RADICAL FUNCTIONS

First thing to know about radical functions is that it involves a radical with a variable in the radicand.


One major rule to keep in mind is that the x-value in the radicand can be any number, as long as it is not negative. Taking the square, cube, fourth, or any root of a negative number is impossible.

Now let's look at a basic radical function.

y = √x

An important thing to do that people tend to forget is to solve for the inequality. Take the value from inside the radical and equate it to zero.

x = 0

Now simply replace the symbol with the inequality symbol.

x ≥ 0

From that, we know that x has to be greater or equal to zero.

The simplest way to represent this radical function on a graph is to use a table of values. There are other methods, such as comparing graphs and solving the equation, but table of values seem to be the easiest method. To make it even easier for yourself, you should pick values for x that you can take the square root out of.

Examples: Values of 0, 1, 4, 9, and 16




 
Irrational numbers such as 3, 5, 6, 7 and so on can also be chosen as the x-value, but the result will turn out with decimals, making it harder for yourself to graph.










Now comes the graphing. Simply plot these points down.





 



It will turn out looking like this!





 


Looking at the graph will help you figure out the domain, range, x-intercept, and y-intercept. However, from solving the inequality earlier, you have already figured out the domain.

Inequality: x ≥ 0
Domain: { x | x ≥ 0, XER }

Range: { y | y ≥ 0, YER}

X-intercept: 0
(Point that crosses x-axis)

Y-intercept: 0
(Point that crosses y-axis)

Natural Base (e) and Natural Logarithm (ln)

Hi guys! Here’s a short review on Natural Logarithms.

• Natural Base

The natural base ‘e’ is another strange number frequently used by  mathematicians which approximately equals to 2.71828

2.718281828x = ex

• Natural Logarithm

Natural Logarithms comes from the fancy Latin name Logarithmus naturali , giving the abbreviation ln. Natural Logarithms is really just the inverse of ‘e’.

e^x = y  —>  lne y = x

or

e^y = x   —>  ln x = y

Understanding Natural Logarithms

Reminder:  All Natural Logarithms use the same patterns and laws as Common Logarithms.

Expanding Natural Logarithmic Expressions

Roots—> Fraction Exponents Division—> Subtraction

Multiplication —> Addition Exponents —> Move to the front

Expand.

Follow expansion rules.

Do not forget to apply negative through.

Simplifying Natural Logarithmic Expressions

Rearrange terms (all negative terms  must be on the end) Fraction Exponents—> Roots

Addition —> Multiplication Move to the front —> Exponents Subtraction —> Division

Simplify.

Rearrange.

Do not forget to FACTOR OUT the negative.

Follow the simplification rules.

Solving Natural Logarithmic Equations

Simplify Convert Logarithm into exponential form (use 7 rule) Solve for x or any missing variable Check answers by plugging answer(s) back into the equation.

Place one logarithm with base e on each side of the eq’n. Using the Laws of Logarithms turn the exponent into a coefficient to the front of its logarithm. If the exponent contains a binomial, use brackets around the exponent. If the exponent is in brackets, multiply through brackets. Collect logarithms on one side and isolate the variable on the other. If there is more than one term containing the missing variable isolate the variable. Solve for the variable with your calculator Check your answer.

Evaluate. Round answer to three decimal place

Follow simplification rules. Rearrange.

Simplify further.

Convert to exponential form using 7 rule.

Solve for x.

Solving Using Cancellation Property

Evaluate

Take the left and right hand side of the equation and raise it as a power of e.

Cancel out same bases.

Raise left and right hand side as a power of e one last time.

Cancel out same bases.

Exponential and Logarithmic Functions Recap

Hello. Here’s a quick re-cap of what we have learned about exponential and logarithm functions before winter break.

Graphing Exponential Functions

Key points to remember:

•   Basic curve: y=c^x
• The same transformation rules apply:

 y=(a)c^b(x-h)+k 

o   a – vertical stretch by a factor of lal about the x axis.

-          ex: y= (5)c^x (Vertical stretch by a factor of 5 about the x axis.)

o   b – horizontal stretch by a factor of 1/b about the y axis.

