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Lesson 11.1 & 11.2: Graphing Exponential & Logarithmic FunctionsLearning Goals:How do we graph an exponential function?How do we find the domain and range of an exponential function?How do we estimate the value of an exponential funciton using a graph?How do we graph a logarithmic function?What is the relationship between an exponential and a logarithmic function?How do we find the domain and range of a logarithmic function?Standard Form of Exponential Functions:y=bx1. Graph the functions:a. y=2xb. y=12xWhat is the domain of the function?What is the domain of the function?D=(-∞,+∞) D=(-∞,+∞) What is the range of the function?What is the range of the function?R=(0, ∞) affected by asymptote at y=0R=(0, ∞)From the graph, estimate the value of From the graph, estimate the value of 212.122.5y=2x=212 y=122.5=12xx=12 and y≈1.5 x=2.5 and y≈0.2 327660038100000c. What does the graph of ex look like? Graph y=ex. State the domain and range of the function.D=(-∞,+∞) R=(0, ∞) From the graph, estimate the value of e2≈7.3891 e is an irrational #!Graph the decimals with mon Characteristics of Graphs of Exponential Functions of the form46767751778000 y=bx, b>0 and b≠1Domain: (-∞, ∞)Range: (0, ∞)y-intercept: (0, 1)x-axis is a horizontal asympototeWhat is the equation of this horizontal asymptote? y=0What is the end behavior of the graph due to the asympotote?x→∞, y→∞ and as x→-∞, y→0 (going towards the asympotote)1. Now let’s compare the graphs of the functions f2x=2x and f3x=3x. Sketch the graphs of the two exponential functions on the same set of axes; then, answer the questions below.4838700429958500a. Where do the two graphs intersect? (0, 1)b. For which values of x is 2x<3x? 0<x<∞ or x>0c. For which values of x is 2x>3x? x<0d. What happens to the values of the functions f2 and f3 as x→∞? y→∞e. What happens to the values of the functions f2 and f3 as x→-∞? y→0f. Does either graph ever intersect the x- axis? Explain how you know. No because of the asymptote at y=0.When real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation y=a1+rtThe quantity 1+r is called the growth factor.2. In January, 1993, there were about 1,313,000 internet hosts. During the next five years, the number of hosts increased by about 100% per year.a. Write a model giving the number h (in millions) of hosts t years after 1993. About how many hosts were there in 1996?b. Graph the model.c. Use the graph to estimate the year when there were 30 million hosts.2676525476250003: On the grid below graph the exponential function y=2x and its inverse. xy=2xxy-2.25.25-2-1.5.5-10110122124423883a. What is the equation of the inverse of y=2x? How would we solve it for y?x=2y Use logs:y=log2xb. In what quadrants does the graph of the exponential function y=2x lie?I & IIc. In what quadrants does the graph of its inverse lie?I & IVd. Is the graph of the inverse of y=2x a function? If so, what is its domain and range? Yes, D=0, ∞ and R=(-∞, ∞)… asymptote at x=04: On the grid below graph the exponential function y=ex and its inverse.a. What is the equation of the inverse of y=ex? How would we solve it for y?x=ey Use logs: y=logexb. In what quadrants does the graph of the exponential function y=ex lie? I & IIc. In what quadrants does the graph of its inverse lie? I & IVd. Is the graph of the inverse of y=ex a function? If so, what is its domain and range? Yes, D=0, ∞ and R=(-∞, +∞)A function f(x) is an exponential function: fx=bx or y=bxDomain =(-∞, ∞) Range =(0, ∞)Asymptote at y=0Its inverse f-1(x) is a logarithmic function: f-1x=logbx or y=logbxDomain =(0, ∞) Range =(-∞, ∞)Asymptote at x=05: Graph the points in the table for the functions fx=logx and gx=log2(x), for the given values. Then, sketch smooth curves through those points and answer the questions that follow.a. What do the graphs indicate about the domain of your functions?b. Describe the x-intercepts of the graphs.c. Describe the y-intercepts of the graphs.d. Find the coordinates of the point on the graph with y-value 1.e. Describe the behavior of the function as x→0.f. Describe the end behavior of the function as x→∞.g. Describe the range of your function.h. Does this function have any relative maxima or minima? Explain how you know.i. For which values of x is log2x<log?(x)Key Features of the graph of y=logbx for b>1.The domain is the positive real numbers, and the range is all real numbers.The graphs all cross the x-axis at (1, 0).A point on the graph is always (b, 1).None of the graphs intersect the y-axis. Asymptote at x=0The have the same end behavior as x→∞, f(x)→∞, and they have the same behavior as x→0, fx→-∞.The functions all increase quickly for 0<x<1, then increase more and more slowly.4714875-7683500As the value of b increases, the graph will flatten as x→∞.There are no relative maxima or minima.Inverse of the exponential function y=bx.6: Graph the points in the table for your assigned function rx=log110(x) and sx=log12(x). Then sketch smooth curves through those points, and answer the questions that follow.a. What is the relationship between your graphs in Example 3 and your graphs from this example.b. Why does this happen? Use the change of base formula to justify what you have observed in part (a).c. For which values of x is log12x<log110(x)?Conclusion: From what we have seen of these sets of graphs of functions, can we state the relationship between the graphs of y=logbx and y=log1bx, for b≠1?if b≠1, then the graphs of y=logbx and y=log1bx are reflections of each other across the x-axis.Key Features of the graph of y=logbx for 0<b<1.The