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This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

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Algebra II Li near, Exponential and Logarithmic Functions

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L inear Functions Exponential Functions Logarithmic Functions Properties of Logs e and ln Growth and Decay Table of Contents Solving Logarithmic Equations click on the topic to go to that section

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Return to Table of Contents Linear Functions

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Goals and Objectives Students will be able to analyze linear functions using x and y intercepts, slope and different forms of equations. Linear Functions

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Why do we need this? Being able to work with and analyze lines is very important in the fields of Mathematics and Science. Many different aspects of life come together in linear relationships. For example, height and shoe size, trends in economics or time and money. Quickly, even these situations become non-linear, but we can still model some information using lines. Linear Functions

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We will begin with a review of linear functions: a) x and y intercepts b) Slope of a line c) Different forms of lines: i) Slope-intercept form of a line ii) Standard form of a line iii) Point-slope form of a line d) Horizontal and vertical lines e) Parallel and perpendicular lines f) Writing equations of lines in all three forms

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Linear Functions Definitions: x and y intercepts Graphically, the x-intercept is where the graph crosses the x-axis. To find it algebraically, you set y = 0 and solve for x. Graphically, the y-intercept is where the graph crosses the y-axis. To find it algebraically, you set x = 0 and solve for y. yx Find the x and y intercepts on the graph to the right. Write answers as coordinates. x-int = y-int = Teacher x-int = (-6, 0) y-int = (0, 8)

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Linear Functions y x Find the x and y intercepts on the graph to the right. Write answers as coordinates. x-int = y-int = Teacher x-int = (5, 0) y-int = (0, 3)

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Linear Functions y x An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept. Examples of lines with a y- intercept of ____ are shown on this graph. What is the difference between them (other than their color)? Teacher Slope Have students then figure out the slope of these three lines. Dark Blue = -2 Green = 0 Red = -1/2 Purple = 4/3

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Linear Functions Def: The Slope of a line is the ratio of its rise over its run. For notation, we use "m." y x run rise You can find slope two ways: Algebraically:Graphically: count Teacher Stress the difference and advantages to both methods.

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y x rise run Find the slope of this line. m = Teacher m = 4/3 Linear Functions

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1Find the slope of the line to the right. y x rise run (Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.) Teacher m = -3/4 Make the rise and run bigger or smaller to emphasize that there are many places to find slope. Linear Functions

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2Find the slope of the line to the right: Linear Functions y x rise run Teacher m = 2/3 Make the rise and run bigger or smaller to emphasize that there are many places to find slope. (Move the rise and the run to fit the slope. Make them bigger or smaller to get an accurate slope.)

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Find the slope of the line going through the following points.

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3Find the slope of a line going through the following points: Linear Functions (-3, 5) and (4, -3) Teacher m = -8/7

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4Find the slope of the line going through the following points. Linear Functions Teacher m = 3/5 (0, 7) and (-5, 4)

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Slope formula can be used to find the constant of change in "real world" problems. Roofs Mountain Highways DistanceHeight Growth And many more... Teacher Get ideas from students where slope is used in the "real world." Linear Functions

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When traveling on the highway, drivers will set the cruise control and travel at a constant speed. This means that the distance traveled is a constant increase. The graph below represents such a trip. The car passed mile- marker 60 at 1 hour and mile-marker 180 at 3 hours. Find the slope of the line and what it represents. Time (hours) Distance (miles) (1,60) (3,180) Linear Functions

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5If a car passes mile-marker 100 in 2 hours and mile-marker 200 after 4 hours, how many miles per hour is the car traveling? Teacher m = 50mph Linear Functions

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6How many meters per second is a person running if they are at 10 meters in 3 seconds and 100 meters in 15 seconds? Teacher m = 7.5 meters per second Linear Functions

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We are going to look at three different forms of the equations of lines. Each has its advantages and disadvantages in their uses. Slope-Intercept Form Standard FormPoint-Slope Form y = mx + b Ax + By = C y - y1 = m(x - x1) Advantage: Easy to find slope, y- intercept and graph the line from the equation. Disadvantage: Must be solved for y. Advantage: Easy to find intercepts and graph. Disadvantage: Must manipulate it algebraically to find slope. Advantage: Can find equation or graph from slope and any point. Disadvantage: Cumbersome to put in another form. Linear Functions

