This week in precalc 11 we learned about Sine Law and Cosine Law.

The formulas are

Sine Law: to use the sine law you need to know two angles and one side of the triangle  or two sides and an angle opposite one of them. It’s used to find a side or an angle of a triangle. It has two versions of the formula, you use it depending on what it is that you’re looking for, a missing angle or a missing side.

or 

Cosine Law: you can use cosine law when the lengths of two sides and the measure of the included angle is known or the lengths of the three sides are known.  It’s used when you need to find a third side of a triangle, when the angle opposite to the side is given. If we also want to find the length of the third side, we can just change the formula to solve for the variable.

Here’s an example:

This week in precalc 11 we learned more about rational expressions and solving. We learned adding and subtracting rational expressions. We also learned how to solve rational equations.

When solving rational expressions, there are three methods you can use
1) Converting to a common denominator: you just multiply both sides of the equation by the common denominator, with this you can eliminate the denominators and turn the rational equation into a polynomial equation.

2) Multiplying through by the common denominator: we convert everything to the common denominator and then compare numerators

3) Cross multiplying: this works only if the equation has exactly one fraction on one side of the equal sign set exactly one fraction on the other side of the equals

When adding/subtracting rational expressions:
-find non-permissible values
-factor (if needed)
-cross out things that are the same or equivalent

-find common denominator
-multiply the fraction to get the common denominator
-add or subtract the numerators

This week in precalc we learned how to add/subtract/multiply/divide rational expressions.

When working with rational expressions, we need to cancel out whats equivalent. To do so we need to figure out our non-permissible values. *Non-permissible values: values that cause the fraction to have a denominator with a value of zero. (In math, you can’t divide by zero)* Then we simplify the numerators and denominators, and finally when everything is simplified then we cancel out identical pairs of numerators and denominators.

A website that helped me understand this topic http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U11_L1_T2_text_final.html

This week in precalc 11 we learned about graphing reciprocal linear  and quadratic functions. When graphing reciprocal linear functions or quadratic functions, we need to understand and try to visualize what most functions will look like. Then we just follow the steps.

Reciprocal Linear:

-graph parent function ( using the slope and y-intercept)

-the x-intercept of the linear graph will become the vertical asymptote of the reciprocal

-find and mark the invariant points (every place where the original graph has a y-value of -1 or 1)

-graph the shape of a hyperbola based on the reciprocals of the y-values of the original function

Reciprocal Quadratic: 

-graph the quadratic function

-x intercepts become the vertical asymptotes

-locate and mark the invariant points. y= 1 or -1

-use a few other points to sketch the graph of the reciprocal

This week in precalc 11 we learned about absolute value functions.
An absolute value function is a function that has an algebraic expression within absolute value symbols in it.
*(the absolute value of a number is its distance from zero on a number line)*
The absolute value parent function is written as f (x) = | x | which is also defined as

f (x) = {x if x >0
0 if x = 0
-x if x < 0

We also learned how to solve equations with absolute values in it. There are some simple steps do to so. First, you isolate the absolute value expression, then you put the quantity inside the absolute notation equal to + and – the quantity on the other side of the equation
You then solve for the variable in both equations, and always remember to always check the answers to make sure you got the right answer and not an extraneous solution.
Here’s an example:
|2x-1| = 5
We write the equations with the different signs + and – and solve them.
|2x-1| = 5 and |2x-1| = – 5

1.) |2x-1| = 5
2x – 1 = 5
2x = 6
x = 3

2.) |2x-1| = – 5
2x – 1 = – 5
2x = – 4
x = – 2

The answers are 3 and – 2. To make sure we got the right answer, we check by substituting these answers into the original equation.

This week in precalc 11 we learned about graphing quadratic inequalities which is in the form of y > a + bx + c.

(the sign could be <,>,  ≤ , ≥)

We also learned how to graph linear inequalities in two variables.

First we need to rearrange the equation into the y = mx + b form. Then we use the y intercept and the slope to graph the points, we then connect these points into a straight line. depending whether the symbol is <> we use a broken line and if its ≥ or ≤ we use a solid line.
Finally we just color the side with the solutions, to do this we just pick a test point and use it on the original equation (usually 0,0 is a good test point), if the statement is true we shade in the side of the graph with the points if not we shade the other side.

This week in precalc 11 we had our chapter 4 unit test and our midterm. Some things I wish I studied a little more for the midterm was unit 2 and 3 where I know I was a little confused.

Then on Friday we had our first lesson on unit 5, where we were introduced to solving and graphing quadratic inequalities with one variable. With inequalities, we just solve the problem like a quadratic equation. First we factor, then we need to find the zeros of the equation, and then finally we need to test the numbers to make sure the statement is right.

Also one thing we were told to remember was that when we divide a negative inequality, you have to switch the sign.

This week in precalc 11 we learned about equivalent forms of a quadratic function. This just means finding different ways or representing a quadratic function to find more information about it. So we basically have

1.) Standard (Vertex) Form: y = a + q 

We can find the vertex

2.) General Form: y = a + bx + c

You can find the Y-intercepts

3.) Factored Form: y = a ( x – x1 ) (x – y2 )

You can find the X-intercepts

We also learned how parent functions transform . Here we have the vertex form y = a + q

The a in the equation will tell us if the parabola opens down or up, and if there’s a stretch or a compression. the p will tell us if there’s a horizontal translation and the q will tell us if there’s a vertical translation.

This week in precalc we learned how to analyze quadratic functions. We also learned how to use desmos to graph and see how parent functions transform. A quadratic function is written y = a + bx + c .

*Note the difference between equations and functions.*

Equations: you can solve and find roots.

Functions: you can graph, and find Y-intercepts and X-intercepts.

The parabola is the curve where any point is at an equal distance from. The vertex is the highest/lowest point, it may be a minimum or maximum point. The line of symmetry is the line that divides the parabola into mirror images.

When analyzing quadratic functions in a graph, we always need to look for the y-intercept, the x-intercept, the domain and range, parabola, vertex, line of symmetry, if it’s opening down or up, and if it’s congruent to the parent function.

How do we know how parent functions transform:

here’s the video and the link to the website that helped me understand how parent functions transform:

https://youtu.be/dJLQ8UOIuZ4  https://www.purplemath.com/modules/fcntrans.htm

On week 7 we learned a little about the discriminant, and the formula used is – 4ac . It helps determining how many solutions and real roots an equation has. The discriminant is pretty much the number you get under the square root in the quadratic formula.

If the answer to the equation is more than 0 there are 2 solutions, two distinctive real roots.

If the answer to the equation is equal to 0, then there is 1 solution (2 equal solutions), one real root.

If the answer to the equation is less than 0, there are no solutions, no real roots.

1.) 2 solutions

2.) 1 solution or 2 equal solutions

3.) no solutions