Exponential Functions

Hello everyone so today's blog is about the first unit we learned

in Unit 4 (Exponents & Logarithms)

Before we start in any examples you would need to know that
y=a^-x - will reflect the basic curve in the y axis

y=-a^x -will reflect the basic curve in the x axis

y=a^x-h+k - you would read h as opposite and k as is

f(x)=4^x

The Horzontal equation is at 0

You can make a table of values if you prefer but it is optional

f(x)=4^x-2 + 1

For next equation you should start out with first half which is 4^x

and plot in the cooridinates. The next step is adjusting cooridinates

coming from the h and k values. Remember that you need to read

the h value as opposite and k as is.

Alright that it's for my blog sorry for the horrible writing and bye.

Sum and Difference Identities Part Two

Hi everyone! This is Steffi. Sorry its late! Here are a few examples of what we did in class.





Sum and Difference Identities Part One :

Hey everyone this is Rajvir, I am going to blog about what we learned on March 15, 2012.

We learned that there are total 6 Identities; luckily we don’t have to remember them, because they are on our formula sheet.

Basic sum Identities:

Basic difference identities:

We also learned:

Note: Undefined values will still occur when the denominator equals to zero.

Example: 1-sin^2(a + B) = cos^2 (a + B)

Basically, we are using these identities to solve different equations.

Example:

Find the exact value of the following using the appropriate formulas

Sin7π                         

     12

Step 1: first we have to write this expression using a grouping of special triangles that we learnt about in the very beginning π/3, π/6, and π/4. (VERY IMPORTANT)

Sin7π     = (   4π   + 3π  )                       

     12            12      12  

    

Step 2: Simplify

                     a     B

       = (   π   + π   )                       

       3      4       

Note: Don’t forget to label which is Alpha and Beta.

Step 3: Plug it in the Sin formula, because we are finding exact value for Sin 7π

                              12   

        =   Sin πCosπ   + Cos πCosπ                        

            3     4            4     3

Step 4: Using your special triangles find the exact value of each. (SOH CAH TOA)

              = (  √3  )  (  1   )  + (  1  )   (  1  )

                      2        √3           2         √2

 =    √3    +     1   

                     2√2      2√2

              =   √3 + 1

                      2√2   

This is how you solve for exact value of sin. As well as solving for Cos and Tan involves the same steps. But the difference is that you’re using a different formula.

In addition, if you are still confused about the whole concept I suggest that you watch this video on Sum and Difference identities.

 http://www.youtube.com/watch?v=v4JrGnWgJLg&feature=related

Hope this was helpful, and that you learned a lot J

Trigonometric Identities by Kulvant

The Absolute Value Function and Creating Equations for Sinusoidal Functions

Hey Guys It's Darian, I'm a little late, but better late then never.
Yesterday We learned about The Absolute Value Function and Other Absolute Value Graphs

This topic was all about absolute values and shifting rules and how they affect a graph.

So say we have a graph that is f (x)= |x| it will look like this

While a graph that is f (x)=x it will look like this

What im trying to get across is that when your dealing with absolute values any negative ooint will be positive Unless your dealing with a question that has a negative sign infront of the x like this

f (x)= -|x| which will turn out like this

We also learned about shifting functions, When a graph is shifted we read horizontal shifts (h values) as opposite of what is given and vertical shifts (k values) as is.


So say we have a question that is asking us to graph f (x) = |x - 3| + 4
You would graph this by first graphing a regular f (x) = |x| graph which looks like the one above, then you would shift it 3 points to the right and 4 points up which would result in the graph looking like this.

We also learned Creating Equations for Sinusoidal Function
f (x) =asinb (x-c)+d    Or      f(x) = acosb (x-c) + d

a - amplitude
                                                
b - affects the period - period= 2π/b

c - horizontal shift

d - vertical shift


Steps:
1. Identify the middle axis. This determines if there is any up or down shifting - the d value.
2.Find the amplitude - the a value.
3.Determine the period and then calculate the b value.
4.Identify the type of original wave - either    y=sin x or y= cos x.
5.Create the first equation including the a,b,c,and d values.
6.Create the second and third equations including the a,b,c,and d values.
7.Double check the horizontal shift symbols.
8.Double check the symbol of the a value in the equation.
Here are some Sinussoidal Functions

Now I know that this was kinda dull so to spice it up.....Heres a picture of a kitty

Hope you enjoyed :)

Graphing Reciprocal Functionss 1 and 2

Hey guys eejay here, im sorry for not updating the blog.

in last week's class, we learned how to Graph Reciprocal Functions I.