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Determine the equation of the line from the given graph. 1) Determine the y-intercept. 2) Graphically find the slope from any two points. 3) Write the equation of the line using the slope-intercept form. Remember, graphically, the y-intercept is where the graph crosses the y-axis. b = -3 y = mx + b y = 2x - 3 click

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7 Which equation does this line represent? Ay = -6x - 6 By = -x - 6 Cy = 6x + 6 D y = x - 6 Linear Functions Teacher D y = x - 6 Ask students which form it is in: Slope-intercept

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8 Which graph represents the equation y = 3x - 2? ALine A BLine B CLine C DLine D Linear Functions A B C D Teacher D line D

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9What equation does line A represent? Ay = 2x + 3 By = -2x + 3 Cy = 0.5x + 3 Dy = -0.5x + 3 A B C D Teacher B

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10What equation does line B represent? Ay = 2x + 3 By = -2x + 3 Cy = 0.5x + 3 Dy = -0.5x + 3 A B C D Teacher D

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Linear Functions Consider the equation 4x - 3y = 6. Which form is it in? Graph it using the advantages of the form. Teacher It is in Standard Form. Graph it using the x and y intercepts. x-int: (3/2, 0) y-int: (0, -2)

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Graph the equation 5x + 6y = -30 using the most appropriate method based on the form. Teacher It is in Standard Form. Graph it using the x and y intercepts. x-int: (-6, 0) y-int: (0, -5)

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Graph the equation 3x - 5y = -10 using the most appropriate method based on the form. Teacher It is in Standard Form. Graph it using the x and y intercepts. x-int: (-10/3, 0) y-int: (0, 2)

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Write the equation in standard form. Multiply both sides of the equation by the LCD (6). Rearrange the equation so that the x and y terms are on the same side together. Make sure that x is always positive.

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11What is the Standard Form of: A3x + 5y = 15 B9x + 15y = 35 C15x - 9y = 35 D5x - 3y = 15 Linear Functions Teacher B: 9x + 15y = 35 Multiply each term by LCD: 15

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12Which form is this equation in? AStandard Form BSlope-Intercept Form CPoint-Slope Form Teacher y - 3 = 4(x + 2)

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13Find y when x = 0. Linear Functions y - 3 = 4(x + 2) Teacher Point-Slope Form y is 11

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14Put the following information for a line in Point-Slope form: m = -3 going through (-2, 5). Ay - 3 = -2(x + 5) By - 2 = 5(x - 3) Cy - 5 = -3(x + 2) Dy + 2 = -3(x - 5) Linear Functions Teacher C y - 5 = -3(x + 2)

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15Put the following information for a line in Point-Slope form: m = 2 going through (1, 4). Ay + 4 = 2(x + 1) By + 4 = 2(x - 1) Cy - 4 = 2(x - 1) Dy - 4 = 2(x + 1) Linear Functions Teacher C y - 4 = 2(x - 1)

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16Put the following information for a line in Point-Slope form: m = -2/3 going through (-4, 3). Ay + 3 = -2/3(x + 4) By - 3 = -2/3(x - 4) Cy + 3 = -2/3(x - 4) Dy - 3 = -2/3(x + 4) Linear Functions Teacher D y - 3 = -2/3(x + 4)

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Horizontal and Vertical Lines Example of a Vertical Line x=3Example of Horizontal line y=2 Horizontal lines have a slope of 0. *Notice that a horizontal line will "cut" the y axis and has the equation of y = m. Vertical lines have an undefined slope. *Notice that a vertical line will "cut" the x axis and has the equation of x = n. Linear Functions

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17 Which equation does the line represent? Ay = 4x By = x + 4 Cy = 4 Dy = x Linear Functions Teacher C y = 4 Ea sy way to remember horizontal and vertical equations: Horizontal lines "cut" the y-axis at the number indicated. Therefore, it is y = n. Vertical lines "cut" the x-axis at the number indicated. Therefore, it is x = m.