graph crosses the x-axis at (1, 0).The graph does not intersect the y-axis.The graph passes through the point 1b, -1.As x→0, the function values increase quickly; that is, fx→∞.As x→∞, the function values continue to decrease; that is, fx→-∞There are no relative maxima or minima.Inverse of the exponential function y=bx.Homework 11.1 & 11.2: Graphing Exponential & Logarithmic Functions1. Sketch the graphs of the functions f1x=12x and f2x=34x below and answer the following questions.a. Where do the two exponential graphs intersect?b. For which values of x is 12x<34x?c. For which values of xis 12x>34x?d. What happens to the values of the functions f1 and f2 as x→∞?e. What are the domains of the two functions f1 and f2?f. What are the ranges of the two functions f1 and f2?2. Graph the function fx=log3(x) and identify its key features. Domain:Range:End Behavior:As x→0, f(x)→As x→∞, f(x)→x-intercepty-interceptGraph passes through point:3. Consider the logarithmic functions fx=logb(x) and gx=log5(x), where b is a positive real number, and b≠1. The graph of f is given.a. Is b>5 or is b<5? Explain how you know.b. Compare the domain and range of functions f and g.c. Compare the x-intercepts and y-intercepts of f and g.d. Compare the end behavior of f and g.4. On the same set of axes, sketch the functions fx=log2x and gx=log2(x3).a. For which values of x is log2x<log2(x3)?b. At what point do the two graphs intersect?c. Describe the end behavior of the function f(x).d. What is the domain and range for the function g(x)?Lesson 11.3: Transformations and Characteristics of Logarithmic and Exponential FunctionsLearning Goals:How do we describe the transformations of a logarithmic or exponential function?What are the characteristics of a logarithmic or exponential function?Warm-Up:The figure below shows graphs of the functions fx=log3x, gx=log5(x), and hx=log11(x).a. Identify which graph corresponds to which function. Explain how you know.b. Sketch the graph of kx=log7(x) on the same axes.c. What happens to the values of the functions fx, gx, and hx as x→∞?d. Do any of the graphs ever intersect the y-axis? Explain how you know.Summary of TransformationsFunctionTransformationfx+aVertical translation, up afx-aVertical translation, down af(x+a)Horizontal translation, left af(x-a)Horizontal translation, right a-f(x)Reflection, negate y, rx-axisf(-x)Reflection, negate x, ry-axisk?fx or 1k?f(x)Vertical, Dilation, multiply y by either k or 1kfk?x or f1k?xHorizontal, Dilation, multiply x by either k or 1kPerforming multiple transformations:If a function has multiple transformations, they are applied in the following order:H orizontal translation D ilation R eflectionV ertical translationSketching Log EquationsExample 1: Sketch the two logarithmic functions g2x=log2(x) and g3x=log3(x) on the axes below; then, answer the following questions. 204787555245002352675-190500a. For which value of x is log2x=log3(x)? x=1 b. For which values of x is log2x<log3x? 0<x<1 c. For which values of x is log2x>log3(x)? x>1d. What happens to the values of the functions g2 and g3 as x→∞? y→∞e. What happens to the values of the functions g2 and g3 as x→0? y→-∞ f. Does either graph ever intersect the y-axis? Explain how you know. No because the asymptote at x=0Example 2: Which statement regarding the graphs of the functions below is untrue?0000gx=(x-.5)(x+4)(x-2) hx=log2(x) Asymptote is the y-axis!jx=-4x-2+3 (1) fx and j(x) have a maximum y-value of 3.(2) fx, hx, and j(x) have one y-intercept.(3) gx and j(x) have the same end behavior as x→-∞.(4) gx, hx, and j(x) have rational zeros.Example 3: Sketch the graph of fx=log2(x) by identifying and plotting at least five key points. Use the table below to get started.Given the above function fx=log2(x), describe each of the following:a. The sequence of transformations that takes the graph of f to the graph of gx=log2(8x). H orizontal D ilation (multiply x by 18)b. The sequence of transformations that takes the graph of f to the graph of hx=3+log2x. V ertical translation (moves up 3)Example 4: Describe the graph of px=log2x4 as a vertical translation of the graph of fx=log2(x). Justify your response.log2x-log24=log2x-2 V ertical translation (moves down 2)Example 5: Describe the graph of g as a transformation of the graph of f.fx=log3x and gx=2log3(x-1). H orizontal translation (right 1) andVertical D ilation (multiply y by 2)Example 6: g(x) is the image of f(x) after a shift four units up followed by a vertical stretch of 3. If fx=log2(x), which of the following gives the equation of g(x)?(1) gx=3log2(x+4)(2) gx=3log2x+12(3) gx=3log2x+4(4) gx=3log213x+4Example 7: Describe the given function as a transformation of the graph of a function in the form fx=bx. So basically, fx=3xgx=3x-2The exponent of x-2 means it is a horizontal translation that moves to the right 2 units.Example 8: Using the function fx=2x, create a new function g whose graph is a series of transformations of the graph of f with the following characteristics:The function g is increasing for all real numbers.The equation for the horizontal asymptote is y=5.gx=2x+5 Example 9: Graph the following function and answer the questions below: fx=3x-24100830265938000400050060198000 a. What is the domain of fx? D=(±∞)b. What is the range of f(x)? R=(-2, ∞)c. What is the equation of the asymptote? y=-2d. Describe the transformation that occurs from gx=3x to f(x). Vertical translation down 2Example 10: Graph the following function and answer the questions below: fx=log3(x+2)3952875189357000 a. What is the domain of fx? D=(-2, ∞)b. What is the range of f(x)? R=(±∞)c. What is the equation of the asymptote? x=-2d. Describe the transformation that occurs from gx=log3(x) to f(x). horizontal translation left 2.Homework 11.3: Transformations and Characteristics of Logarithmic and Exponential Functions1. Describe the graph of g as a transformation of the graph of f.a. fx=log2x & gx=log2(x-3)b. fx=log2x & gx=log28xc. fx=log5x & gx=-log5(5x+2)d. Describe the given function as a transformation of the graph of a function in the form fx=4x for gx=4x-2+5.2. Relative to the graph of gx=log2(x), what is the shift of the graph of fx=3log2(x+5)?