Pre-Calculus. Linear Functions. Slide 1 / 305 Slide 2 / 305. Slide 3 / 305. Slide 4 / 305. Slide 5 / 305. Slide 6 / 305.

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1 Slide 1 / 305 Slide 2 / 305 Pre-alculus Parent Functions Slide 3 / 305 Slide / 305 Table of ontents click on the topic to go to that section Step, bsolute Value, Identity and onstant Functions Exponential Functions Logarithmic Functions Properties of Logs ommon Logs e and ln Growth and ecay Logistic Functions Trig Functions Power Functions Positive Integer Powers Negative Integer Powers Rational Powers Rational Functions Return to Table of ontents Slide 5 / 305 Slide / What is the y-intercept of this line? 2 What is the x-intercept of this line?

2 n 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's the difference between them (other than their color)? Slide 7 / 305 onsider this Slide 9 / 305 The red line has a positive slope, since the line rises from left to the right. Slide / 305 The Slope of a Line rise 2 0 Slide / 305 run The Slope of a Line The Slope of a Line The orange line has a negative slope, since the line falls down from left to the right. 2 0 rise run The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis. 2 0 Slide 11 / 305 The Slope of a Line The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis. rise 0 = undefined 2 0 While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line. Slide 12 / 305 Measuring the Slope of a Line rise 2 0 run

3 Slide 13 / 305 Slide 1 / 305 Measuring the Slope of a Line Measuring the Slope of a Line The slope of the line is just the ratio of its rise over its run. The symbol for slope is "m". So the formula for slope is: slope = rise run run rise 2 0 slope = rise run The slope is the same anywhere on a line, so it can be measured anywhere on the line. rise 2 0 run For instance, in this case we measure the slope by using a run from x = 0 to x = +: a run of. uring that run, the line rises from y = 0 to y = : a rise of. slope = rise run m = m = 3 ut we can also start at x = 3 and run to x = : a run of 3. uring that run, the line rises from y = 3 to y = 7: a rise of. slope = rise run m = 3 Slide 15 / 305 Measuring the Slope of a Line 2 0 Slide 17 / 305 run rise Measuring the Slope of a Line 2 0 rise run ut we get the same result with a run from x = 0 to x = +3: a run of 3. uring that run, the line rises from y = 0 to y = : a rise of. slope = rise run m = 3 ut we can also start at x = and run to x = 0: a run of. uring that run, the line rises from y = to y = 0: a rise of. slope = rise run m = m = 3 Slide 1 / 305 Measuring the Slope of a Line 2 0 Slide 1 / run 0 rise run Measuring the Slope of a Line rise

4 Slide 19 / 305 Slide 20 / 305 Slope formula can be used to find the constant of change in a "real world" problem. 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 istance increase. (miles) The graph at the right represents such a trip. The car passed mile-marker 0 at 1 hour and milemarker at 3 hours. Find the slope of the line and what it represents. m= miles0 miles = 120 miles = 3 hours-1 hours 2 hours 0 miles hour (1,0) Time (hours) (3,) If a car passes mile-marker 0 in 2 hours and mile-marker 200 in hours, how many miles per hour is the car traveling? Use the information to write ordered pairs (2,0) and (,200). So the slope of the line is 0 and the rate of change of the car is 0 miles per hour. Slide 21 / 305 Slide 22 / If a car passes mile-marker 90 in 1.5 hours and mile-marker 150 in 3.5 hours, how many miles per hour is the car traveling? How many meters per second is a person running if they are at meters in 3 seconds and 0 meters in 15 seconds? Slide 23 / 305 Slide 2 / Which equation does the graph represent? y = x y = -1x+ y = x+ y = x 2 0 Which equation does the graph represent? y = x y = x+ y = y = x 2 0

5 Slide 25 / 305 Slide 2 / Which equation does the graph represent? y = -x + y = -3x+ y = -3x+3 y = 2 Which equation does the graph represent? y = 2x y = x+3 y = x-3 y = -3x Slide 27 / 305 Slide 2 / Which equation does the graph represent? y = x+ y = x+ y = 5x+ y = -5x+ 2 0 Which graph represents the equation y = 3x? line line line line 2 0 Slide 29 / 305 Slide 30 / Which graph represents y = -1/2x+3? line line line line 2 0 line can be graphed by using the x- and y- intercepts. The technique of using intercepts works well when an equation is written in STNR FORM. Standard form looks like x + y =, where,, and are integers and >0. and the Greatest ommon Factor of,, and is 1.

