Warm Up MM2A5. 1. Graph f(x) = x 3 and make a table for the function. 2. Graph g(x) = (x) and make a table for the function

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1 MM2A5 Warm Up 1. Graph f(x) = x 3 and make a table for the function 2. Graph g(x) = (x) and make a table for the function 3. What do you notice between the functions in number 1? 4. What do you notice between the functions in number 2? inverse functions.tns Aug 25 2:37 PM 1

2 MM2A5 Warm Up Continued 1. Graph f(x) = x 2 and make a table for the function 2. Graph g(x) = (x) and make a table for the function 3. What do you notice between the functions in number 1? 4. What do you notice between the functions in number 2? inverse functions.tns Aug 25 2:37 PM 2

3 MM2A5 Inverse Function Two relations are inverse relations if and only if one relation contains the element (b, a) whenever the other relation contains the element (a, b). If f(x) denotes a function, then f 1 (x) denotes the inverse of f(x) What does this mean? Aug 20 2:56 PM 3

4 MM2A5 Inverse Function Characteristics of Inverse function and relations. If (x, y) is a coordinate in the graph of f(x), then (y, x) is a coordinate in f 1 (x) If the function f(x) has an inverse function, then f(x) must be one to one Each x is mapped to exactly one y and each y is mapped to exactly one x The function passes the horizontal line test. If f(x) and g(x) are inverses of each other, then f(g(x)) = x and g(f(x)) = x Steps to solve an inverse function Step 1: Change f(x)= to y= Step 2: switch x and y Step 3: Use reverse order of operations to solve for y Step 4: Change y= to g(x)= Aug 25 2:37 PM 4

5 MM2A5 f(x) = x 2 x y Oct 25 7:40 PM 5

6 MM2A5 y 1 g(x) = x x Oct 25 7:40 PM 6

7 MM2A5 Domain: Range: intercepts: zeros: relative extrema: absolute extrema: intervals of increase: intervals of decrease: end behavior Lets Graph y = 1 / x Is there any other characteristic that can be used to describe this functions? What would its inverse be? Sep 8 11:13 AM 7

8 MM2A5 Domain: Lets Graph y = 1 (x 3) Range: intercepts: zeros: relative extrema: absolute extrema: intervals of increase: intervals of decrease: end behavior Is there any other characteristic that can be used to describe this functions? What would its inverse be? Sep 9 9:02 AM 8

9 MM2A2 Lets Graph (Use a table to help) Domain: Range: intercepts: zeros: relative extrema: absolute extrema: intervals of increase: intervals of decrease: end behavior Sep 1 2:45 PM 9

10 MM2A2 Exponential Equations Exponential equations are of the form y = a x, where a is a constant and greater than zero, and x is an exponent What else can I say about x or the domain of the function? What can I say about the range of the function? Sep 1 2:37 PM 10

11 Exponential Rules When exponentials have the same base, pull down the exponents If you are multiplying the same bases, then you add the exponents. If you are dividing the same bases, then you subtract the exponents. If you are raising a power to another power, then you multiply the two powers. When you have a negative exponent, flip the fraction to make it positive. When any power is raised to zero, it equals 1. Jan 25 10:32 AM 11

12 MM2A2 Lets not Graph, Lets Think y = 2 x y = 2 x +4 Domain: Range: intercepts: zeros: relative extrema: absolute extrema: intervals of increase: intervals of decrease: end behavior Domain: Range: intercepts: zeros: relative extrema: absolute extrema: intervals of increase: intervals of decrease: end behavior Sep 1 2:58 PM 12

13 MM2A2 Interest Anyone? Oct 28 9:28 AM 13

14 MM2A2 What is e? Like π, e is a constant that is an irrational number. Where does e come from? Remember: A = P(1 + r) t This is the equation for calculating the amount A of money when investing principal P with simple interest. T is time in years and r is the interest rate This is the equation for calculating the amount A of money when investing principal P with compound interest. N is the number of times interest is compound each year, t is time in years, r is the interest rate. Lets focus on Let r =1 and t =1 What will that equal as n approaches infinity? This is the equation for calculating the amount A of money when investing principal P with continuously compounding interest. R is interest rate and t is time in years Sep 1 3:42 PM 14

15 MM2A2 Examples If you invest $1000 for 3 years at 4% interest, what will that amount be if interest is compounded Yearly Semiannually Monthly Daily Continuously Sep 2 2:57 PM 15

16 MM2A2 How long? If you invest $10 for 17 years at 6% interest, what will that amount be if interest is compounded bimonthly? New 3.49% Used 3.99% Mortgage 6.5% Sep 1 3:18 PM 16

17 MM2A2 Exponential Functions From the Folding Paper Activity and the M&M Activity, which mathematical model would represent an: exponential growth: exponential growth and decay exponential decay: How much interest would you earn, if you invested $567 at 4.15% interest for 7 years compounded: monthly? continuously? Oct 30 4:21 PM 17

18 MM2A2 Continuous Growth or Decay A = Ne kt Sep 1 3:05 PM 18

19 Piecewise Functions A piecewise function is defined by at least two equations, each of which applies to a different part of the function's domain. Point on the graph of a function in which there is a break, hole, or gap are called points of discontinuity. A step function is a piecewise function that is defined by a constant value over each part of its domain. Its graph resembles a set of stairs. Extrema are the maximums and minimums of a function. Extrema can be relative (within a given part of the domain) or absolute (within the entire domain). Feb 21 8:11 AM 19

20 MM2A1 1. Graph y = x Piecewise Functions 2. Is the graph curved? explain 3. If you draw a vertical line through the vertex, what type of function would make up each half? Is it linear, quadratic, etc. 4. Create a function for each half of the graph 5. What x values will you use for the left half and for the right half? Nov 3 3:12 PM 20

21 Creating Piecewise from Absolute Value 1. Use Reverse Order of Operations to solve to the absolute value bars. Example: x+4 = 4 2. Once you get to the absolute value bars. Write the problem twice, but instead of the bars put parentheses. 3. Make one equation positive and one negative. 4. Solve for x. Feb 15 9:16 AM 21

MM2A2 Postal Rates Pounds Rate 0< x 5 $5.00 5< x 10 $10.00 10< x 15 $15.

22 MM2A2 Postal Rates Pounds Rate 0< x 5 $5.00 5< x 10 $ < x 15 $ < x 20 $20.00 What would this look like on a graph? Nov 6 7:36 AM 22

23 MM2A1 Make into Piecewise Functions 1. y = x y = 2x 2 3. y = x+2 3 Nov 4 2:05 PM 23

24 MM2A1 Graph Nov 3 3:25 PM 24

25 MM2A1 f(x) = { 2x 5<x<1 x+2 1 x<3 Warm Up Mar 23 7:49 AM 25

26 MM2A1 Warm Up. Work on until I return from GHSGT Graph 1. x+4 = 3 2. x 3 = 8 3. x 5 3 = 1 Mar 24 7:55 AM 26

27 Attachments inverse functions.tns rational functions.tns Piecewise functions.tns

You can graph the equation, then have the calculator find the solutions/roots/zeros/x intercepts.

To find zeros, if you have a quadratic, x 2, then you can use the quadratic formula. You can graph the equation, then have the calculator find the solutions/roots/zeros/x intercepts. Apr 22 10:39 AM Graphing