Radical and Rational Functions

Since I haven' t done it yet and I wanted to try.....

Radical and Rational Functions

Radical Function is a function that includes a radical with a variable in the radicand

Example:  √3x

Graphing Radical Functions

For this lesson it is important to remember the concepts that we learned during the transformations unit 2.

In the graph above transformations were done only on the y values and the x values stay the same.

It is also important to remember that the number inside the radicand in the cases above cannot be < 0.

Example:

y=2√x Vertical translation by a factor of 2

y= -2√x Vertical translation by a factor of 2 and reflection on the x axis

Solving the Equation of Radical and Linear Functions

G(x) = x-5

We can solve this by using the table of values. Since there is no radicand there is no restriction on the domain and range of the function.

F(x) = (√x-3)

There is a restriction on the domain since the value inside the radicand cannot be less than 0.

 x≥3

Another way to solve this is algebraically.

x-5 = (√x-3)

1.      Square both sides to get rid of the radicand symbol in f(x).

(x-5)² = x-3

2.      Square the equation (x-5)².

x²-10x+25 = x-3

3.      Solve for x.

x²-11x+28 = 0

(x-7)(x-4) = 0

(x-7) =0  or   (x-4) = 0

x= 7    or x= 4

Check to see if the solutions are correct.

x-5=(√x-3)                                 x=4

x=7                                       4-5=(√4-3)

7-5=(√7-3)                               -1=√1

2=√4                                        -1≠1

2=2

Thank you for a great class and semester, Mr.P!

Converting logarithmic functions to  

exponential logarithmic functions

In class, we learned how to convert logarithmic functions into 

exponential logarithmic functions using Mr. P's "7 Rule"method.

For example,

Evaluate:

Solution:

 By using Mr. P's seven rule method, we know that:
-3 is the new base
-"x" becomes an exponent and 81 is the answer

Final Answer:

Trigonometric Identities

Trigonometric Identities
 



Trigonometric identities is a form of proof that all permissible values is true of the variable for which both sides of the equation are defined.

Reciprocal Identities



  In calculation we may replace either member of the identity with the other.

Tangent and Cotangent Identities Pythagorean Identities   Verifying the identity or proving that the given equation is an identity, it is show that the left hand side of the equation is equal to the right hand side.  Example:  Sum and Difference Formulas    Evaluate sin 15°. Solution. sin 15°

 

Determining Equations for Sinusoidal Functions

In order to determine the equation of a sinusoidal function, you need to find:

a - amplitude
b - affects the period
Period -    2π
                 /b/
c - horizontal shift
d - vertical shift

f(x)=asinb(x-c) + d           OR             f(x)=acosb(x+c) + d

The equation could either be for a sine or cosine function

Example 
State a sine equation for the function graphed below.

Steps

1. First, we need to identify the Middle Axis. This determines if there is any up or down shifting - the d value. 

                  Middle Axis = Maximum + Minimum
                                                           2
                  Middle Axis = 3 + (-1)  = 1
                                                2
                  Middle Axis is at  y  =  1

We can see that there is a vertical translation 1 unit up therefore,  d = 1

     2. Next, find the amplitude or the a value. The amplitude is the distance between the middle axis and the maximum value. In this graph, the amplitude is 2.

     3.  Next find b. But in order to find b, we must determine the period of the function. To find the period, measure the distance along the x-axis between two maximum points or two minimum points.

Once you got the period, you can calculate for b.

           period = 2π                         b =  2π          b = 2π        b = 1

                                                           period             2π



4. Lastly, let's find the phase shift (horizontal shift) or c. This is the trickiest value to find since there are many choices possible for c. But the question is asking for a sine equation. so we know that at
 x = 0, the y-intercept is on the middle axis.

Let's recap the information we've determined:
a = 2
b = 1
c = π/2
d = 1

 Since the question is asking for an equation for a sine function, we will use the form    
 f(x)=asinb(x-c) + d


Just plug the a, b, c and d values into the equation.