6 Slide 31 / 305 Slide 32 / Is x -3y = in standard form? Yes No 13 Is x + 3y = 7 in standard form? Yes No Slide 33 / 305 Slide 3 / Is x - y = in standard form? Yes No 15 The standard form of y = x +7 is 2x - y = 7 2x + y = 7 x - y = 7 x + y = -7 Slide 35 / 305 Slide 3 / The standard form of y = 3x -7 is 3x - y = 7 17 The standard form of y = - 2 / 3x + 5 is 2x - 3y = 15 3x + y = -7 2x + 3y = 15-3x - y = 7 x - 3y = 15-3x + y = -7 x + 3y = 15

7 Slide 37 / 305 Slide 3 / The standard form of y - = 1 / 2(x +7) is x - 2y = -15 x + 2y = 15 x - 2y = 15 x + 2y = -15 Given the equation x-3y=12 Find the x-intercept. x-intercept: Let y=0: x-3(0)=12 x=12 x=3 so x-intercept is (3,0) Find the y-intercept. y-intercept: Let x=0: (0)-3y=12-3y=12 y= so y-intercept is (0) Slide 39 / 305 Slide 0 / 305 Given the equation x-3y=12 we found the x-intercept is (3,0) and the y-intercept is (0,). Graph the intercepts and then the line that passess through them. 2 What are the x- and y- intercepts of y=3x-9? 0 Slide 1 / 305 Slide 2 / Given the equation y = 1/2x-7, what is x when y=0? 20 Given the equation y = 1/2x-7, what is y when x=0?

8 Slide 3 / Given the equation y = (x+2), what is x when y=0? Slide / Given the equation y = (x+2) what is y when x=0? Slide 5 / 305 Slide / 305 Horizontal and Vertical Lines VERTIL line goes "up and down". It has the equation x=a number, the x-intercept. Notice no y in the equation. n Example of a Vertical Line x=3 Return to Table of ontents HORIZONTL line goes "sideways". It has the equation y=a number, the y-intercept. Notice no x in the equation. Horizontal line y=2 Slide 7 / 305 Slide / 305 horizontal line has a slope of 0, as opposed to a vertical line which has an undefined slope. n Example of a Vertical Line x=3 23 Is the following equation that of a vertical line,a horizontal line, neither, or cannot be determined: y= 2 points on the horizontal line are 2 0 (0,2) and (5,2): m= = = points on the vertical line are -9-5 (3,) and (3,9): m= = =undefined because you can't divide by Horizontal line y=2 Vertical Horizontal Neither annot be determined

9 Slide 9 / 305 Slide 50 / Is the following equation that of a vertical line, a horizontal line, neither, or cannot determine: x+2y = 9 25 Is the following line that of a vertical, a horizontal, neither, or cannot be determined: x = 3 Vertical Horizontal Neither annot be etermined Vertical Horizontal Neither annot be etermined Slide 51 / 305 Slide 52 / Is the following equation that of a vertical line, a horizontal line,neither, or cannot be determined: 2x-3 = 0 Vertical Horizontal Neither annot be etermined 27 The intercepts method of graphing could not have been used to graph which of the following graphs? E There is more than 1 answer. E G H F G H F Slide 53 / 305 Slide 5 / Which of the following equations can't be graphed using the intercepts method? There are multiple answers. y=3 y = 1/2(x+9) y = -3x x = E y = x+7 F 3x - y =12 G x = 2y H y=x I 2y = x J y+ = (x+1) Parallel and Perpendicular Lines Return to Table of ontents

10 Slide 55 / 305 Slide 5 / 305 The lines at the right are parallel lines. Notice that their slopes are all the same. Parallel lines all have the slopes because if they change at different 2 rates eventually they would 0 intersect. h(x)=x+ This also works for vertical and horizontal lines. q(x)=x+2 r(x)=x-1 s(x)=x-5 29 Which line is parallel to y = -3x-17? y = -3x+1 y = 1/3x y = 5 x = 2 Slide 57 / 305 Slide 5 / Which line is parallel to y = 0? y = -3x+1 y = 1/3x y = 5 x = 2 31 Which line is parallel to y + 1 = -3(x)? y = -3x+1 y = 1/3x y = 5 x = 2 Slide 59 / 305 Slide 0 / 305 In the diagram the 2 lines form a right angle, when this happens lines are said to perpendicular. Look at their slopes. This time they are not the same instead they are opposite reciprocals and -3x 1 / 3 = -1. h(x)=-3x-11 g(x)= 1 /3x ) y=x is perpendicular to ) y=- 1 / 5x+1 is perpendicular to ) y=- 1 / (x-3) is perpendicular to ) 5x-y= is perpendicular to E) y= 1 / x is perpendicular to F) y-9=-5(x-.) is perpendicular to G) y=(x+2) is perpendicular to Perpendicular Equation ank (rag the equation to complete the statement.) x+y= y= 1 / 5x 1 / 5y=x y=- 1 / 5x+9 y= 1 / x y=x+1 y=- 1 / x-3

