Category: Math 11

Sine Law

This week in Precalc 11 we learned a new trigonometry formula known as the sine law. Sine law is used to find the angles and the side length of triangles that don’t contain a ninety-degree angle. To use this formula you need 3 pieces of information, 2 of which must correspond to one another. For example, if you have angle B you need the side length b.

while the Cosine Law is used when the Sine Law cannot be used. The cosine law can calculate a missing side length or angle. The formula for finding a missing side length is and vice-versa

 

Here is a video to help: https://www.youtube.com/watch?v=VjmFKle7xIw

Adding and subtracting rational expressions with monomial numerators 

First, we should mention what a rational expression is that has a binomial denominator. A rational expression is that has a binomial denominator is an expression that has a binomial, in the place of the denominator. The binomial can be any numbers in addition or subtraction of any variable of any power.

This week in math, we learned how to multiply and divide monomials. The first step is always to factor if possible, we always need to take a look at what we’re doing; either multiplication or dividing. From previous math, we know that if it’s dividing then we need to reciprocate the fraction. The next step is to simplify all like terms, you always need to find what X can’t equal to, because the denominator cannot equal 0.

In the examples below, i forgot my non-permissible values, for the first one it’s b cannot equal 0, and same for the second one, except B and A cannot=0

Here is a video to help : https://www.youtube.com/watch?v=f1ajvLpb32E

 

Multiplying and dividing rational expressions

This week in Pre Cal, we learned how to multiply and divide rational expressions. Rational expressions are equations or quotients that have polynomials on most likely both the bottom and top, each with their own variables. It would be very difficult if not impossible to divide a polynomial by another polynomial without using a calculator. This is why we have to take steps in finding and making binomials that are easy to cancel out on both the bottom and top.

Disclaimer: When canceling out polynomials or numbers, one must be on the top of the division and the other on the bottom. It does not matter if the polynomial or number is apart of the same quotient, as long as one is somewhere on the top, and the other somewhere on the bottom. Also, if you must divide by another fraction/quotient, flip the fraction around to make it a multiplication. Ex. 1/2 ÷ 2/3 –>  1/2 x 3/2

First, we must find the “non-permissible” values, which are essential ‘x’. Contrary to the last unit, instead of ‘x’ equalling only specific numbers, in this chapter, ‘x’ cannot equal specific number. Therefore, ‘x’ would be 0 making, for example, the equation 3/0 which is impossible. Next, you must find alike polynomials, mostly binomials, by factor them. We try and make it so that a polynomial on the top and the bottom both have the same polynomial so that they can cancel each other out, making the equation easier to solve. Watch out for negatives (-) or any small details because when you cancel out polynomials, they both have to be the exact same. Then after canceling out all that is possible, you should be left with regular, real numbers. If so, you are allowed to cancel those out too. Once finished those steps, you should be left with an easy multiplication that will give you your answer.

Graphing reciprocals of linear functions

During Week 13 of Pre Cal 11, we learned about how to graph reciprocals of both linear functions and quadratic functions. Today, I will be explaining how to graph and calculate linear reciprocals. When starting, we should try and find the original linear function (y=fx). Once we find the original, we must graph it so we can keep it in relation to the reciprocal that we will soon graph. Once we find the original function (y=fx), we must turn it into a reciprocal function. See Image 1 to find out how. Then, find the x-intercepts for y=a and y=-1. These will be your invariant points, they help you graph your reciprocal function. After this, you will need to find your asymptotes, both horizontal and vertical. See Image 2 for more info. Then depending if x is negative or not, it is now time to graph the reciprocal. See image 3 for info.

Solving systems of equations using substitution

This week in Pre Calc 11, I learned how to solve systems of equations using substitution. A method of how to solve a linear system is to use the substitution method. You use the substitution method by substituting one y-value in an equation with the other. While using the substitution method, you first substitute y in the second equation with the first equation since y = y. After substituting y into the equation and solving for x, the value of x can then be used to find y by substituting the number you found, with x. While using the substitution method you can also start by substituting x in the second equation with the first equation.