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18Describe the slope of the following line: y = 4 (Think about which axis it "cuts.") AVertical BHorizontal CNeither DCannot be determined Linear Functions Teacher BHorizontal

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19Describe the slope of the following line: 2x - 3 = 4. AVertical BHorizontal CNeither DCannot be determined Linear Functions Teacher A Vertical

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Parallel and Perpendicular Lines Parallel lines have the same slope: h(x) = -3x - 11 g(x) = 1/3x - 2 Perpendicular lines have opposite, reciprocal slopes. h(x) = x + 6 q(x) = x + 2 r(x) = x - 1 s(x) = x - 5 Perpendicular lines have opposite, reciprocal slopes.

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A) y = 4x - 2 is perpendicular to _____________. B) y = -1/5x + 1 is perpendicular to _____________. C) y - 2 = -1/4(x - 3) is perpendicular to _____________. D) 5x - y = 8 is perpendicular to _____________. E) y = 1/6x is perpendicular to _____________. F) y - 9 = -5(x -.4) is perpendicular to _____________. G) y = -6(x + 2) is perpendicular to _____________. Drag the equation to complete the statement. Linear Functions Teacher Have students come to the board and drag. A) y = -1/4x - 3 B) 1/5y = x - 2 C) y = 4x - 1 D) y = -1/5x + 9 E) 6x + y = 10 F) y = 1/5x G) y = 1/6x - 6

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How to turn an equation from standard form to slope-intercept form. Rearrange the equation so that the y term is isolated on the right hand side. Make sure that y is always positive. How to turn an equation from point-slope to slope-intercept form. Distribute Rearrange the equation so that the y term is isolated on the right hand side.

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Are the lines 2x - 3y = 7 and 4x - 6y = 11 parallel? Turn the equations into slope-intercept form. Both slopes are 2/3. Yes the lines are parallel.

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Are the lines 2x + 5y = 12 and 8x + 20y = 16 parallel? Turn the equations into slope-intercept form. Teacher Both slopes are 2/5. Yes the lines are parallel.

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20Which line is perpendicular to y = -3x + 2? Ay = -3x + 1 B Cy = 5 Dx = 2 Linear Functions Teacher B

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21Which line is perpendicular to y = 0? Ay = -3x + 1 By = x Cy = 5 Dx = 2 Linear Functions Teacher D x = 2

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Linear Functions Writing Equations of Lines Remember the three different forms of the equations of lines. Brainstorm when each form would be easiest to use when having to write an equation of the line. Slope-Intercept Form Standard FormPoint-Slope Form y = mx + b Ax + By = C y - y1 = m(x - x1) Teacher Slope-Intercept Form: Easiest to use if you have slope and y- intercept. If not, you will have to solve for b. Standard Form: Not very useful to find equations, but easy to find intercepts and graph with. Point-Slope Form: Easiest form to use in general. The downside is most questions will ask for another form.

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Writing an equation in Slope- Intercept Form: From slope and one or two points: 1. Find the slope. Use either given information or the formula: 2. Plug the coordinates from one point into x and y respectfully. 3. Solve for b. 4. Write as y = mx + b. From slope and y-intercept: 1. Just plug into slope into m and y-intercept into b. 2. Write as y = mx + b Linear Functions

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Write an equation in slope-intercept form through the given points. Find the slope. Plug the coordinates from one point into x and y respectfully.

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Linear Functions Write the equation of the line with slope of 1/2 and through the point (2,5). Leave your answer in slope-intercept form. Teacher

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Linear Functions Write the equation of the line through (-3,-2) and perpendicular to y = - 4/5x + 1 in slope-intercept form. Teacher

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22Write the equation of the line going through the points (1, 2) and (3, -4). Your answer should be in slope-intercept form. A B C D Linear Functions y = -3x + 5 y = -3x - 3 y = -3x + 2 Teacher

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23What is the equation of the line with a slope parallel to 3x + y = 2 going through the point (4, -2) Ay = -3x + 10 By = -3x - 14 Cy = 3x + 10 Dy = -3x + 14 Linear Functions Teacher

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Writing an equation in Point-Slope Form: Linear Functions From slope and one or two points: 1. Find the slope. Use either given information or the formula: 2. Plug the coordinates from one point into x1 and y1 respectfully. 3. Write as y - y1 = m(x - x1).