11 Slide 1 / 305 Slide 2 / 305 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? 32 Which line is perpendicular to y = -3x+2? y = -3x+1 y = 1/3x y = 5 x = 2 Slide 3 / 305 Slide / Which line is perpendicular to y+2 = -3(x)? 3 Which line is perpendicular to y = 0? y = -3x+1 y = 1/3x y = 5 x = 2 y = -3x+1 y = 1/3x y = 5 x = 2 Slide 5 / 305 Slide / 305 Graphing Using the Point-Slope Equation of a Line The point-slope equation for a line is y - y 1 = m (x - x 1 ) where m is the slope and (x 1,y 1) is a point on the line. (x 1,y 1) can be graphed and then applying m, a second point can be graphed. The line containing all of the (x,y) can be graphed. Return to Table of ontents 1) Plot (x 1,y 1) 2) From that point, use the slope (m) to plot a second point. 3)Graph the line through these 2 points. This line represents all of the points that are solutions to the equation.

12 Slide 7 / 305 Slide / 305 The point-slope equation for a line is y - y 1 = m (x - x 1 ) where m is the slope and (x 1,y 1) is a point on the line. 35 What is the slope of y-3 = (x+2)? Notice that the signs of (x 1,y 1) are changed from the formula. 1 2 (-7,3) 2 Given the equation y - 3 = 2 (x + 7) the line passes through (-7,3) and has a slope of 2. 0 Now the graph can be drawn. Slide 9 / 305 Slide 70 / Which line represents y+5 = - 1 / 3(x)? line line line line Which is the slope and a point on the line y-1 = 1/3(x)? m=1/3; (-1,0) m= -1/3; (0,-1) m=1/3; (0,1) m is undefined; (0,1) Slide 71 / 305 Slide 72 / 305 Given the equation y = 5(x -1) etermine the point on the line and the slope. Given the equation y -1 = 2 / 5 (x +5) etermine the point on the line and the slope. Now graph the line. Graph the line representing the equation.

13 Slide 73 / 305 Slide 7 / What is the slope and a point on the line y+5 = -3(x)? 39 Which line represents y+ = -3(x)? m=-3; (,-5) m=-3; (,5) m=3; (,-5) m=3; (,5) line line line line 2 0 Slide 75 / 305 Slide 7 / Which line represents y+5 = -3(x)? line line line line Which point is on the line y-3 = (x+2)? (-3,2) (3,) (2,-3) (,3) Slide 77 / 305 Slide 7 / 305 Given the equation y + = 1 / 3 (x +2) etermine the point on the line and the slope. Graph the line representing the equation. To write an equation in point-slope form: First find the slope: -the slope can be given:for example "the slope is"or"m=" y2-y1 -given two points, use the slope formula: m= x2 -x1 -given a parallel line, use the same slope -given a perpendicular, use the opposite reciprocal slope fter finding the slope use a point on the line to write the equation. If the directions ask for a different form, like y=mx+b, convert point slop into the desired form.

14 Slide 79 / 305 Slide 0 / 305 Write the equation of the line with slope of 1 / 2 and through the point (2,5). Write the equation of the line with slope -3 and containing the point (1,2) in slope-y-intercept form. Slide 1 / 305 Slide 2 / 305 Write the equation of the line through (5,) and (7,1). Write the equation of the line through (3,1) and (,5) in standard form. Slide 3 / 305 Slide / 305 Write the equation of the line with x-intercept of 5 and y-intercept of. Write the equation of the line through (,1) and parallel to the line y=3x.

