General form – Ax + By + C = 0
Vertex form – Y = a (x-p)^2+ q
Example: y=3x^2+18x+20
Step 1: Factor out the leading coefficient. 
Steps 2 & 3: Complete the square, at the same time as completing the square multiply the number that is not part of the equation by the coefficient 
step 4: simplify 
This week in Precalc 11 we started the intro to quadratic functions. The graph of a quadratic function is a parabola, which is a U-shaped curve. The direction and openness of the parabola depend on the sign of the coefficients. If a > 0, the parabola opens upwards, and if 0>a the parabola opens downwards.
Vocabulary :
parent function:
Vertex: the turning point of the parabola (also known as the most important point on a parabola)
Maximum of quadratic function: the highest point on a graph when a parabola opens down
Minimum of quadratic functions: the lowest point on a graph when a parabola opens up
The axis of symmetry: the equation of a line that divides the figure into two equal parts where one is the mirror image of the other.
The standard form: 
Here is an example: f(x)= (x-2)^2+3 
The blue parabola is the parent function, and the green parabola is the example. In the green parabola we can see that the minimum value is y=3 but in the blue parabola it is 0. this is because in the equation we have a vertical translation of +3 this made the parabola move up 3 units. Another difference is that the green parabolas axis of symmetry is at x=2 rather than 0. Again this is due to the horizontal translation. Even though it says -2 in the brackets we are moving to the right when a negative sign is seen. If a positive sign is seen in the brackets we move to the left.
This week in Precalc 11, in addition to reviewing our past units, we learned about the Quadratic formula. The quadratic formula allows us to solve any quadratic equation that’s in the form ax^2 + bx + c = 0. This is useful because you can use this formula anytime you don’t want to do grouping, inspection, etc.
In an equation, this is where the ABCs are:
To solve an equation you simply plug in the numbers and solve:
In the end, if you are left with a radical that can be simplified, make sure to simplify. Then you work out the equations till you get your answers.
In this example I have simplified the radical to a mixed radical, the index, the denominator and the constant can all be simplified so we simplify. 
This week in Precalculus, we learned how to factor equations with radicals. It’s an important skill since radicals pop up in different units, and knowing how to factor them helps in solving a variety of problems.
Example 1: 
step 1: isolate the radical
step 2: get rid of the radicand by squaring everything. This removes the radical sign and squares the other side. Be careful because if we have a binomial on the other side we need to write it out twice and foil.
step 3: move everything to one side so we have 0 on one side. If your squared variable is positive on one side it’s better to move everything there because negative squared variables don’t look nice.
step 4: factor by prefered method (mine is grouping)
step 5: find out which number makes the equation equal to zero. we might have 1 answer or 2 answers.
This week in Precalc 11 we covered the basics of factoring and we also got to touch on solving quadratic equations using factoring.
What makes something a quadratic equation:
Example 1:
step 1: Factor – you want to first look for any common variables or numbers, if there aren’t any you can factor by your prefered method (boxing, grouping, inspection, etc) the numbers that are on the other side of the zero. (during factoring quadratic quations we do not touch the zero it just stays there, because essentially the whole point is to get the none zero side to equal zero.)
step 2 : apply the zero product law – applying the zero-product property helps us find the values of that make each factor equal to zero, thus giving us the solutions to the quadratic equation. For example -8 x 8 = 0
A great way to check to see if you have done your work correctly, is to just plug back in the number to the original equation.
Example 2 : As we can see in this example, we didn’t have all the number on one side and zero on another, but thats okay because all we need to do is to move them to one side. The reason as to why the answers are fractions is because when applying the zero product law, we are taking the equation and isolating x.
Example 3: in this example we are dealing with fractions. After moving the constant to the left hand side, we multiply everything by 6 because 6 is the least common multiple. then we factor, group and apply the zero product law.
This week in Precalc 11 I learned how to factor polynomials by grouping them. This is a good skill to have as polynomials can look scary sometimes and factoring them isn’t easy.
Example: x^{2}+2x+6x+12
Step 1: Start by drawing a square and dividing it into 4 equal boxes inside. Then, place numbers in each box. It’s crucial to arrange these numbers in a specific order so that when you examine them diagonally, the products of the diagonals are equal. Additionally, ensure that similar variables are positioned diagonally opposite each other. For example, if you have 2x in one box, place 6x in the box diagonally opposite to it.