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Write an equation in point-slope form through the given two points. Find the slope. Use either point.

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Linear Functions Write the equation of the line through (5,6) and (7,1) using Point-Slope form. Teacher or

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Linear Functions Write the equation of the line with slope of 1/2 and through the point (2,5) in Point-Slope form. Teacher

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24Which of the following is the Point-Slope form of the line going through (-2, 4) and (6, -2)? A B C D Linear Functions Teacher

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25 Which of the following is the Point-Slope form of the line going through (0, 3) having the slope of 2? Ay - 3 = 2(x - 3) By = 2(x - 3) Cy - 3 = 2x Dy = 3 Linear Functions Teacher

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Linear Functions Write the equation of the line with m= 4, through (3,1) and (4,5) in Standard Form. Standard Form Teacher 4x - y = 11 Have a class discussion about the usefulness of this form and how easy this was to do. Put the equation into point-slope form. Then turn it into standard form. Distribute Rearrange the equation so that the x and y terms are on the same side together. Make sure that x is always positive. Rearrange the equation so that the x and y terms are on the same side together. Make sure that x is always positive. Distribute

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How to turn an equation from point-slope to standard form. Write the equation of the line through (-1,-5) and (-4,-2) in Standard Form. Teacher x + y = -6 Have a class discussion about the usefulness of this form and how easy this was to do.

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Write the equation of the line through (-7, -2) and (1, 6) in Standard Form. Teacher x - y = 5 Have a class discussion about the usefulness of this form and how easy this was to do.

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Linear Functions Write the equation of the line with x-intercept of 5 and y-intercept of 10. Which form would be easiest to use? Teacher y = -2x + 10 or y - 10 = -2x or y = -2(x - 5) There is nothing that states what form you must leave answer in, so many answers are correct.

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Linear Functions Write the equation of the line through (4,1) and parallel to the line y = 3x - 6. Which form is easiest to use in this case? Teacher y - 1 = 3(x - 4) y = 3x -11 3x - y = 11

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26 Given a point of (-3, 3) and a slope of -2, what is the easiest form to write the equation in? ASlope-intercept Form BPoint-Slope Form CStandard Form Linear Functions Teacher This is an opinion question that fuels discussion about different forms and their purpose.

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27Given two points on a line, (3, 0) and (0, 5), what is the easiest form to write the equation in? ASlope-intercept Form BPoint-Slope From CStandard Form Linear Functions Teacher This is an opinion question that fuels discussion about different forms and their purpose.

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28Which of the following is/are the equation(s) of the line through (1, 3) and (2, 5) in Point-Slope form? Ay - 5 = 2(x - 2) By - 3 = 2(x - 1) C y = 2x + 1 D2x - y = -1 Linear Functions Teacher All of the answers are equations of the line, but only A and B are in Point-Slope form.

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Return to Table of Contents Exponential Functions

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Goals and Objectives Students will be able to recognize and graph exponential functions. Exponential Functions

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Why do we need this? Most situations that people study do not have linear relationships. Population growth is now heavily studied around the world. Is this a linear function? Why do we study population growth? Exponential Functions

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We have looked at linear growth, where the amount of change is constant. This is a graph of the amount of money your parents give you based on the number of A's that you receive on your report card. XY Can we easily predict what you would get if you have 7 A's? Number of A's $$

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Exponential Functions The equation for the previous function is y = 10x. What if you asked your parents to reduce the amount per A to $3, but then asked them to use the following function: y = 3x Graph it!! What would you get for 7 A's? Teacher Capitalize on this problem to emphasize the difference between linear and exponential relationships. Which equation is better for your pocketbook? But is it realistic?

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Exponential Functions Here is the graph of your function! This is an example of exponential growth. y = 3x Can you find the amount for 7 A's? Would that provide some motivation?

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Exponential Functions Now, let's look an opposite problem: Suppose you are given 80 M&M's and each day you eat half. What does this curve look like? This is an example of exponential decay. X Y Teacher Draw the curve on the graph. Ask students if the curve will ever go below the x-axis and why?