15 Slide 5 / 305 Slide / 305 Write the equation of the line through (-3,) and perpendicular to y=- / 5x+1 in slope-y-intercept form. 2 Which is the equation of the line with slope of -3 and through (1,)? y = -3(x+1) y+ = 3(x-1) y+ = -3(x-1) y+ = -3(x+1) Slide 7 / 305 Slide / Which is the equation of the line through (1,3) and (2,5) in slope y-intercept form? y-3 = 2(x-1) y = 2x+1 y-5 = 2(x) y = 2x+3 Which is the equation of the line through (,5) and perpendicular to y-7 = -1/3(x+9)? y-5 = -1/3(x+2) y-5 = -3(x+2) y-5 = 3(x) y-5 = 3(x+2) Slide 9 / 305 Slide 90 / 305 Step, bsolute Value, Identity, and onstant Functions There are special functions that have there own names and graphs. onstant Function Identity Function Return to Table of ontents y = b omain: Reals Range: Reals y = x omain: Reals Range: Reals

16 Slide 91 / 305 Slide 92 / 305 bsolute Value Function y = a cx -h + k omain: Reals Range: < y To Graph an bsolute Value Graph 1) Set the value inside of the absolute value sign equal to zero and solve. This is x-value of the vertex of the graph. 2) reate a table. Use the solution to step one as a middle value by picking a couple of points smaller and a couple larger. omplete the table. 3) Graph the points. ) onnect the points. 5) s a check, if the number in front of the absolute value sign is positive the "V" opens up, if its negative it opens down. X Y Slide 93 / 305 Slide 9 / 305 :Reals; R: < y Graph y = -3 2x Graph y = 2 x X Y X Y : ; R: : ; R: Slide 95 / 305 Slide 9 / What is the x value of the vertex of y = 2x Which of the following is the correct graph of y = x+ - 2? 1 1/2 2

17 Slide 97 / 305 Slide 9 / Which of the following is the correct graph of y = x - - 2? Which of the following is the correct graph of y = 3x ? Slide 99 / 305 Slide 0 / Graph y = 2x What is the domain of the graphed function? Set of Integers Set of Reals x > -3 x < -3 Slide 1 / 305 Slide 2 / What is the range of the graphed function? 52 What is the domain of the graphed function? Set of Integers Set of Integers Set of Reals Set of Reals y > -3 x > 3 y < -3 x < 3

18 Slide 3 / 305 Slide / What is the range of the graphed function? Greatest Integer Functions Set of Integers Set of Reals y > 3 y < 3 [2] = 2 [2.1] = 2 [2.3] = 2 [2.5] = 2 [2.75] = 2 [2.999] = 2 [3] = 3 [] = [.1] = -3 [.3] = -3 [.5] = -3 [.75] = -3 [.999] = -3 [-3] = -3 The [ ] tell you to round to the preceding integer. Think round to the left on a number line. [ ] are a grouping sign and the inside should be simplified before rounding. Slide 5 / 305 Slide / 305 Evaluate 5 Evaluate [2.] [3.5 +.] [3.7 -.] [ 2-2.1] 3[2. +.2] [3(2.) +.2] 3[2.] +.2 [2.1-2] 2 Slide 7 / 305 Slide / Evaluate [5+2] 5 Evaluate [. ]

19 Slide 9 / 305 Slide 1 / Evaluate [.1 ] 5 Evaluate 3[2. +.5] 2 Slide 111 / 305 Slide 112 / 305 Graphing a Greatest Integer Function Graphing a Greatest Integer Function It also called a Step Function because of the shape of its graph. omain: Reals Range: Integers 1) Find the values of x that don't have to be rounded. The inside of [ ] determines that. 2) Make a table. Pick values around the integer values in step 1. Remember our graph will look like steps so once we know the height and width of each step we can repeat the pattern. 3) Graph. ontinue the pattern to complete. (graph is on next page) X Y Slide 113 / 305 Slide 11 / 305 Graph y = [x +1] X Y X Y

20 Slide 115 / 305 Slide 11 / 305 Graph y = [x] Graph y = 2[x -3] X Y X Y Slide 117 / 305 Slide 11 / 305 Graph y = [.5x] Graph y = 2[.5x + 1] X Y X Y Slide 119 / 305 Slide 120 / What is the domain of the graphed function? Set of Integers Set of Reals Set of Odd Integers Set of Even Integers 0 What is the range of the graphed function? Set of Integers Set of Reals Set of Odd Integers Set of Even Integers

21 Slide 121 / 305 Slide 122 / What is the domain of the graphed function? 2 What is the range of the graphed function? Set of Integers Set of Integers Set of Reals Set of Reals Set of Odd Integers Set of Odd Integers Set of Even Integers Set of Even Integers Slide 123 / 305 Slide 12 / 305 Exponential Functions We have looked at linear growth, where the amount of change is constant. Exponential Functions X Y 1 3 Return to Table of ontents If x = 5 what is y? Slide 125 / 305 Slide 12 / 305 Exponential Functions Exponential Functions When the rate of growth increases as time passes, the function is said to be exponential. We will also looking at exponential decay. Think of it as you m&m's and each day you eat half. X Y X Y How many will be left of the 5th day? If x = 5 what is y?