(1) 5 left (2) 5 up (3) 5 right (4) 5 downLesson 11.4: Solving Log Equations GraphicallyLearning Goals:How do we solve log equations graphically?How do we graph a system of log equations?Do Now: Answer the following questions in order to prepare for today’s lesson.284797525019000a) Sketch the graph of this pair of functions on the same coordinate axes.fx=12xgx=-12x+4666759080500b) Describe the graph of gx as a series of transformations on the graph of fx. rx-axis, then up 4 with a new asymptote at y=4c) Describe the end behaviors of f(x) and of g(x).x→-∞, y→∞ and x→∞, y→0 x→-∞, y→-∞ and x→∞, y→4 d) Find the solution to the equation fx=g(x). -1, 2 is where they intersect!Solving a System of Log Equations GraphicallyReview from the last unit:Solve the following equation algebraically for all values of x: logx=log2x“drop a log”:x1=2xx2=2x=±2≈1.4Model Problem: Solve the following equations graphically for all values of x: logx=log2x31623001256665001. Solve the following equation graphically for all values of x: logx=log?(x-2)952503145155We can extend this idea to any unfamiliar equation. You can use this graphing method if:The question does not specify to solve algebraically.The question tells you to round your answers.It is a multiple choice question where no work is required.020000We can extend this idea to any unfamiliar equation. You can use this graphing method if:The question does not specify to solve algebraically.The question tells you to round your answers.It is a multiple choice question where no work is required. 2. When fx=2x+2 and hx=logx+1+3 are graphed on the same set of axes, which coordinates best approximate their point of intersection?(1) -0.9, 1.8 (2) (-0.9, 1.9) (3) (1.4, 3.3) (4) (1.4, 3,4)3. Given the two functions below:fx=18x3+78x2-1511x+1gx=-0.5x+4State the solutions to the equation fx=g(x), rounded to the nearest hundredth.4. Solve for x: 30-2x=x-3 Justify your solution(s).30-2x=(x-3)230-2x=x2-6x+90=x2-4x-210=x-7x+3x=-7 & x=3No solution!Homework 11.4: Solving Log Equations Graphically1. Solve the following equation graphically 2. Solve the following equation for all values of x: logx=log2xgraphically for all values of xto the nearest hundredth:14x-2=log4(x)318135015303500317182513970003. Given the graph of the following function on the coordinate axes, describe the intercepts and end behavior of the graph fx=12x+34. Use the relationship between the graphs of exponential and logarithmic functions to sketch the graphs of the functions g1x=log12x and g2=log34(x) on the same sheet of graph paper. Then, answer the following questions.242887528194000a. Where do the two logarithmic graphs intersect?b. For which values of x is log12x<log34(x)?c. For which values of x is log12x>log34(x)?d. What happens to the values of the functions g1 and g2 as x→∞?e. What are the domains of the two functions g1 and g2? 5. If fx=3(5)x, determine the coordinates of the y-intercept of the function defined by fx+4.(1) (0, 9) (2) (7, 0) (3) (0, 7) (4) (0, 5)6. Given the two functions below:fx=12x2-25x+3gx=-4 3x-2State the solutions to the equation fx=g(x), rounded to the nearest hundredth.7. If fx=x2-2x-8 and gx=14x-1, for which value of x is fx=g(x)?(1) -1.75 and -1.438 (2) -1.75 and 4 (3) -1.438 and 0 (4) 4 and 0Lesson 11.5: Inverses of Logarithmic and Exponential FunctionsLearning Goals:How can we find the domain of a logarithmic function algebraically?How do we find the inverse function of a logarithmic equation? How do we find the inverse function of an exponential equation?Do Now: How do you find the inverse of a function? Switch the x and y-values!Find the inverse of each of the following:1. fx=3logx2 for x>0y=3logx2 x=3logy2 inversex=32logy x6=6logy6 x6=logy “loop” and your base =10y=10x6 2. fx=2x-3y=2x-3 “anti-loop”x=2y-3 inverselog2x=y-3logx=log2y-3 add logs to both sidesy=log2x+3logxlog2=y-3log2log2 y-3=logxlog2 y=logxlog2+3 3. Consider the function fx=2x+1, whose graph is shown below.40576501841500a. What are the domain and range of f? D=-∞, ∞ and R=(1, ∞) b. Sketch the graph of the inverse function g on the graph. What type of function do you expect g to be? Find points on f(x) and switch x and y-values. Log Function!c. What are the domain and range of g? How does that relate to your answer in part (a)? D=1, ∞ and R=(-∞, ∞), really just swap domain and range of fd. Find the formula for g. gx is the inverse of fx.y=2x+1 x=2y+1 “anti-loop”x-1=2y log2x-1=ylogx-1log2=ylog2log2 add logs to both sides.y=logx-1log2 3667125514350004. The graph of a function f is shown below. Sketch the graph of its inverse function g on the same axes.a. Explain how you made your sketch.