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Exponential Functions Number of M&M's Days Here is a graph of the M&M problem. 1. Where does the graph cross the y-axis? 2. How many M&M's do you have on day 2, 3, 4 and 5? 3. When are all of the M&M's gone? Teacher 1. It crosses at 80. That is the amount we start with. 2. Day 2: 20, Day 3:10, Day 4: 5, Day 5: Either 2, 2.5, or 3. Have a discussion about how you would deal with the issue of "half of 5" M&M's. 3. Bring up asymptotes and discuss the possibility that the M&M's may never be gone if we take half of one, then half of the half, etc... Relate it to asymptotes.

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Exponential Functions Now we are going to identify an Exponential Function both graphically and algebraically.

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Exponential Functions The exponential function has a curved shape to it. Y-values in an exponential function will either get bigger or smaller very, very quickly. Domain: (x values) Range: (y values) Graphically Exponential GrowthExponential Decay Teacher Stress and explore domain and range. Why does the domain run from negative infinity to positive infinity? Why does the range run from 0 to positive infinity?

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29Which of the following are graphs of exponential growth? ( You can choose more than one.) Exponential Functions A B C D E FG H Teacher A and F

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30Which of the following are graphs of exponential decay? (You can choose more than one.) A BCD EFGH Exponential Functions Teacher D

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where x is the variable and a, b, and c are constants. The general form of an exponential function is Exponential Functions b is the growth rate. If b > 1 then it is exponential growth If 0 < b < 1 then it is exponential decay y = c is the horizontal asymptote (0, a + c) is the y-intercept remember, to find the y-intercept, set x = 0. Teacher Students will struggle with b. Make sure you stress the difference between b > 1 and 0 < b < 1 related to growth and decay. Discuss why 1 is not included.

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31Does the following exponential equation represent growth or decay? AGrowth BDecay Exponential Functions Teacher A Growth

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32 For the same function, what is the equation of the horizontal asymptote? Ay = 2 By = 3 Cy = 4 Dy = 5 Exponential Functions Teacher C y = 4

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33Now, find the y-intercept: A(0, 3) B(0, 4) C(0, 7) D(0, 9) Exponential Functions Teacher C (0, 7)

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34 Now consider this new exponential equation: Does it represent growth or decay? AGrowth BDecay Exponential Functions Teacher B Decay

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35 Which of the following is the equation of the horizontal asymptote? Ay = 0.2 By = 1 Cy = 3 Dy = 4 Exponential Functions Teacher C y = 3

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36Find the y-intercept in the same function: A(0, 0.2) B (0, 1) C (0, 3) D (0, 4) Exponential Functions Teacher D (0, 4)

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37Does the following exponential function represent growth or decay? AGrowth BDecay Exponential Functions Teacher B Decay. Rewrite the function with a positive exponent: H ave a discussion on what is different about this function.

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38Find the horizontal asymptote for the following function: Ay = 0 By = 1 Cy = 3 Dy = 4 Exponential Functions Teacher A y = 0

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39For the same function, what is the y-intercept? A(0, 0) B(0, 1) C(0, 3) D(0, 4) Exponential Functions Teacher D (0, 4)

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To sketch the graph of an exponential function, use the values for a, b and c. 1) Identify horizontal asymptote (y = c) 2) Determine if graph is decay or growth 3) Graph y-intercept (0,a+c) 4) Sketch graph Exponential Functions Try it! Teacher Note: A horizontal asymptote is the horizontal line at y = c that the exponential function cannot pass.

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Exponential Functions Graph by hand: Teacher

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Exponential Functions Graph by hand: Teacher

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40Which of the following is a graph of: AB C D Exponential Functions Teacher C

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41Which of the following is a graph of: AB CD Exponential Functions Teacher B

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Logarithmic Functions Return to Table of Contents

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Goals and Objectives Students will be able to rewrite exponential equations as logarithmic equations, rewrite logarithmic equations as exponential equations, manipulate logarithmic expressions, solve exponential equations and logarithmic equations. Logarithmic Functions

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Why do we need this? Logarithms are used to simplify calculations. They make certain exponential equations much easier to solve and allow us to study how exponents affect functions.

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Logarithmic Functions A logarithmic function is the inverse of an exponential function. += Graphically, the inverse of any function is the reflection of that function over the line y = x. Essentially, the x and y coordinates become switched.