Basic Shape: f(x) = 1/x Shifted Shape: f(x) = (1/ x-h ) + k *read h as opposite, k as is*

There will be an asymptote at the value for x that makes the function undefined - this is called a vertical asymptote. In basic form the VA is always at x = 0

There will also be an asymptote at the value for y that is no longer able to occur due to the unacceptable value for the x (VA) - this is called a horizontal asymptote. Basic form the HA is always at y = 0

If the graph is shifted, we have to read the horizontal shifts ( h values ) as opposite of what is given and vertical shifts ( k values ) as is.

If the graph is shifted, the VA will always be at x = h and the HA will always be at y = k

*ASYMPTOTES HAS TO BE DASHED LINES

Next thing we learned was Graphing Reciprocal Functions II: Reciprocal Trig. Functions.

f(x) = csc x = 1/sin x → there will be undefined values when sin x = 0, assuming the period is 2π, the vertical asymptotes will occur at x = 0, π, 2π... etc

• To graph the basic sin x shape, place the asymptotes at the x - intercepts which will become undefined once reciprocated, and then flip the remaining curves OR create a table of values.

f(x) = sec x = 1/cos x → there will be undefined values when cos x = 0, assuming the period is 2π, vertical asymptotes will occur at x = π/2, 3π/2... etc

• To graph the basic cos x shape, place the asymptotes at the x - intercepts which will become undefined once reciprocated, and the flip the remaining curves OR create a tables of values.

f(x) = cot x = 1/tan x = cos x/ sin x → there will be undefined values when tan x = 0, assuming the period is π, vertical asymptotes will occur at x = 0, π, 2π...etc

• To graph cot x we can create a table of values using quadrantals and π/4's. The asymptotes of cot x will occur at different x - values if the b -values (period) changes.

Symmetry, Reflections and Inverses

Hi! it's Demerie :) and Jihoo :P



















Today, we learned about Symmetry, Reflections and Inverses!

 When x is replaced with its reciprocal (-x) in the equation of a function y= f(x), it's graph is expressed in the y-axis.

For example,

f(x)= x³

f(-x) = (-x³)

 

 The reflection on the y-axis --> make the x-values negative!

When y is replaced with -y in the equation of a function y= f(x), its graph is reflected in the x-axis.

For example:

f (x)= x²

-f (x) = (x²) --> f(x)= -(x²)

 Reflection on the x-axis --> make y-values negative!

 When x is interchanged with y in the equation of a function y= f(x), it;s reflected in the mirror line y=x. This is called an inverse function.

The process of finding the Inverse Function:

1) Replace f (x) with y.

2) Switch x and y.

3) Solve for y.

4) Replace  with f -¹ (x)

Graph the function f (x)= 2x+2 and its inverse. Determine algebraically the equation of the f -¹ (x).

Reflection in the mirror line --> switch x and y values!

1)  y= 2x+2

2) 
Remember: y= mx+ b


m= rise/ run 
b= y-intercept


(1,4) -> (4,1)  

(0,2) -> ( 2,0)

(1,0) -> (0,1)

(-2,-2) -> (-2,-2) 

 3)  
f(x)= 2x+2 
y= 2x+2
x= 2y +2
x-2/2= 2y/2
y= x-2/2
y= 1/2 x-1
f -¹ (x) = 1/2 x-1
 

Transformations	Effects on Graph
-f (x)	Reflection in x-axis
f (-x)	Reflection in y-axis
f-¹ (x)	Reflection in y=x

 Symmetry
A graph is said to be symmetrical through an axis or the origin if either side is the mirror image of the other.

A function f(x) is even if for any value "x" f(-x) or -f(-x)=-f(x). Even functions are symmetric about the y-axis. This means that positive and negative x-values result in the same y-value. Even functions would be symmetrical between quadrants 1 and 2 or quadrants 3 and 4. (example is a vertical parabola)

A function f(x) is odd if a any value "x" f(-x) = -f(x) or f(x)=-f(-x). Odd functions are symmetric about the origin. This means that positive and negative x-values result in different y-values. Odd functions would be symmetrical between quadrants 1 and 3 or quadrants 2 and 4 (example is a vertical cube) 

Determining if a shape is a function

When is a relation a function?

A relation is a function if each x values has 1 y values.

 

There are two tests to check:

VLT --> Vertical line test 

HLT--> Horizontal line test

Use the VLT test first if it passes and then use HLT.

VLT- if a vertical line test crosses through the graph only once then the graph is a function, IF more than once it's not a function

HLT- if a horizontal line crosses through the graph only once and it has already passed the VLT then the graph is a one-to-one function.

 

Not a function:

It touched the line twice while performing a vertical line test

Don't forget to do our homework!

Mr. P gave us an Inverse Function sheet and Exercise 9, Questions 1-20, Omit 5a iii & iv, 5b, 10, 16, and 17