-          ex. y=c^2x (Horizontal stretch by a factor of ½ about the y axis.)

o   h – horizontal translation

-          ex. y=c^(x+2) (Horizontal translation by 2 units to the left)

o   k – vertical translation

-          ex. y=c + 3 (Vertical translation by 3 units up)

 

Steps:

1.  Determine the horizontal asymptote. (The horizontal asymptote is related to the vertical translation, k.)
2.  Table of Values; pick x values.
3.   Plug in the x values into the equation in order to find the corresponding y values.
4.  Graph.

y = 2^z

HA @ y=0

Table of Values:

         

Graph:

Graphing Logarithmic Functions

Key points to remember:

•   A logarithm function is the inverse of an exponential function.

Steps

1.  Convert logarithmic function to exponential function using the 7 Rule.
2.   Determine the vertical asymptote.
3. Table of Values; pick y values.
4. Plug in the y values into the equation in order to find the corresponding x values.
5.  Graph.

f(x) = log₂x -> 2^y = x

VA @ x=0

Table of Values

          

Graph:

Expanding and Simplifying Logarithmic Expressions

Key points to remember when expanding:

•   Roots -> fraction exponent
•  Division -> subtraction
•  Multiplication -> addition
•  Exponent -> coefficient

Ex.

 

Key points to remember when simplifying:

•  Move all negative terms to the end
• Coefficient -> exponent
•  Addition -> multiplication
•  Subtraction -> division
•  Fraction exponent -> roots

 

                                                       

 

 

Proving Trigonometric Identities

Hi friends! Yesterday we finished up the lesson on Proving Trig Identities. I'm going to give quick a run through on how to prove identities. ♪~ ᕕ(ᐛ)ᕗ

To prove an identity:

• Simplify each expression on either side of the equal sign in your equation given, so that both sides are equal. This means that the expression on the left hand side is equal to the expression on the right hand side. Not to be mistaken for verifying an identity, where you substitute values that make the statement true.

 Let's start simple, given the example 

sin²x + cos²x  = 1
↑
left hand side

Looking at the left hand side, we know that sin²x + cos²x is also equal to 1. This can then be simplified to 1, since they both represent the same thing. The equation then turns into:

1 = 1

When you make sure both sides of the equation are simplified, you make sure that they are both equal to eachother. In this case both sides need to equal 1, which is correctly stated above.

This equation also happens to be the identity sin²x + cos²x  = 1. When you see an expression that can be simplified, using these known identities make working with the equation easier.

Examples of Identities

Other things to take note of to make proving identities easier like the one above is to
Tip 1: Multiply the fraction by the conjugate (opposite signs) of an expression.
Tip 2: Factor
Tip 3: Rewrite the expression in terms of sine and cosine.

Tip 2

Tip1

Tip 3

Graphing Sine Functions

Hi guys! In class we are currently learning how to graph in our circular functions unit. I will be talking about how to graph sine functions. When graphing on a Cartesian plane it is known as “unrolling” the Unit Circle.

The general form for sine is:

y = a sin b(x – c) + d

In order to graph a sine or cosine function you must determine the period and amplitude.

PERIOD is the length of one cycle that can either be in degrees or radians. The period for sinx or cosx functions can be found with the formula  2π / |b| 


AMPLITUDE  is the distance from the middle axis to the highest or lowest point for a sinx or cosx function. Amplitude can be found by taking the absolute value of "a"  

Amplitude = |a|    or    Amplitude = max-min / 2

Now that you have a better understanding of what period and amplitude is lets try an example: 

Sketch the graph of y = 2sin2x for 0 ≤ x ≤ π

1. First find the period by using the formula 2π / |b|

     

       This means the length for one full cycle of y = 2sin2x is π

2. Now lets find the amplitude by taking the absolute value of "a"

3. You are now ready to graph the sine function! On a graph plot the period on the x-axis and the amplitude on the y-axis. 

  We split the period into 4 sections because of the 4 quadrants in the unit circle (remember we are "unrolling" the Unit Circle).

4. Since we are graphing a sine function plot your points starting from (0, 0). When graphing a cosine function you will start from (0, a). The given equation is positive therefore the next point will be in the positive side at (π/4, 2) The graph should look like a wave.

* Make sure to check for restrictions* 

In this example the restrictions are 0 ≤ x ≤ π since our graph fits the restrictions it is complete! 

I hope my example did not confuse you but instead helped you better understand how to graph a sine 

function. Thank you for reading!