-1905045974000Switch x and y-value points on fx.b. The function f graphed above is the function fx=log2x+2 for x>0. Find a formula for the inverse of this function.y=log2x+2 x=log2y+2 x-2=log2y “loop”2x-2=y 5. Find the inverse of the following function. Then find the domain and range of both the original function and its inverse.fx=ex-5D=-∞, ∞ and R=(0, ∞)y=ex-5 x=ey-5 add logslnx=(y-5)lne lnx+5=y D=0, ∞ and R=-∞, ∞ -1047750If two functions whose domain and range are a subset of the real numbers are inverses, then their graphs are reflections of each other across the diagonal line given by y=x in the Cartesian plane.If f and g are inverses of each other, thenThe domain of f is the same set as the range of g.The range of f is the same set as the domain of g.In general, to find the formula for an inverse function g of a given function f:Write y=f(x) using the formula for f.Interchange the symbols x and y to get x=f(y).Solve the equation for y to write y as an expression in x.Then, the formula for g is the expression in x found in step (iii).The functions fx=logbx and gx=bx are inverses of each other.00If two functions whose domain and range are a subset of the real numbers are inverses, then their graphs are reflections of each other across the diagonal line given by y=x in the Cartesian plane.If f and g are inverses of each other, thenThe domain of f is the same set as the range of g.The range of f is the same set as the domain of g.In general, to find the formula for an inverse function g of a given function f:Write y=f(x) using the formula for f.Interchange the symbols x and y to get x=f(y).Solve the equation for y to write y as an expression in x.Then, the formula for g is the expression in x found in step (iii).The functions fx=logbx and gx=bx are inverses of each other.Practice: Find the inverse of each of the following functions: 6*. fx=7log1+9x7*. fx=lnx-lnx+1y=7log?(1+9x) y=lnx-lnx+1x=7log?(1+9y) x=lny-lny+1x7=log?(1+9y) x=lnyy+110x7=1+9y ex=yy+110x7-1=9y exy+1=y10x7-19=y exy+ex=yex=y-exyex=y(1-ex)ex1-ex=y or ex-1=y8*. fx=2x2x+19. fx=2x+1y=2x2x+1 y=2x+1x=2y2y+1 x=2y+1x2y+1=2y x-1=2yx2y+x=2y logx-1=ylog2x=2y-2yx log?(x-1)log2=yx=2y(1-x) xx-1=2y logxx-1=ylog2 logxx-1log2=y Homework 11.5: Inverses of Logarithmic and Exponential Functions1. Find the inverse of each of the following functions. In each case, indicate the domain and range of both the original function and its inverse.a. fx=6log?(1+2x)b. fx=lnx+5-ln?(x+1)c. fx=8+ln?(5+3x)d. fx=25-8xLesson 11.6: Identifying Exponential Growth and DecayLearning Goal: How do we determine if a situation is exponential growth or decay and what the growth/decay rate is?Do Now: Answer the following questions in order to prepare for today’s lesson.1. A population of wolves in a county is represented by the equation Pt=801.12t, where t is the number of years since 1998. How many wolves will there be in the year 2010? t=2010-1998=12P12=801.1212=311, so it is a growth!2. A population of wolves in a county is represented by the equationPt=800.90t, where t is the number of years since 1998. How many wolves will there be in the year 2010? t=2010-1998=12P12=800.9012=22, so it is a decay!Which question above represents growth? Which question represents decay? Why? See aboveWhat part of the formula causes the growth or decay?Pt=801.12t Pt=800.90tIn the formula, what does the "80" represent? Initial number of wolves!y=final amountt=timea=initial amount1+r or 1-r=rate of changeDirections: Determine whether each function represents exponential growth or decay. Then identify the rate of change.a. y=51.07t growth! 1.07>1b. ft=20.98t decay! .98<11+r=1.07 1-r=.98r=.07=7% -r=-.02r=.02=2%c. y=12t decay! 12<1d. pt=1002t growth! 2>11-r=12 1+r=2-r=-12 r=1=100%r=12=50% Rewriting Exponential Functions to Determine Growth or Decayy=a1+r2 t=a(1+r)2tDirections: Rewrite each function to determine whether each function represents exponential growth or exponential decay. Then identify the rate of change.a. ft=31.0210tb. y=1000.92t4ft=31.0210t y=1000.9214tft=3(1.21899442)t y=100(.9793703613)tft=3(1.22)t growth! 1.22>1y=100(.98)t decay! .98<11+r=1.22 1-r=.98r=.22=22% -r=-.02r=.02=2%c. pt=80?496412 t d. y=1.02-2 t pt=80496412t y=1.02-2tpt=8078t y=.9611687812t pt=80.88t decay! .88<1y=.96t decay! .96<11-r=.88 1-r=.96-r=-.12 -r=-.04r=.12=12% r=.04=4%Rewriting Exponential Functions in Terms of Monthly Rate of Growth or Decayy=a1+rt=a1+r1212 t=a1+r11212 tDirections: Rewrite each function in terms of monthly rate of growth or decay. Round decimals to the nearest thousandth.a. Pt=5001.12tb. y=12tPt=500(1.12)1212 t y=121212 tPt=5001.1211212t y=1211212 tPt=500(1.009)12 t (monthly)y=(.944)12 t (monthly)Determining Growth or Decay of Difficult Exponential FunctionsBased on today’s lesson, how do you think we can answer the following questions? 1) identify the initial amount (t=0); 2) pick another value for time (t=1)1. The function M(t) represents the mass of radium over time, t, in years.Mt=100eln12t1590Determine if the function M(t) represents growth or decay. Explain your reasoning.t=0, M=100 are the initial amounts use t=1 Mt=100eln12(1)1590 Mt=99.95641534 Decay because it is less than the initial amount2. The function f(t) represents a colony of bacteria over time, t, in years.ft=10eln2004 tDetermine if the function f(t) represents growth or decay. Explain your reasoning.t=0, 10 are the initial amounts use t=1 ft=10eln2004 (1) Mt=37.60603093 Growth because it is more than the initial amountHomework 11.6: Identifying Exponential Growth and Decay1. Use the properties of exponents to identify the percent rate of change (to the nearest hundredth) of the functions below, and classify them as representing exponential growth or decay.a. ft=1001.02tb. ft=21.0112 tc. ft=0.97td. ft=50?2312 t2. Milton has his money invested in a stock portfolio. The value, v(x), of his portfolio can be modeled with the function vx=30,0000.78x, where x is the number of years since he made his investment. Which statement describes the rate of change of the value of his portfolio?