22 Slide 127 / 305 Slide 12 / 305 Exponential Functions From a Graph The exponential function has a curved shape to it. How can we recognize an Exponential function? : {reals} R: {positive reals} Exponential Growth Exponential ecay Slide 129 / 305 Slide 130 / 305 Exponential Functions 3 Which of these are exponential growth graphs? E Exponential Functions Which of these are exponential decay graphs? E F G H I F G H I Slide 131 / 305 Slide 132 / 305 Exponential Functions Exponential Functions The general form of an exponential is where x is the variable and a, b, and c are constants. b is the growth rate. If b > 1 then its exponential growth If 0< b < 1 then its exponential decay 5 onsider the following equation, is it exponential growth or decay? growth decay c is the horizontal asymptote a + c is the y-intercept

23 Slide 133 / 305 Slide 13 / 305 Exponential Functions Exponential Functions onsidering the following equation, what is the equation of the horizontal asymptote? 7 onsidering the following equation, what is the y-intercept? y=2 (0,3) y=3 (0,) y= (0,7) y=5 (0,9) Slide 135 / 305 Slide 13 / 305 Exponential Functions Exponential Functions onsider the following equation is it exponential growth or decay? growth 9 onsidering the following equation, what is the equation of the horizontal asymptote? y=.2 decay y=1 y=3 y= Slide 137 / 305 Slide 13 / 305 Exponential Functions Exponential Functions 70 onsidering the following equation, what is the y-intercept? (0,.2) 71 onsider the following equation is it exponential growth or decay? growth (0,1) decay (0,3) (0,)

24 Slide 139 / 305 Slide / 305 Exponential Functions Exponential Functions 72 onsidering the following equation, what is the equation of the horizontal asymptote? 73 onsidering the following equation, what is the y-intercept? y=0 (0,0) y=1 (0,1) y=3 (0,3) y= (0,) Slide 11 / 305 Slide 12 / 305 Exponential Functions Exponential Functions Sketching the graph of an exponential requires using a, b, and c. Graph 1) Identify horizontal asymptote (y = c) 2) etermine if graph is decay or growth 3) Graph y-intercept (0,a+c) ) Sketch graph Example: Step 1 Step 2 Step 3 Step y = 2 growth (0,5) Slide 13 / 305 Slide 1 / 305 Logarithmic Functions Logarithm functions are the inverses of exponential functions. Logarithmic Functions Exponential Log Return to Table of ontents Logs have the same domain as the exponential had range, that is a you cannot take the log of 0 or a negative.

25 Slide 15 / 305 Slide 1 / 305 Logarithmic Functions Rewrite the following in exponential form. Slide 17 / 305 Slide 1 / 305 Logarithmic Functions 75 Which of the following is the correct logarithmic form of? Slide 19 / 305 Slide 150 / 305 Logarithmic Functions 7 Which of the following is the correct exponential for? Logarithmic Functions 77 Which of the following is the correct exponential form?

26 Slide 151 / 305 Slide 152 / 305 Logarithmic Functions 7 Solve Slide 153 / 305 Slide 15 / 305 Logarithmic Functions 79 Solve Slide 155 / 305 Slide 15 / 305 Properties of Logs Return to Table of ontents

27 Slide 157 / 305 Slide 15 / 305 Properties of Logs Properties of Logs Properties of Logs Examples: Use the Properties of Logs to expand These rules may seem strange but recall that logs are a way of dealing with exponents, so when we multiplied like bases we added the exponents. Just like rule 1. Slide 159 / 305 Slide / 305 Properties of Logs Properties of Logs 2 Which choice is the expanded form of the following 3 Which choice is the expanded form of the following Slide 11 / 305 Slide 12 / 305 Properties of Logs Properties of Logs Which choice is the expanded form of the following 5 Which choice is the expanded form of the following