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Logarithmic Functions Logarithmic functions switch the domain and range of exponential functions, just like points are switched in inverse functions. *Notice that 0 and any negative are not in the domain of a logarithm. Therefore, you cannot take the log of 0 or a negative. Domain: Range:

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Logarithmic Functions Logarithms are a way to study the behavior of an exponent in an exponential function. Here is how to rewrite an exponential equation as a logarithmic equation (and visa versa): baseexponent = answer logbaseanswer = exponent *A log that appears to have no base is automatically considered base 10. For example:

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Logarithmic Functions baseexponent = answer logbaseanswer = exponent Rewrite the following in logarithmic form: Teacher

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Logarithmic Functions baseexponent = answer logbaseanswer = exponent Rewrite the following in exponential form. Teacher

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42 Which of the following is the correct logarithmic form of: A B C D Logarithmic Functions Teacher A

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43 Which of the following is the correct logarithmic form of: A B C D Logarithmic Functions Teacher D

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44 Which of the following is the correct exponential form of: AB CD Logarithmic Functions Teacher B

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45 Which of the following is the correct exponential form of: A B C D Logarithmic Functions Teacher C

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46What is the exponential form of: A B C D Logarithmic Functions Teacher A

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Logarithmic Functions Convert each of the following to exponential form in order to either simplify the logarithmic expression or to find the value of the variable. Teacher

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47Find c: Logarithmic Functions Teacher

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48Find x: Logarithmic Functions Teacher

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49Evaluate: Logarithmic Functions Teacher

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50Evaluate: Logarithmic Functions Teacher 4

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51Evaluate: Logarithmic Functions Teacher 3

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52Evaluate: Logarithmic Functions Teacher -4 Use this as a reminder of negative exponents and what they mean.

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53Evaluate: Logarithmic Functions Teacher -3 Use this as a reminder of negative exponents and what they mean.

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54Evaluate: Logarithmic Functions Teacher No solution. You cannot take the log of a negative number.

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55Evaluate: Logarithmic Functions Teacher You need a calculator to do this one. We will cover it in the next few slides.

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Logarithmic Functions Take out a calculator and find "log". What base is it? Use the calculator to find the following values. Round off to three decimal places. Teacher

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Logarithmic Functions Evaluate: Since the log button on your calculator is only base 10, we need a change of base to get the value of something other than base 10. Teacher Make sure that students close parentheses and put them in the right place.

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56Evaluate: Logarithmic Functions Teacher Make sure that students close parentheses and put them in the right place.

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57Evaluate: Logarithmic Functions Teacher Make sure that students close parentheses and put them in the right place.

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58Evaluate: Logarithmic Functions Teacher Students will question you about the negative, as there are not logs of negative numbers. Stress that this is returning a negative VALUE, and is not the log of a negative number.

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59Evaluate: Logarithmic Functions Teacher Students will question you about the negative, as there are not logs of negative numbers. Stress that this is returning a negative VALUE, and is not the log of a negative number

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Logarithmic Functions The change of base formula and converting to logarithms will now allow you to solve more complicated exponential equations. Algebraically, how would you solve: Teacher This can be done with a graph, but logarithms make it easier. Put it in logarithmic form and use a calculator to solve.

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Logarithmic Functions Solve the following equation: Teacher

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60Solve the following equation. Logarithmic Functions Teacher

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61Solve: Logarithmic Functions Teacher

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62Solve: Logarithmic Functions Teacher

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Logarithmic Functions Practice using the same conversion to solve logarithmic equations again. Solve: Teacher

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63Solve: Logarithmic Functions Teacher

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64Solve: Logarithmic Functions Teacher

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65Solve: Logarithmic Functions Teacher

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Properties of Logarithms Return to Table of Contents

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Properties of Logarithms Basic Properties of Logs

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Properties of Logarithms WHAT?

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Properties of Logarithms Relate logs to exponents... Take: The property says that: In exponential form, it means: And...

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Properties of Logarithms Relate logs to exponents... Take: The property says that: In exponential form, it means: And...

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Properties of Logarithms Relate logs to exponents... Take: The property says that: In exponential form, it means: And...