(1) It decreases 78% per year.(2) It decreases 22% per year.(3) It increases 78% per year.(4) It increases 22% per year.3. The growth of a certain organism can be modeled by Ct=101.02924 t, where C(t) is the total number of cells after t hours. Which function is approximately equivalent to C(t)?(1) Ct=240.08324 t(2) Ct=10.083t(3) Ct=101.986t(4) Ct=2401.986t244. A study of the annual population of the red-winged blackbird in Ft. Mill, South Carolina, shows the population, B(t), can be represented by the function Bt=7501.16t, where the t represents the number of years since the study began. In terms of the monthly rate of growth, the population of red-winged blackbirds can be best approximated by the function.(1) Bt=7501.012t(2) Bt=7501.01212 t(3) Bt=7501.1612 t(4) Bt=7501.16t125. The function f(t) represents a herd of llamas over time, t, years.ft=1000eln24 tDetermine if the function f(t) represents growth or decay. Explain your reasoning.Lesson 11.7 & 11.8: Writing Exponential Growth and Decay Functions & Exponential and Logarithmic RegressionLearning Goals: How can we write an exponential growth/decay equation and define its variables?How do we write an equation that models an exponential or logarithmic function.Do Now: Answer the following questions in order to prepare for today’s lesson.1. Given the exponential functions below, identify the initial amount, the rate of change, and if it is growth or decay.a. fx=1251.035xb. fx=50000.82xInitial Amount =125Initial Amount =5000Rate of Change =3.5%Rate of Change =36%1+r=1.035 fx=5000.82x=5000(.64)xr=.035=3.5% (Growth)1-r=.64-r=-.36r=.36=36% (Decay)Writing Exponential Growth and Decay Functionsy=final amounta=initial amount1+r or 1-r=rate of changet=time1. The inaugural attendance of an annual music festival is 150,000. The attendance increases by 8% each year.Write an exponential function that represents the attendance after t years.How many people will attend the festival in the fifth year? Round your answer to the nearest thousand.Find a formula…Growth! y=a(1+r)ty=150000(1+.08)t y=150000(1+.08)5 y=150000(1.08)5 y=220399.2115≈220,000 2. The value of a car is $21,500. Its value depreciates by 12% each year.Write a function that represents the value, in dollars, of the car after t years.Use this function to find how long it will take, to the nearest tenth of a year, for the car to be valued at $15,000Find a formula…Decay! y=a(1-r)ty=21500(1-.12)t 15000=21500(1-.12)t 15000=21500(.88)t 1500021500=21500(.88)t21500 3043=(.88)t log3043=tlog(.88) or “anti-loop”log.883043=tt=2.8 3. An apartment purchased 4 years ago for $80,000 was just sold for $105,000. Assuming exponential growth, approximate the annual growth rate, to the nearest percent.Write an exponential growth formula! y=a(1+r)t105000=80000(1+r)4 10500080000=80000(1+r)480000 1.3125=(1+r)4 raise both sides to 14 or 4 both sides!1.312514=(1+r)414 1.070347571=1+r r=.070347571≈7% -161925311785What if the word problem does not give you the rate of change as a percent?Exponential growth – look for keywords such as doubles, triples, quadruples, …Exponential decay – look for keywords such as halves (half-life), a third, ...00What if the word problem does not give you the rate of change as a percent?Exponential growth – look for keywords such as doubles, triples, quadruples, …Exponential decay – look for keywords such as halves (half-life), a third, ...Writing More Difficult Exponential Growth and Decay Functions4. An initial population of 20 mice triples each year. Write an equation that models this population after t years.Exponential growth (triples each year)!y=a(1+r)t1+r=3 Initial Amount =20y=20(3)t 5. The half-life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient’s bloodstream. A certain antibiotic has a half-life of one hour. A patient takes 500 milligrams of the medication. Write a function A(t) that models the amount of the medication in the patient’s bloodstream over time.Exponential decay (half-life)!y=a(1-r)t1-r=12 Initial Amount =500A(t)=50012t What if the word problem mentions the growth or decay rate over a specific period of time!a. The half-life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient’s bloodstream. A certain antibiotic has a half-life of one hour. A patient takes 500 milligrams of the medication. Divide by 1At=50012t b. The half-life of a medication is the time it takes for the medication to reduce to half of its original amount in a patient’s bloodstream. A certain antibiotic has a half-life of about 3 hours. A patient takes 500 milligrams of the medication.Divide by 3At=50012t3 6. Suppose that a water tank is infested with a colony of 1,000 bacteria. In this tank the colony doubles in number every 4 days. Determine a formula, A(t), for the number of bacteria present in the tank after t days.Exponential growth (doubles in number)!y=a(1+r)tEvery four days means to divide the time by 41+r=2 Initial Amount =1000A(t)=1000(2)t4 7. Lead- 209, a radioactive isotope, decays to nonradioactive lead over time. The half-life of lead-209 is 8 days. Suppose that 20 milligrams of lead-209 are created by a particle physics experiment. Write an equation for the amount of lead-209 present t days after the experiment.Exponential decay (half-life)!y=a(1-r)tEvery eight days means to divide the time by 81+r=12 Initial Amount =20y=2012t8 8. Titanium-44 is a radioactive isotope such that every 63 years, its mass decreases by half. For a sample of titanium-44 with an initial mass of 100 grams, write a function