There are a number of new things that I learned this year in Pre-calculus 11 that really stuck with me and made way for those “ah ha” moments. This is a list of the top 5 things that I learned in Pre-calculus 11.
Acronyms: One of the things that i learned that really helped me in Pre-calculus 11 were a number of different acronyms. The acronyms allowed for me to easily remember what steps i needed to take when dealing with specific questions. They made things a lot easier for me. Two of the acronyms that I learned that really helped me are CDPEU, and CAST.
- The CAST Rule: Starting in quadrant 4 and moving in a counter clockwise manner, will tell you which trigonometric ratios will be positive in the certain quadrants. This makes it easier for you if you are trying to find if your value will be positive or negative, since it depends on what quadrant it’s in.
Ex. Find what quadrants the terminal arm of the angles could lie.
–> Quadrants: 2 and 4
To find what possible quadrants a terminal arm could lie you use the CAST law. If we know that the tangent angle is negative then we know that it will have to lie in either quadrant 2 or 4. In quadrant one, all angles are positive, and in quadrant three, tangent is positive.
- CDPEU (Can Divers Pee Easily Underwater): This short acronym helps you remember the steps that you must follow when dealing with factoring. First you must look for anything that may be Common.
ex.
+8x [both have a common factor of 2x]
- 2x(x+4)
Then you check if it is a Difference of Squares. These only occur in binomials, and only work if the terms are separated by a negative.
ex.
- (x+3)(x-3)
Next you would check if there is a Pattern. The pattern that is being referred to is the pattern of 
ex.
+5x+4 [because this follows the pattern this is quadratic and needs to be factored]
- (x+4)(x+1)
The next letter in the acronym stands for Easy. This is when the expression has a factor of one in front of the
ex.
- (x+4)(x-5)
The last letter in the acronym stands for Ugly. Ugly, refers to the expressions where the
Finding the Discriminant: The discriminant is taken from the quadratic formula and really tells you a lot about an expression. It allows you to find how many “zeros”/root/solutions an expression or equation has as well as if it is extraneous of or not. Finding the discriminant made questions a whole lot easier because I could just find the discriminant as one way of making sure that my answer(s) is correct. If the value of the discriminant is:
Positive: then that means that you will have to solutions/Negative: then that means that there are no solutions/Equal to zero: then that means that there is one solution
ex.
- Since the equation is already written in the proper format of (
), we need to write down the values for a, b, and c. (a = 2, b = 9, c = -4)
- Next, you input the values into the formula.
- Use BEDMAS to simplify the equation until you are left with a value.
- Since the number you are left with are 113, and it’s positive, then we know that this quadratic equation will have two possible solutions.
Finding the Vertex: Being able to find the vertex of a quadratic function can tell you a lot about it. From the axis of symmetry to even the range, the vertex gives you a lot of information on the function. The vertex can be found by a number of different way. One method that really stuck with me however was completing the square.
Reciprocal Functions: Reciprocal functions are one of the top 5 things that I learned because it was something that was really new to me that I learned how to deal with. At first I found that its concept was really hard for me to grasp, but after a bit of practice they became easier for me to understand. I think that they are something worth talking about and are super interesting.
Ex.
- First, we graph the linear function as we normally would if it wasn’t reciprocated. To do this we would write down everything we know about the original function and plot it on the graph. Since the original function is 2x+6, we know that the function will be positive and have a slope of 2, we also know that the y-intercept will be 6.
- Next, you circle your invariant points which will be when y is at both 1 (_,1) and at -1 (_,-1). To do this go to where 1 is on the y-axis and then slide your finger vertically along the graph until you touch the linear function. You would do the same thing for -1.
- Next you draw in the asymptotes. Since the horizontal asymptote will always be y=0, we draw a horizontal broken line along that part of the graph. Also, since the vertical line is half way in between the two invariant points, we know it will be when x= -3. Another way to find what the vertical asymptote is without looking at a graph is by making the original linear function equal to zero and solving for x.
- After drawing in the asymptotes, you can now draw the hyperbolas. To do this you would take of the points along the linear function, reciprocate it and plot its reciprocal. You would then draw the hyperbola through the invariant and the new reciprocated points.
Note: when writing the domain or range x and y can be any real number except they can’t be equal to the corresponding asymptote.
Special Triangles: I think that the special triangle were very helpful. There are two special triangles , the 30°-60°-90° and the isosceles right triangle (45°-45°-90°), and since we know the sides of these triangles, we know their trigonometric ratios. If you come across any of these angles you will be able to refer to these two special triangles for your ratios instead of trying to calculate. Note: If one of the sides was multiplied by 2, you would need to do the same to the rest of them.