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Properties of Logarithms Relate logs to exponents... And... Take: The property says that: In exponential form, it means:

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Relate logs to exponents... Properties of Logarithms And... Take: The property says that: In exponential form, it means:

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Properties of Logarithms Examples: Use the Properties of Logs to expand: Teacher

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66Which choice is the expanded form of the following A B C D Properties of Logarithms Teacher D

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67Which choice is the expanded form of the following A B C D Properties of Logarithms Teacher C

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68Which choice is the expanded form of the following A B C D Properties of Logarithms Teacher A

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69Which choice is the expanded form of the following A B C D Properties of Logarithms Teacher C

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Properties of Logarithms Examples: Use the Properties of Logs to rewrite as a single log. Teacher

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70Contract the following logarithm: A B C D Properties of Logarithms Teacher A

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71Contract the following logarithmic expression: A B C D Properties of Logarithms Teacher C

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72Which choice is the contracted form of the following: A B C D Properties of Logarithms Teacher B

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73Which choice is the contracted form of the following: A B C D Properties of Logarithms Teacher A

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Solving Logarithmic Equations Return to Table of Contents

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Solving Logarithmic Equations To solve a logarithmic equation, it needs to be put into one of the following forms: *After the equation is in this form, you may need to convert to exponential form. *After the equation is in this form, a and c must be equal. Therefore, you may remove the logarithms and solve.

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Solving Logarithmic Equations Before we solve, should we put this equation into a logarithm on one side or a logarithm on both sides? Teacher This equation can be contracted on one side to get one logarithm.

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Solving Logarithmic Equations Solve: Teacher The solution r = -1 is actually an extraneous solution as you cannot take a log of x ≤ 0.

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Solving Logarithmic Equations STOP! Extraneous Solutions: Remember you cannot take a log of x ≤ 0. ALWAYS check to see if your solution(s) work. Teacher Stress that students must check for extraneous solutions when solving logarithmic equations.

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Solving Logarithmic Equations 74Solve the following equation: Teacher

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75Solve the following equation: Solving Logarithmic Equations Teacher

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76Solve the following equation: Solving Logarithmic Equations Teacher

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77Solve the following equation: Solving Logarithmic Equations Teacher,

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78Solve the following equation: Solving Logarithmic Equations Teacher

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79Solve the following equation: Solving Logarithmic Equations Teacher

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Solving Logarithmic Equations How can we use these concepts to solve: Teacher It is actually legal to take the log of both sides. You can then apply the properties of logarithms to solve.

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Solving Logarithmic Equations Try: Teacher It is actually legal to take the log of both sides. You can then apply the properties of logarithms to solve.

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80Solve: Solving Logarithmic Equations Teacher

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81Solve: Solving Logarithmic Equations Teacher

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82Solve: Solving Logarithmic Equations Teacher

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e and ln Return to Table of Contents

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Properties of Natural Logarithms Basic Properties of Natural Logs

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e and ln The letter e, shows up quite often when dealing with exponential functions. It is a number that models things like the growth of a bacteria colony, the spread of an oil spill and even calculating compound interest. Formally, e is defined to be: Simply, as a number: *e is similar to pi in sense that it will never repeat and never ends...

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e and ln As with exponential functions, we can find the inverse of a function with base e. This is called The Natural Log and is noted: *Find ln on your calculator. This is loge.

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e and ln The graphs of e and ln are similar to our other functions. The domain and range also remain the same.

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e and ln e and natural logs have all of the same properties that other exponentials and logarithms have. For example: because

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e and ln Write the following in the equivalent exponential or log form. Teacher

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e and ln Put the following into a single logarithm: Teacher

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e and ln Expand the following logarithms: Teacher

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83Put into a single logarithm: A B C D e and ln Teacher D

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84Expand the following logarithm: A B C D e and ln Teacher B

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85Expand the following logarithm: A B C D e and ln Teacher B

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86Put into a single logarithm: A B C D e and ln Teacher A

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e and ln Solve the following equations: Teacher *Sometimes you will be asked to leave answers as exact numbers.

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e and ln Solve the following equations: Teacher *Only 60.6 or are solutions because you cannot take the log of a negative.