that will give the mass of the sample remaining after any amount of time.Exponential decay (decreases by half)!y=a(1-r)tEvery 63 years means to divide the time by 631+r=12 Initial Amount =100A(t)=10012t63 LOCATION OF REGRESSIONS IN THE CALCULATORThese non-linear regressions are also found using the graphing calculator. All types of regressions on the calculator are prepared in a similar manner.Your regression options can be found under STAT→CALC (scroll for more choices)9. A box containing 1,000 coins is shaken, and the coins are emptied onto a table. Only the coins that land heads up are returned to the box, and then the process is repeated. The accompanying table shows the number of trials and the number of coins returned to the box after each trial. Write an exponential regression equation, rounding the calculated values to the nearest ten-thousandth. stat→edit→L1,L2 stat→calc→?:ExpReg y=a(b)x y=1018.2839(.5969)x Decay!10. An object at a temperature of 160° is removed from a furnace and placed in a room at 20℃. The table shows the temperatures d at selected times t (in hours) after the object was removed from the furnace. Write a logarithmic regression equation for this set of data, rounding coefficients to 3 decimal places.stat→edit→L1,L2 then stat→calc→9:LnReg y=a+blnx t=.863-6.447lnd 33528002800350011. Jean invested $380 in stocks. Over the next 5 years, the value of her investment grew, as shown in the accompanying table. Write the exponential regression equation for this set of data, rounding all values to two decimal places.y=a(b)x y=379.92(1.04)x Growth!Using this equation, find the value of her stock, to the nearest dollar, 10 years after her initial purchase.y=379.92(1.04)10=$562 x=10 and y=?Using this equation, also find how many years, to the nearest tenth, it will take for her investment to grow to reach $900. x=? and y=900 900=379.92(1.04)x 900379.92=379.92(1.04)x379.92 2.368919773=1.04x “anti-loop” or log both sidesx=22.0 32766003575050012. The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10℉.Write the logarithmic regression equation for this set of data, rounding coefficients to the nearest ten thousandth.y=a+blnx y=13.0134-7.3135lnx Based on your equation, if the wind chill factor is 0, what is the wind speed, to the nearest mile per hour? y=0 and x=? 0=13.0134-7.3135lnx -13.0134=-7.3135lnx -13.0134-7.3135=-7.3135lnx-7.3135 lnx=1.779366924 “loop” with a base =ee1.779366924=x x=6 48291751987550013. The following set of data shows U.S. gas prices in recent years. The table below represents the U.S. price of gas from 2005 to 2015, where t=1 represents year 2005.a) Based on the table what was the average rate of change in the price of gasoline from 2005 to 2014, to the nearest thousandth?b) What is the exponential regression for the data in the table, rounding coefficients to the nearest thousandth? c) Based upon your regression, what is the average rate of change in the price of gasoline from 2005 to 2014, to the nearest thousandth?d) Why is there a difference between your answers using the table and using the regression equation?Homework 11.7 & 11.8: Writing Exponential Growth and Decay Functions & Exponential and Logarithmic Regression1. Doug drank a cup of tea with 130 mg of caffeine. Each hour, the caffeine in Doug’s body diminishes by about 12%. Write a formula to model the amount of caffeine remaining in Doug’s system after each hour. How long, to the nearest hour, should it take for the level of caffeine in Doug’s system to drop below 30 mg.2. A tree planted 10 years ago at 3 feet tall now measures 25 feet tall. Assuming exponential growth, approximate the annual growth rate, of the tree to the nearest percent.3. Which function represents exponential decay:(1) y=20.3t (2) y=1.23t (3) y=12-t (4) y=5-t4. The number of carbon atoms in a fossil is given by the functiony=51000.95x, where x represents the number of years since being discovered. What is the percent of change each year? Explain how you arrived at your answer.5. Last year, the total revenue for Home Style, a national restaurant chain, increased 5.25% over the previous year. If this trend were to continue, which expression could the company’s chief financial officer use to approximate their monthly percent increase in revenue?(1) 1.0525m (2) 1.052512m (3) 1.00427m (4) 1.00427m12373380047625006. The price of a postage stamp in the years since the end of World War I is shown in the scatterplot. The equation that best models the price, in cents, of a postage stamp based on these data is:(1) y=0.59x-14.82(2) y=1.041.43x(3) y=1.431.04x(4) y=1.040.96x7. The number of ticks in an infested field is decreasing with the use of a new pesticide. Below is a table which shows the number of ticks in the field after x applications of pesticide.x, number of applications01234y, number of ticks50004000320025602048a) Determine an exponential equation of the form y=a?bx that models this data, rounding all values to two decimal places.b) Using this equation, determine the number of ticks, to the nearest whole number, in the field after 9 applications of the pesticide.c) Using this equation, find the number of applications, to the nearest whole number, necessary to reduce the number of ticks to 1000.8. The height of a tree increases with age. At the time of planting, the tree was 1 year old. The table below shows the height h (in feet) of the tree at specific ages t (in years).a) Use a graphing calculator to find a logarithmic model of the form h=a+b ln t that represents the data. Round all values to the nearest hundredth.b) Estimate the