Step 2: Take out the greatest common factor (GCF) or common variable from each row. This process will result in having two numbers at the top of the box and two numbers at the left of the box. If there’s no apparent GCF or common variable in a row, simply write 1, as 1 is divisible by everything. This step helps simplify the expressions within each row, making further calculations easier.
Step 3: Next, let’s group the numbers together. The two expressions at the top of the box should be grouped together, as well as the two expressions on the left. Ultimately, you should have two binomials formed, where the product of these binomials equals the polynomial we began with.
(x+2)(x+3)
This week in precalculus 11 I was introduced to diving radicals, with this comes rationalizing the denominator. Rationalizing the denominator means eliminating irrational radicals from the denominator of a fraction, thereby converting it into an expression with only rational numbers (fractions or integers) in the denominator. This makes diving radicals way easier. To start we need to know what a conjugate is. Conjugate is changing the sign of a variable and conjugates are used to rationalize the denominator.
Example: if we have 234+47 the conjugate of this number is 234-47 (just changing the sign)
To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator.
Another example
we know that the conjugate of sqrt(2) – sqrt(3) is sqrt(2) + sqrt(3), so we multiplied the fraction by that. then we simplified the expression more by 2-3 and then we realized we have a negative denominator which is not good so we multiply everything by -1 so the negative denominator cancels out and we have a nicer number.
Always remember that you need to rationalize the denominator if the denominator is an irrational number
This week in pre-calc 11 I learned how to add and subtract radicals. I think adding and subtracting radicals is an important skill to have because it makes working with radicals easy at the end of the equation. To start, we need to know that radicals are treated the same as variables in this lesson, now if we have 2x + 3y + 3x we get 5x + 3y, this is because only the x’s can be added together since they are the same. to start easily, since we know how to add variables, we can get into adding radicals.
Like variables, radicals that have the SAME radicand go together. here is an example.
Now as you can see we only added the coefficients together, you do not touch the radicand, that is just there for you to know which numbers go together.
The same case goes for subtracting radicals. Numbers with the same radicand go together. Here is an example.
However, if we have something like √18 – √8 the roots are the same (both are square roots), but the radicands are different. We’ll start by simplifying each radical. √18 can be simplified to √(9 * 2), and since 9 is a perfect square, it simplifies to 3. So, √18 becomes 3√2. Similarly, √8 simplifies to √(4 * 2), which is 2√2. Now, we have 3√2 – 2√2. Since they have the same root and radicand, we can simply subtract the coefficients: (3 – 2)√2 = √2. So, √18 – √8 simplifies to √2.
This week in Pre calc 11 I learned how exponents are radicals in disguise. I was familiar with exponents and exponent laws from past math years, but when exploring radicals this year, I learned how exponents and radicals have so much in common. This is an important skill because it was difficult to imagine radicals and work with them in my head but now that I have exponents to help me, it has become way easier to work with radicals. to understand this topic, it is important to remember the exponent laws: 
Now that we have recalled the exponent laws we can get into radicals. If we have an example like 4^1/2, we can recognize that the fraction is really 0.5.
what will happen if we put the calculation in the calculator
now isnt 2 the square root of 4? so with this we can conclude that 2 which is the denominator is the root, and 1 which is the numerator is the power of that number
With the knowledge that we have right now with exponents and radicals, we can apply exponent laws ro radical problems and solve them without the need of a calculator. Remember that the even though we are working with radicals the SAME exponent laws apply to the problem.
This project aims to dive into the potential of commonly overlooked materials, typically seen as waste and reimagine them as art pieces by recycling them. This genre of art has been explored before by artists such as Bill Thomson and Pablo Picasso who invented the idea of collage, pasting scraps of newspaper, photographs, and other items together to make a picture. I’m driven by the desire to challenge myself to dive deep into my imagination and finding a use for items that will go to waste. By embracing this creative piece, I’m aiming to expand my artistic skills and personal growth in illustrating art through different materials.
To begin the project, I’m planning on finding items in my house (or even the classroom) that are of no use anymore, and working with them to create a piece of art. Some challenges that I might face during this journey, might be the lack of materials or not finding good use of them. To resolve this issue, I’m planning on asking my peers and teachers for feedback and help. For example, I could ask my friends to envision what a pistachio shell could become beyond just being a shell or trash.