28 Slide 13 / 305 Slide 1 / 305 Slide 15 / 305 Slide 1 / 305 Slide 17 / 305 Slide 1 / 305

29 Slide 19 / 305 Slide 170 / 305 Slide 171 / 305 Slide 172 / 305 Slide 173 / 305 Slide 17 / 305 Properties of Logs 93 Solve the following equation:

30 Slide 175 / 305 Slide 17 / 305 Slide 177 / 305 Slide 17 / 305 Properties of Logs 9 Solve the following equation: Slide 179 / 305 Slide / 305 Properties of Logs 9 Solve the following equation: ommon Logs Return to Table of ontents

31 Slide 11 / 305 Slide 12 / 305 ommon Logs Solving Exponential Equations Using ommon Logs We can solve the variable as exponent using common logs Slide 13 / 305 Slide 1 / 305 ommon Logs 0 Solve the following equation. Slide 15 / 305 Slide 1 / 305 ommon Logs ommon Logs 1 Solve the following equation. 2 Solve the following equation.

32 Slide 17 / 305 Slide 1 / 305 Slide 19 / 305 Slide 190 / 305 Slide 191 / 305 Slide 192 / 305 e and ln e has a constant value of about e and ln e is the number used when something is continually growing, like a bacteria colony or an oil spill. The Natural Log is the inverse function of a base e function. Return to Table of ontents

33 Slide 193 / 305 Slide 19 / 305 e and ln e and ln Work with e and ln the same way you did and log. For example: Write the following in the equivalent exponential or log form. Slide 195 / 305 Slide 19 / 305 Slide 197 / 305 Slide 19 / 305 e and ln e and ln The amount of money in a savings account,, can be found using the continually compounded interest formula of =Pe rt, where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years. gain,using the compounded continually formula of =Pe rt If $500 is invested at %, how long until the account balance is doubled? If $500 is invested at % for 2 years, what will account balance be?

34 Slide 199 / 305 Slide 200 / 305 e and ln 7 Find the value of x. Slide 201 / 305 Slide 202 / 305 e and ln Find the value of x. Slide 203 / 305 Slide 20 / 305 e and ln 111 Find the value of x.

35 e and ln Slide 205 / The amount of money in a savings account,, can be found using the continually compounded interest formula of =Pe rt, where P is the principal (amount deposited), r is the annual interest rate (in decimal form),and t is time in years. If $00 is invested at % for 3 years,what is the account balance? e and ln Slide 20 / The amount of money in a savings account,, can be found using the continually compounded interest formula of =Pe rt, where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years. If $00 is invested at %, how long until the account balance is doubled? Slide 207 / 305 Growth & ecay Slide 20 / 305 Growth and ecay Formulas to remember: Simple Interest ompounded Interest(annually) ompounded Interest bbreviations I = interest P= principal (deposit) r= interest rate (decimal) t= time n= number of compoundings in one unit of t Return to Table of ontents ompounded ontinually (instantaneously) Slide 209 / 305 Slide 2 / 305 Growth & ecay Example: bacteria constantly grows at a rate of % per hour, if initially there were 0 how long till there were 00? Growth & ecay new car depreciates in value at a rate of % per year. If a 5 year old car is worth $20,000,how much was it originally worth? (Hint: Since the value of the car is going down, the rate is -.0)

36 Slide 211 / 305 Slide 212 / 305 Growth & ecay Growth & ecay The local bank pays % monthly on its savings account, how long would it take for a deposit, left untouched, to double? certain radioactive material has a half-life of 20 years. If 0g were present to start, how much will remain in 7 years? Use half-life of 20 years to find r. Slide 213 / 305 Slide 21 / 305 Growth & ecay 11 If you need your money to double in years, what must the interest rate be if is compounded continually? Growth & ecay 115 If you need your money to double in years, what must the interest rate be if is compounded annually? Growth & ecay Slide 215 / If you need your money to double in years, what must the interest rate be if is compounded quarterly? Growth & ecay Slide 21 / 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?