height, to the nearest tenth, when the tree is 3 years old.Lesson 11.9: Calculating Monthly PaymentsLearning Goal: How do we determine a monthly payment for a loan using any given formula?Warm-Up: You are 25 years old and hold a steady job. One day on the way to work, you see that your dream car is on sale at a car dealership. You go to the dealer to see about buying it. The car dealer tells you that he has this great deal on the car. Apparently, only for today, he is selling the car for $45,000 with a down payment of $3,000. He says he did some calculations and gives you three car loan options which are shown below:What will be the monthly payments for each option? Which will you take? Can you afford any of them? If you end up buying it, how much spending money will you have left over each month for each option? How much will you end up paying back (with interest) after the loan is paid off? Is there a better deal out there? Since we are talking math here, there definitely is a better way to figure all this out than just signing the agreement and hoping for the best.Many people take out a loan to purchase a car and then repay the loan on a monthly basis. Today we will figure out how banks determine the monthly payment for a loan in today’s class. As seen in the scenario above, when deciding to get a car loan, there are many things to consider. For car loans, a down payment is a very common requirement. A down payment is the amount of money the person will pay towards the cost of the car before the loan is taken.For example: For the scenario above, he is selling the car for $45,000 with a down payment of $3,000. What will the loan amount be if he decides to choose one of the three loan options?When calculating loans and monthly payments, there are a few different formulas that can be used. The formula depends on the type of loan and what is included in the loan. Below are some examples of monthly loan formulas.-95250429260Tips when working with monthly loan formulas:Read the question twice. Be sure to pay attention to what each variable stands for.The second time, write down the numbers that correspond to each variable.Determine the variable that they are asking you to find.Substitute the values in the given formula.Try to simplify any expressions within the formula before solving for the remaining variable.If you are solving for an exponent, you will have to use logs.A down payment is the amount you would pay prior to taking out a loan. This amount gets subtracted from the total purchase price.00Tips when working with monthly loan formulas:Read the question twice. Be sure to pay attention to what each variable stands for.The second time, write down the numbers that correspond to each variable.Determine the variable that they are asking you to find.Substitute the values in the given formula.Try to simplify any expressions within the formula before solving for the remaining variable.If you are solving for an exponent, you will have to use logs.A down payment is the amount you would pay prior to taking out a loan. This amount gets subtracted from the total purchase pounding periods indicate how often the interest is calculated:Annually (n=1), Semi-annually (n=2), Quarterly (n=4), Monthly (n=12)1. Monthly mortgage payments can be calculated according to the formula, A=Mpn t(1-p)1-pn t, where M is the size of the mortgage, n is the number of compounds per year, t is the length of the mortgage, in years, and p=1+rn where r is the interest rate as a decimal.a. What would the monthly mortgage payments be on a $180,000, 15 year mortgage with 6% interest, compounded monthly, to the nearest dollar?b. Determine the length of the mortgage, to the nearest year, in order for the monthly payment to be $1,080.A=?M=180000n=12t=15r=.06 p=1+rn=1+.0612=1.005 A=Mpn t(1-p)1-pn tA=1800001.00512?15(1-1.005)1-1.00512?15A=-2208.684206-1.454093562=$15192. Monthly mortgage payments can be found using the formula below:M=Pr121+r12n1+r12n-1M= monthly payment, P= amount borrowed, r= annual interest rate, n= number of monthly paymentsThe Bank’s family would like to borrow $120,000 to purchase a home. They qualified for an annual interest rate of 4.8%. Algebraically determine the fewest number of whole years the Banks family would need to include in the mortgage agreement in order to have a monthly payment of no more than $720.n=?(monthly)M=720P=120000r=.048720=120000.048121+.04812n1+.04812n-1720=4801.004n1.004n-1“cross multiply”720(1.004n-1)=4801.004n720(1.004n-1)720=4801.004n7201.004n-1=231.004nlet x=1.004nx-1=23x13x=1x=3sub the x back in1.004n=3add logs or “anti-loop”nlog1.004=log3 or log1.0043=nn=275.2020158 (months)Fewest number of years!! Must divide by 12n12=22.93350 always round up so 23 years!3. Using the formula below, determine the monthly payment on a 3-year car loan with a monthly percentage rate of 0.75% for a car with an original cost of $25,000 and a $2,000 down payment, to the nearest cent.Pn=P M T1-1+i-niPn= present amount borrowed, n= number of monthly payments, P M T= monthly payment, and i= interest rate per monthPMT=?Pn=25000-2000=23000n=3*12=36i=.75100=.007523000=P M T1-1+.0075-36.007523000=P M T (31.44680525)731.3938512=$731.39=P M TThe affordable monthly payment is $350 for the same time period. Determine an appropriate down payment, to the nearest dollar.PMT=350Pn=25000-DPn=3*12=36i=.75100=.0075 25000-DP=3501-1+.0075-36.007525000-DP=11006.38184DP=$139944. Jim is looking to buy a vacation home for $172,600 near his favorite southern beach. The formula to compute a mortgage payment, M, is M=P?r1+rN1+rN-1 where P is the