Tangent

Tangent is a very interesting operation. It is the brother of cosine and sine, and involves both of them in its composition. Sine over cosine equals tangent, or opposite over adjacent. As can be assumed from the previous blog post, tangent is found in right triangles. Even more interesting, the tangent graph has asymptotes. Its counterpart is cotangent, or 1/tan on a calculator or in a calculation. However, tangent is also a term used to mean touching at one point.

Sine and Cosine

Sine and cosine are two of the 6 important operations that one can perform on triangles. However, the only type of triangles one can use these two on is right triangles. Right triangles have one angle of 90 degrees, and using sine and cosine, we can find missing sides or angles. Sine and cosine on a calculator are represented by sin and cos. Sine is equal to the opposite side over the hypotenuse, whereas the cosine is equal to the adjacent side over the hypotenuse. Additionally, on a unit circle and also cartesian plane, cosine is equal to the x value, whereas the sine value is equal to the y value. We can use these to substituted x and y values and find specific lengths.

Chapter 3 Summary

Chapter 3 was all about functions. This involved rational functions, complex numbers, and factoring. We used many methods to solve and factor, like the foiling method, quadratic formula. Also, we found zeroes, which are the solutions of a function. An example of a quadratic function is f(x) = 4x^2 + 2x - 6.
The zeroes of this function can be found using the quadratic formula, or [-b +/- sqr(b^2 - 4ac)]/2a. When you plug in the values for the function, you get two solutions because of the +/- sign. This is where 4 = a, 2 = b, and -6 = c. Overall this was a very interesting chapter, although it is review.

Rational Functions

Hello,

Rational functions, the amazing nature of them. First, we learned about Rational functions in a test. This is called the Rational Zero Test, and is represented in p/s = factors of constant/factors of leading coefficient. The constant is the number at the end of function, and the leading coefficient is the coefficient of the biggest degree of the variable. When one of these works in f(x), you can use synthetic division to find the equation and eventually factors. Next, we used the midpoint equation to estimate a point to the utmost accuracy. Lastly, we understood rational functions and finding certain points and parts to a function. We found holes, asymptotes, x and y intercepts.

Thanks,
Kaili Chiu

Zeroes of Functions

Hello,

The zeroes of a function are known by many names. Some of these names include the x-intercept, x value. However, they all boil down to one thing. Solving for x. In a function f(x), if f(x) = x - 1, the zero would be 1, as when one solves for x, by adding a 1 to both sides, one can isolate the x. Zeroes are a key unit to solving any math equation, and are often asked for in many math problems inside and outside the classroom. Finding the zeroes is incorporated in algebra, calculus, math analysis, and geometry, and trigonometry. Pretty much every aspect of math.

Thanks,

Kaili Chiu

Piecewise Functions

Hi,

Today in Mathland, I learned about Piecewise functions. The funny thing about these is that they are multiple functions put together in the Cartesian Plane. We found the domain of the functions, and used it to plot each of them. Later, we discovered whether they were continuous or discontinuous. This can be found just by looking at the visuals of the plotted functions. If it is discontinuous, we would state where and at which points it was discontinuous. In addition, we also learned how to plot and identify [(x)] functions.

Thanks,
Kaili Chiu

Superhero Functions

Hi,

In the assignment for the f(x) men, we used specific functions to complete missions. This project allowed us to learn about transformations in functions, and how they are represented on the Cartesian plane. We used specific functions as superheroes to defeat the evildoers by transforming the function so that the function would hit certain points where the targets were. In certain graphs, we had to combine heroes and eliminate in the least amount of moves possible, and in others, we could only move along a certain axis. Some of these functions were 2^x, linear functions, parabolas, cubic functions, absolute value functions. The hardest mission was "The Outer Demon - Magic Foxes", where we had to finish the mission using negative functions. Overall, this was an interesting assignment that allowed us to understand transformations of functions.

Thanks,

Kaili Chiu