37 Slide 217 / 305 Slide 21 / 305 Growth & ecay 11 NS 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 & ecay 119 If the half-life of an element is 50 years, at what rate does it decay? Growth & ecay Slide 219 / 305 Growth & ecay Slide 220 / If the half-life of an element is 50 years, how much of the element is left in years? 121 If the half-life of an element is 50 years, how much of the element is left in 15 years? Slide 221 / 305 Slide 222 / 305 Growth & ecay 122 If the half-life of an element is 50 years, how much of the element is lost between years and 15? Logistic Functions Return to Table of ontents

38 Slide 223 / 305 Slide 22 / 305 Logistic Functions Logistic Functions To visualize a logistic function graph think of how a rumor spreads around the school. It starts with a couple of people then a few more and continues to spread till everyone has heard a version of it. The graph would look something like this: # who heard time Slide 225 / 305 Slide 22 / 305 Logistic Functions Ex: new strain of flu shows up at a school of 00 people one day. The determines that people brought the flu in and that the rate of growth is 20% per day. Write an equation. Make a graph. Identify the point where the spread is increasing the fastest? What happens to the rate of increase after the point in the previous question? Slide 227 / 305 Slide 22 / 305 Trig Functions Trigonometric Ratios Trig Functions The fundamental trig ratios are: Sine; called "sin" for short osine; called "cos" for short Tangent; called "tan" for short Return to Table of ontents ngles are usually named θ: "theta" So you'll usually see these as: sinθ; cosθ ; and tanθ

39 Slide 229 / 305 Slide 230 / 305 Trig Functions Trig Functions opposite side Trigonometric Ratios hypotenuse adjacent side θ These ratios depend on which angle you are calling θ; never the right angle. You know that the side opposite the right angle is called the hypotenuse. The leg opposite θ is called the opposite side. The leg that touches θ is called the adjacent side. θ adjacent side Trigonometric Ratios hypotenuse opposite side There are two possible angles that can be called #. Once you choose which angle is #, the names of the sides are defined. You can change later, but then the names of the sides also change. Slide 231 / 305 Slide 232 / 305 Trig Functions Trig Functions θ adjacent side Trigonometric Ratios hypotenuse sinθ = opposite side = hypotenuse opp hyp cosθ = adjacent side adj hypotenuse = hyp tanθ = opposite side adjacent side = opp adj θ.5 opposite side SOH-H-TO.0 Slide 233 / 305 Slide 23 / 305

40 Slide 235 / 305 Slide 23 / 305 Trig Functions θ 1 1 Slide 237 / 305 Slide 23 / 305 Trig Functions Trigonometric Ratios If you have the sides trig ratios let you find the angles. ut if you have a side and an angle, trig ratios also let you find the other sides. Slide 239 / 305 Slide 20 / 305 Trig Functions Trig Functions Trigonometric Ratios Trigonometric Ratios x 7.0 For instance, let's find the length of side x. The side we're looking for is opposite the given angle; x 7.0 sinθ = opposite side = hypotenuse sinθ = opp hyp opp = (hyp) (sinθ) opp hyp 30 o and the given length is the hypotenuse; so we'll use the trig function that relates these three: 30 o x = (7.0)(sin(30 o )) x = (7.0)(0.50) x = 3.5 sinθ = opposite side = hypotenuse opp hyp

41 Slide 21 / 305 Slide 22 / 305 Trig Functions Trig Functions Trigonometric Ratios 9.0 x 25 o Now, let's find the length of side x in this case. The side we're looking for is adjacent the given angle; and the given length is the hypotenuse; so we'll use the trig function that relates these three: cosθ = adjacent side adj hypotenuse = hyp Trigonometric Ratios 9.0 x 25 o cosθ = adjacent side adj hypotenuse = hyp adj cosθ = hyp adj = (hyp)(cosθ) x = (9.0)(cos(25 o )) x = (9.0)(0.91) x =.2 Slide 23 / 305 Slide 2 / 305 Trig Functions Trig Functions Trigonometric Ratios Trigonometric Ratios Now, let's find the length of side x in this case. tanθ = opposite side adjacent side = opp adj o The side we're looking for is adjacent the given angle; and the given length is the opposite the given angle; o tanθ = opp adj opp = (adj)(tanθ) x = (9.0)(tan(50 o )) x so we'll use the trig function that relates these three: opposite side tanθ = adjacent side = opp adj x x = (9.0)(1.2) x =. Slide 25 / 305 Slide 2 / 305

42 Slide 27 / 305 Slide 2 / 305 Slide 29 / 305 Slide 250 / 305 Trig Functions Graphing a Trig Function There are two ways to approach graphing trig functions. 1) is to graph cosine and sine in how they relate to each other will yield a circle, which is not a function. 2) is to graph the value of the function for a given theta. will yield a curve that is a function Graphing is usually done in radians. Slide 251 / 305 Slide 252 / 305 Trig Functions onvert degrees to radians onverting between Radians and egrees onvert radians to degrees