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e and ln 87Find the value of x. Teacher

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e and ln 88Find the value of x. Teacher

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e and ln 89Find the value of x. Teacher

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e and ln 90Find the value of x. Teacher

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e and ln 91Find the value of x. Teacher

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e and ln 92Find the value of x. Teacher

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Growth and Decay Return to Table of Contents

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Growth and Decay Goals and Objectives Students will be able to model growth and decay problems with exponential equations.

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Growth and Decay Why do we need this? What will the population of the world be in 2050 at the current rate of growth? How long will it take the radioactive material near the nuclear reactors in Japan to dissipate to harmless levels after the destructive tsunami? These are important questions that we need to answer to plan for the future! Growth and decay can be modeled and analyzed with exponential and logarithmic functions.

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Growth and Decay And...these functions will also model problems dealing with something we ALL need to learn to work with... Money!

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Growth and Decay This formula represents the amount (A) of money in a savings account if the interest is continually compounded. P = the principal (amount deposited) r = the annual interest rate (in decimal form) t = time in years If $500 is invested at 4% for 2 years, what will account balance be? Teacher $ in 2 years.

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Growth and Decay If $500 is invested at 4%, compounded continually, how long until the account balance is doubled? Teacher It will take about 17.3 years for the money to double.

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93 If $1000 is invested at 4% for 3 years, compounded continually, what is the account balance? Growth and Decay Teacher

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94 If $1000 is invested at 3.5%, how long until the account balance is doubled? Growth and Decay Teacher 19.8 years

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Growth and Decay I = interest P = principal (deposit) r = interest rate (decimal) t = time in years n = number of times compounded per unit of t A=account balance after the interest is included S imple Interest: Account Balance with Simple Interest SiSsS sss A ccount Balance with Compound Interest: There are other types of interest that you may encounter... Teacher Spend time discussing parts of each type of interest.

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Growth and Decay Calculate the Simple interest if you were to buy a car from your parents for $4000 and pay for it over 4 years at 3% interest. Teacher I = $480

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Growth and Decay Calculate the simple interest if you were to purchase a car from a used car dealer for 4000 at 10% interest over 4 years. Teacher I = $1600 Ask students which is the better deal.

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Growth and Decay Using the formula for compound interest, calculate the amount in your account if you invest $1300, over 15 years, at 4.6% interest compounded monthly. Teacher

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Growth and Decay How long would it take for an investment of $10,000 to increase to a total of $25,000 compounded quarterly at a rate of 3.5%? Teacher

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95 How many years did you take to pay off a $5000 car at 7% simple interest if you paid a total of $6200 for the car? Growth and Decay Teacher

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96What was the rate of your investment if you invested $5000 over 10 years, compounded continuously and you made $2200 in interest? Growth and Decay Teacher

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97How much was originally invested if you have $63, in an account generating 4% interest (compounded monthly) over 15 years? Growth and Decay Teacher

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Growth and Decay The same formulas can be used to model growth or decay in other situations. P = Population (Initial) r = rate or growth or decay(decimal) t = time Teacher Discuss the different uses of the same formulas.

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Growth and Decay Example: A bacteria constantly grows at a rate of 10% per hour, if initially there were 100 how long till there were 1000? Teacher It will take the colony 23 hours to increase its population from 100 to 1000.

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Growth and Decay A new car depreciates in value at a rate of 8% per year. If a 5 year old car is worth $20,000,how much was it originally worth? How will we write the rate? Teacher The original value of the car was about $30,345.

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Growth and Decay A certain radioactive material has a half-life of 20 years. If 100g were present to start, how much will remain in 7 years? Use half-life of 20 years to find r. Teacher In 7 years, about 78.5g remain.

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98 If an oil spill widens continually at a rate of 15% per hour, how long will it take to go from 2 miles wide to 3 miles wide? Growth and Decay Teacher It would take ≈ 2.7 hours

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99 If you need your money to double in 8 years, what must the interest rate be if is compounded quarterly? Growth and Decay Teacher

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100 NASA calculates that a communications satellite's orbit is decaying exponentially at a rate of 12% per day. If the satellite is 20,000 miles above the Earth, how long until it is visible to the naked eye at 50 miles high, assuming it doesn't burn up on reentry? Growth and Decay Teacher

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Growth and Decay 101 If the half-life of an element is 50 years, at what rate does it decay? Teacher

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