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Here is a video to help you:

This week we learned how to graph linear inequalities in two variables. A linear equation is a graph that splits into two sections.  A linear inequality looks very similar to a linear equation, the difference between the two is that a linear equation has a “equals” symbol and a linear inequality is divided into 4 groups: greater than, greater than and/or equal to, less than, less than and/or equal to. When writing or understanding a graph of a linear inequality, she will shade in or color in the side where the linear inequality is true. A solid line represents _____ and/or equal to. A checkered line represents ____ than. To find the region that will satisfy the inequality we choose a point (0,0 would be easiest) on its appropriate side and plug it into the linear inequality. Then solve the linear inequality. If the inequality sign is true to the numbers then shade in the region, if not shade in the opposite region on the graph.

Below is an example of how to graphs such equations:

Since the midterm was coming up, or rather has already passed, I decided to focus on what I had somewhat forgotten, which is what we first learned; geometric and arithmetic sequences. Since we learned the until months ago, I had forgotten some key information when having to solve for a series or sequence. I mostly forgot the naming and letters in the formulae. Being that Physics 11 has so many abbreviations, I got caught up in all of the ones I had already known. It was also good to review this chapter because when we don’t practice something constantly, we will eventually forget it.

Here is a link to help understand: http://discretetext.oscarlevin.com/dmoi/sec_seq-arithgeom.html

I studied thoroughly on infinite series and the rules and restrictions that come with it.  The sum of an infinite arithmetic sequence is a negative infinite number if d is less than 1, and a positive infinite if d is bigger than 1. There are two ways to find the sum of a finite arithmetic sequence. To use the first method, you must know the value of the first term (a₁), the rate at which it multiplies (r) and the value of the last term (a_n).  Then all you have to do is punch in the numbers into the equation. S_∞ = a₁ / (1-r ).

See this link to learn more : http://home.windstream.net/okrebs/page133.html

Finding Vertex in Factored form

I learned this week in Pre Calc 11 is how to find the vertex of a quadratic equation that is in factored form. The first step is to find the x-intercepts or “zeros” of the equation. Once you’ve determined the zeros/ x-intercepts, you find the average of the zeros by adding them and dividing them by two. Once you find that, you plug it in as X in the equation in order to find Y. The x and y coordinates are the coordinates to your vertex.

This can help you if above is not detailed enough

http://www.gilbertmath.com/uploads/1/4/2/7/14279231/3c_u2_d6.pdf

Parabolic Graphing

The standard type of a quadratic equation is y=ax² + bx + c , where a, b, and c are real numbers, and that a can never equal 0. Each term in the condition has its own significance. The vertex of the parabola is its highest or lowest point. A determines the size of the parabola, b determines the x-int, and c determines the y-int. The vertex may be a minimum point or a maximum. If a is less than 0, than the vertex would be a maximum and the parabola would open downwards. If a is bigger than 0, than the vertex would be the minimum and the parabola would open upwards. The axis of symmetry is the line that runs vertically through the vertex, showing the starting point on a line when the parabola has moved away from the original graph lines. The domain would be all possibilities than run along the x-axis, the range would be all possibilities that run along the y-axis. The Parabola can be of different sizes and different directions(up or down), depending on the a value. 4x² would produce a more compressed parabola in the upwards direction, while 1/4x² would give a wider parabola.

Here is some help. https://www.youtube.com/watch?v=7QMoNY6FzvM

Discriminant

A discriminant is the part of the quadratic formula in the square root, which is (b² – 4ac). The complete quadratic formula is  The discrimant tells us how many solutions there are, aswell as what x means and its properties. It helps us to determine a shortened answer without going through mulyiple calculations, which only works in some situations. A positive discriminant means that the quadratic has two real number solutions. A discriminant of zero means that the quadratic has a one real number solution. A negative discriminant means that there are no solutions.

To start, lets start with x²+5x+6=0. In this case, a = 1, b = 5 and c = 6. When you plug them in the equation for finding the discriminant, it becomes 5² – 4(1)(6).

Then it becomes, 25 – 24

Then the asnwer is, 1, meaning there would be 2 answers.

For more info check out : https://www.purplemath.com/modules/quadform2.htm