The final answer will be f(x) = 2sin(x-π/2) + 1


Thank you for reading my post. I hope this helped you a bit and I wish you all the best in our final exam. Good luck and have a nice day!

Graphing Sine function

Sketch the graph y= 3sinΘ for 0 < x < 2π

In order to graph sine or cosine function, you need to determine:

·        -  Amplitude (a)

·         - Period (b)

General Form:
     y= asinb(Θ-c)+d

Determine the Amplitude of a sine Function
- Amplitude is the distance from the center of the axis to the highest or lowest point for a sinΘ or cosΘ function.

     Amplitude = Max- min
                                 2

     Amplitude= IaI

So

 Determine the period of sine function

- To find the period, you first need to determine b

Then plug it in the formula of a period which is    2π
                                                                                IbI

- the period of the sin function is 2π, which means the function repeats itself every 2π

You are now ready to graph a sine function!    

1.      Plot the period (make sure to have 5 coordinates)

2.      The amplitude will be the maximum and minimum y-coordinates

3.      If the equation is positive, after plotting zero the next point will be plot on the positive side. If the equation is negative, it will be on the negative side.

4.      then, the next point will be on the x-axis since the amplitude on the positive and negative side should equidistant, the graph should look like a wave

 

 

y= 3sinΘ

 

Solving Trigonometric Equations Part 1

        Hi Pre Calc. class! Last time you guys learned about the unit circle, and all of it's properties, special angles, etc. Well now you are ready to continue to the next step, which is solving Trigonometric Equations with a specified interval! How exciting right? I'll walk you through the the steps though, so don't be worried.

THINGS TO REMEMBER:
- Special triangles
-Quadrantal Angles
SOH CAH TOA
-CAST rule
- ΘR angles

Example 1
To solve for  Θ:
1) See if the interval is set in either radians or degrees. That will determine how you will write your final answer. 

2)Simplify the equation by isolating the unknown, which is the theta, and factor.

3)Find  ΘR 

and find the quadrants that the answer is in using the CAST rule.

4)Find values for  Θ in degrees or radians. 

Since the interval is in degrees, I will keep it my answer in degrees. 

You see how there are two answers? That's because there are two quadrants that my answer for  Θ could be in! 

* If there is a particular restriction, such as " Where sin  Θ> 0" in  a question such as " tan Θ= -1/√ 3 " tan would be possible in TWO quadrants, Q. 2 and 4. BUT because of restriction, sin cannot be in Q. 4, only Q. 2 therefore there will only be ONE answer.*

Here is the full question for you! 

So I hope that this gave you a little more understanding about solving trigonometric equations with a specific interval. Try to review your note as well, as they are very useful and important! 

Enjoy my fellow Pre calc mates!

Trigonometric Ratios

Hello class. I am going to be teaching you all about trigonometric ratios.. Just kidding Mr.P already taught us. I'm just going to be recapping what we may or may not already know. Bad joke? Yeah I know, anyways lets get started.

The first thing you'll need to remember are these: 

The 6 Trigonometric Ratios

Also SOH CAH TOA will come very handy in this unit. 

So for example, lets say the given points are A(-3/5, -4/5), we need to find 6 trigonometric ratios for theta(θ)

So first off we're going to need a diagram, 

Then you'll need to look at the given points again which is (-3/5,-4/5) and figure out which quadrant point A would go when using the CAST rule. You also need to remember that cos is x and sin is y. 

So since cos and sin are both negative, the point would go in quadrant 3. 

Cos is adjacent/hypotenuse which makes -3 the adjacent and sin is opposite/hypotenuse which makes -4 is opposite and the hypotenuse is 5

If you understand the CAST rule then you are able to plug the points in sin, cos and tan plus their reciprocals. 
So here are the following answers:

This is just one of many examples that are almost exactly the same as this. So now that you know this, the world is in your hands... haha just kidding but in all seriousness, you will be able to figure out any trigonometric ratio question that comes your way.