Week 18 – Top 5 things I have learned in Precalc 11

I’ve learned so many new things in precalculus 11. Here are my top 5 things that I have learned this semester:

Series and sequences:

In an arithmetic sequence, the difference between the terms is constant. The value is called the common difference. The general equation for $t_{n}$ is $t_{n}=$ $t_{1}+ (n-1)d$
An arithmetic series is the sum of the terms in an arithmetic sequence. The formula to find the sum of n terms is $S_n=\frac{n}{2}(t_{1}+t_{n})$ or $S_n=\frac{n}{2}(2t_{1}+(n-1)d)$
A geometric sequence is formed by multiplying each term after the 1st term by a constant, to determine the next term. The constant is called the common ratio, r. 4,12, 36, 108…, is an infinite geometric sequence because it continues forever. 4,12, 36, 108 is a finite geometric sequence because the sequence is limited to a fixed number of terms. The general term is $t_{n}=ar^{n-1}$
A geometric series is the sum of the terms of a geometric sequence. The formula to find the sum of a geometric series is $S_n=\frac{a(r^n-1)}{r-1}$ . The formula for the sum of an infinite geometric series is $s \infty=\frac{a}{1-r}$

You can find the sum of a series and also find a term number which are super helpful in real world problems!

Factoring

This is the order in which you need to look for when factoring.

Ex. 2x² – 2x – 40

A common factor in this is 2. Take the 2 out. 2(x² – x – 20).

Find two numbers whose product is -20 and whose sum is -1: 4 and -5

So, 2(x-5)(x+4)

Ex. x² – 81

This is a difference of squares. This factors to (x-9)(x+9)

CDPEU has made factoring way more easier.

Graphing quadratic functions

A quadratic function is any function that can be written in the form y = ax²+bx+c. This is called general form.
- In this form, we know if the parabola opens up or down
- y-intercept
Another form is standard form, y = a(x – p)²+ q
- Convert general form to standard form by completing the square
- In this form, we know the vertex
- axis of symmetry
Factored form,
- We know the x-intercepts
- axis of symmetry

The discriminant

The expression, b² – 4ac, is called the discriminant of the quadratic equation
It tells us how many solutions an equation has
The quadratic equation ax² + bx + c = 0 has:
- two real roots when b² – 4ac > 0
- one real root when b² – 4ac = 0
- no real roots when b² – 4ac < 0

Trigonometry

Two new things that I learned in the trigonometry unit is the sine and cosine law.

Sine law
- The sine law formula is $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ or $\frac{Sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$
- It can be used for any triangle to determine the length of a side or to find an angle.
Cosine law
- When two sides and an angle are known, the sine law cannot be used to determine the measure of the other sides and angles, so we use the cosine law.
- The formula is a² = b² + c² – 2bccos A