principal amount of the loan, r is the monthly interest rate, and N is the number of monthly payments. Jim’s bank offers a monthly interest rate of 0.305% for a 15-year mortgage. With no down payment, determine Jim’s mortgage payment, rounded to the nearest dollar.M=?P=172,600r=.305%=.00305N=15*12=180M=172600.00305(1+.00305)180(1+.00305)180-1=1247.49339Algebraically determine the fewest number of whole months that Jim needs to make in order for his mortgage payment to be no more than $1,100.M=1100N=?P=172600r=.00305 1100=172600.00305(1.00305)N(1.00305)N-11100172600=.00305(1.00305)N(1.00305)N-1 cross multiply5.2643(1.00305)N=11(1.00305N-1)Let x=1.00305N5.2643x=11x-15.2643x=11x-11-5.7357x=-11x=1.9178129961.917812996=1.00305NN=213.828861 so 214 months Homework 11.9: Calculating Monthly Payments1. Monthly mortgage payments can be found using the formula below:M=Pr121+r12n1+r12n-1M= monthly payment, P= amount borrowed, r= annual interest rate, n= number of monthly paymentsThe Ranallo family just bought a house. They qualified for an annual interest rate of 5.2% and their monthly payment was $800 for a 30 year mortgage. Algebraically determine the amount of their loan to buy the house.2. Using the formula below, determine the original loan value, to the nearest dollar, Sam would have taken out for a 5 year car loan with a monthly percentage rate of 2.5% if he is paying $300 a month.Pn=P M T1-1+i-niPn= present amount borrowed, n= number of monthly payments, P M T= monthly payment, and i= interest rate per monthLesson 11.10: Compound Interest FormulaLearning Goals:How do we determine the value of an investment using the compound interest formula?How do we determine the amount of time it takes for an investment to reach a certain value?Warm-Up: Use the formula A=P er t to answer the following question.Jackie’s parents invested money at an interest rate of 6% compounded continuously, to save toward Jackie’s college education.How much money did the parents invest when Jackie started first grade to accumulate $20,000 after 12 years? A=P er t for compounded continuouslyA=2000 A=P er tP=? 2000=P e.06?12 r=.06 2000=2.054433211 Pt=12 P=$9735.05Compound Interest FormulaA=P1+rnn t(memorize)Compounding periodsA= final amountAnnually (n=1)P=initial amount (investment)Semi-annually (n=2)r=interest rateQuarterly (n=4)n=compound periodsMonthly (n=12)t=time 1. Robert invests $800 in an account at 1.8% interest compounded annually. He will make no deposits or withdrawals on this account for 3 years. Which formula could be used to find the balance, A, in the account after the 3 years.(1) A=8001-.183 (2) A=8001+.183 (3) A=8001-.0183 (4) A=8001+.0183A=P1+rnn t A=8001+.01811?3(P=800 A=8001+.0183r=.018 n=1 t=3 2. Duane plans to make a one-time investment and wants to reach $10,000 in thirty-six months for Kira’s wedding. He finds an investment with 3.5% interest compounded monthly. How much should he invest?A=10000 A=P1+rnn t P=? 10000=P1+.0351212?3 r=.035 10000=P1.00291666736n=12 10000=P(1.110540876)t=3 years P=9004.623. A home owner invests $15,000 at an annual interest rate of 4%. What will be the value of this investment after 6 years if the investment is compounded quarterly?A=P1+rnn t=150001+.0444?6=19046.024. Linda invests $6,000 into an account. After 5 years her investment reaches $7,200. If the interest was compounded annually, determine the rate to the nearest tenth of a percent.A=7200 A=P1+rnn t P=6000 7200=60001+r11?5 r=? 7200=60001+r5n=1 72006000=60001+r56000t=5 years 1.2=(1+r)5get rid of exponent1.215=(1+r)5151.037137289=1+rr=.037137289=3.7%5. Jack and Jill are twins that received $2,000 each for their 16th birthday. They decided to separately invest their money. Jack is going to invest his money at 3.7% interest compounded quarterly. Jill is going to invest her money at 3.6% interest compounded monthly.a. Write a function of Jack’s option and Jill’s option that calculates the value of each account after t years.A=P1+rnn tJack: A=20001+.03744 tJill: A=20001+.0361212 tb. Determine who will have more money by the time they graduate high school in two years, to the nearest cent. t=2Jack: A=20001+.03744 ? 2Jill: A=20001+.0361212 ? 2Jack: A=$2152.88Jill: A=$2149.08c. Algebraically determine, to the nearest tenth of a year, how long it would take for Jill’s initial investment to triple.A=2000×3=6000 6000=20001+.0361212 ? tt=? 60002000=20001.00312 ? t20003=1.00312tlog3=12tlog1.003log312log1.003=12tlog1.00312log1.003t=30.6Finding Growth Rate6. Find the rate of growth to the nearest ten-thousandth for the amount of money accumulated if $6,000 is invested at 8% interest compounded quarterly.7. If $10,000 is invested in an account at 6.5% compounded monthly, determine the growth rate to the nearest thousandth.Homework 11.10: Compound Interest Formula1. Krystal was given $3,000 when she turned 2 years old. Her parents invested it at a 2% interest rate compounded annually. No deposits or withdrawals were made. Which expression can be used to determine how much money Krystal had in the account when she turned 18?(1) 30001+0.0216 (2) 30001-0.0216 (3) 30001+0.0218 (4) 30001-0.02182. Nelson’s financial goal is to invest a sum of money at 3.5% interest compounded monthly. If his goal is to have $10,000 at the end of 3 years, how much will he need to invest? (Assume rounding to the nearest penny.)3. Katie invests $4,500 in an account that pays 3.75% interest compounded quarterly. How long, to the nearest tenth of a year, will it take for Katie’s investment to double? ................
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