?. Thermometers are calibrated using the so-called triple point of water, which is 7. K on the Kelvin scale and. C on the Celsius scale. A one-degree difference on the Celsius scale is the same as a one-degree difference on the Kelvin scale, so there is a linear relationship between the temperature T C in degrees Celsius and the temperature T K in kelvins. (a) Find an equation that relates T C and T K. (b) Absolute zero ( K on the Kelvin scale) is the temperature below which a bod s temperature cannot be lowered. Epress absolute zero in C. 7. To the etent that water can be assumed to be incompressible, the pressure p in a bod of water varies linearl with the distance h below the surface. (a) Given that the pressure is atmosphere ( atm) at the surface and.9 atm at a depth of m, find an equation that relates pressure to depth. (b) At what depth is the pressure twice that at the surface? 8. A resistance thermometer is a device that determines temperature b measuring the resistance of a fine wire whose resistance varies with temperature. Suppose that the resistance R in ohms ( ) varies linearl with the temperature T in C and that R =. when T = C and that R =.9 when T = C. (a) Find an equation for R in terms of T. (b) If R is measured eperimentall as 8., what is the temperature? 9. Suppose that the mass of a spherical mothball decreases with time, due to evaporation, at a rate that is proportional to its surface area. Assuming that it alwas retains the shape of a sphere, it can be shown that the radius r of the sphere decreases linearl with the time t. (a) If, at a certain instant, the radius is.8 mm and das later it is.7 mm, find an equation for r (in millimeters) in terms of the elapsed time t (in das). (b) How long will it take for the mothball to completel evaporate? 7. The accompaning figure shows three masses suspended from a spring: a mass of g, a mass of g, and an unknown mass of W g. (a) What will the pointer indicate on the scale if no mass is suspended? (b) Find W. g Figure E-7 mm mm mm g W g 7. The price for a round-trip bus ride from a universit to center cit is $., but it is possible to purchase a monthl commuter pass for $. with which each round-trip ride costs an additional $.. (a) Find equations for the cost C of making round-trips per month under both pament plans, and graph the equations for (treating C as a continuous function of, even though assumes onl integer values). (b) How man round-trips per month would a student have to make for the commuter pass to be worthwhile? 7. A student must decide between buing one of two used cars: car A for $ or car B for $. Car A gets miles per gallon of gas, and car B gets miles per gallon. The student estimates that gas will run $. per gallon. Both cars are in ecellent condition, so the student feels that repair costs should be negligible for the foreseeable future. How man miles would the student have to drive before car B becomes the better bu?
G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs
- Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
. Slope of a Line (-) 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points
. Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name
8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
Writing Linear Functions.1 Writing Equations in Slope-Intercept Form. Writing Equations in Point-Slope Form.3 Writing Equations in Standard Form. Writing Equations of Parallel and Perpendicular Lines.5
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending
Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged
2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
- Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiple-choice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse
87533_01_ch1_p001-066 1/30/08 9:36 AM Page 1 STRAIGHT LINES AND LINEAR FUNCTIONS 1 THIS CHAPTER INTRODUCES the Cartesian coordinate sstem, a sstem that allows us to represent points in the plane in terms
Chapter Linear and Quadratic Functions. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that
8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be
ALGEBRA Pupils should be taught to: Generate points and plot graphs of functions As outcomes, Year 7 pupils should, for eample: Use, read and write, spelling correctl: coordinates, coordinate pair/point,
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
- Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
7.1 The Slope-Intercept Form 7.1 OBJECTIVES 1. Find the slope and intercept from the equation of a line. Given the slope and intercept, write the equation of a line. Use the slope and intercept to graph
. Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,
.: Graphing Linear Functions STARTER. What does it mean to INTERCEPT a pass in football? The path of the defender crosses the path of the thrown football. In algebra, what are - and -intercepts? What are
Chapter 1 Test Do ou know HOW? Identif a pattern and find the net three numbers in the pattern. 1. 5, 1, 3, 7, c. 6, 3, 16, 8, c Each term is more than the previous Each term is half of the previous term;
8.8 Linear Inequalities, Sstems, and Linear Programming 481 8.8 Linear Inequalities, Sstems, and Linear Programming Linear Inequalities in Two Variables Linear inequalities with one variable were graphed
90 (7-) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
Math 152, Intermediate Algebra Practice Problems 1 Instructions: These problems are intended to give ou practice with the tpes Joseph Krause and level of problems that I epect ou to be able to do. Work
CONDENSED L E S S O N 5.1 A Formula for Slope In this lesson ou will learn how to calculate the slope of a line given two points on the line determine whether a point lies on the same line as two given
STRSS-STRAIN RLATIONS Strain Strain is related to change in dimensions and shape of a material. The most elementar definition of strain is when the deformation is along one ais: change in length strain
ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
CONDENSED L E S S O N 6.1 Solving Sstems of Equations In this lesson ou will represent situations with sstems of equations use tables and graphs to solve sstems of linear equations A sstem of equations
8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive - and -directions.
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram
Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
Precalculus Worksheet P. 1. What is the distance formula? Use it to find the distance between the points (5, 19) and ( 3, 7).. What is the midpoint formula? Use it to find the midpoint between the points
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
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