43 Slide 253 / 305 Slide 25 / 305 Slide 255 / 305 Slide 25 / 305 Trig Functions 13 onvert radians to degrees: Slide 257 / 305 Slide 25 / 305 Trig Functions Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat. cos x x

44 Slide 259 / 305 Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat. sin x Slide 20 / 305 Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat. tan x x x Slide 21 / 305 Slide 22 / 305 Slide 23 / 305 Slide 2 / 305 Trig Functions Which choice(s) below satisfy the following condition: Increasing E F G H I J none of the above

45 Slide 25 / 305 Slide 2 / 305 Trig Functions 11 Which choice(s) below satisfy the following condition: oncave Up Trig Functions 12 Which choice(s) below satisfy the following condition: has a relative max F G H F G H I I E J none of the above E J none of the above Slide 27 / 305 Slide 2 / 305 Trig Functions 13 Which choice(s) below satisfy the following condition: ecreasing E F G H I J none of the above Power Functions Return to Table of ontents Slide 29 / 305 Slide 270 / 305 Power Functions power function is in the form of: Positive Integer Powers Where n is a real number epending on the value of n we can get a wide variety of graphs. Return to Table of ontents

46 Slide 271 / 305 Slide 272 / 305 Power Functions Power Functions Let's first consider n to be a positive integer. n is odd n is even The end behaviors of positive integer powers of a power function follow the rules from the last unit. Slide 273 / 305 Slide 27 / 305 Power Functions Power Functions 1 Identify the function(s) that satisfy the following: 15 Identify the function(s) that satisfy the following: E F G H I J E F G H I J Slide 275 / 305 Slide 27 / 305 Power Functions Power Functions 1 Identify the function(s) that satisfy the following: 17 Identify the function(s) that satisfy the following: always decreasing E F G H I J E F G H I J

47 Slide 277 / 305 Slide 27 / 305 Power Functions Negative Integer Powers Now let's consider n to be a negative integer. Recall what a negative exponent means: Return to Table of ontents Slide 279 / 305 Slide 20 / 305 Power Functions Power Functions f(x)=x 2 g(x)= 1 / x 2 n is odd n is even ompare and contrast: end behaviors and as the function goes to zero What is f(x)g(x)=? Slide 21 / 305 Slide 22 / 305 Power Functions Power Functions 1 Identify the function(s) that satisfy the following: E F G H I J

48 Slide 23 / 305 Slide 2 / 305 Power Functions Power Functions 19 Identify the function(s) that satisfy the following: 150 Identify the function(s) that satisfy the following: F G H F G H E I J E I J Slide 25 / 305 Slide 2 / 305 Power Functions 151 Identify the function(s) that satisfy the following: an odd function E F G H I J Rational Powers Return to Table of ontents Slide 27 / 305 Slide 2 / 305 Rational Powers Rational Powers is defined as the n th root of x

49 Slide 29 / 305 Slide 290 / 305 Rational Powers Rational Powers 152 Select the choice that satisfies the following:# the greatest value at x = 153 Select the choice that satisfies the following? the greatest value at x = Slide 291 / 305 Slide 292 / 305 Rational Powers 15 Select the choice that satisfies the following? the least value at x = Rational Powers 155 Select the choice that satisfies the following? the least value at x = Slide 293 / 305 Slide 29 / 305 Rational Functions In a previous unit, the end behaviors of rational functions was discussed. Rational Functions 3 cases m>n: m<n: Return to Table of ontents m=n:

50 Slide 295 / 305 Slide 29 / 305 Rational Functions Since a rational function was defined as: Rational Functions Removable There are 2 kinds of disconitunity: Essential there may be values of x that make g(x)=0, which would make h(x) undefined These values of x are discontinuities of h(x). Slide 297 / 305 Slide 29 / 305 Rational Functions Removable iscontinuity when x = a, x - a =0 so (x - a) is a factor of both f(x) and g(x). factor (x-a) out of both f(x) and g(x) re-evaluate h(a) if h(a)= # / 0, then x=a is a vertical asymptote if h(a)=c, then (a,c) is the location of the hole Slide 299 / 305 Slide 300 / 305 Rational Functions Examples: graph the following Rational Functions Examples: graph the following

51 Slide 301 / 305 Slide 302 / 305 Rational Functions Examples: graph the following Slide 303 / 305 Slide 30 / 305 Rational Functions 157 There is a hole at x = c for the equation Find c. Slide 305 / 305