Week 9- Analyzing general form of quadratic equation

In week 9 of pre-cal 11 We learned how to analyze the general form of quadratic equation along with many things.

With analyzing, I mean you factor the equation if it’s factorable. And apply the knowledge I learned from last unit to find the two or 1 roots, and using logical thinking such as the axis of symmetry must have the same distance on the x axis to both roots and etc to find more information without changing the equation into standard form.

Week 8 – Graphing Quadratic Equations

In week 8 of pre-cal 11, we learned how to graph quadratic equations with the general form and the standard form.

ax^2+bx+c, is the general form of the equation, with this form the equation we can find limited information comes to graphing; we can determine the direction and the compression ratio of the graph from a, and y-intercept from c.

But we can easily change this equation to standard form by completing the square; y=a(x-p)^2+q  from this form of the equation we can determine the position of the vertex with q being the y-axis(+ going up,- going down), p being the x-axis(- going right, +going left) and the direction of opening and the compression ratio of the graph by looking at a,( the ratio of the graph will be congruent to ax^2=y, a determines the direction of the graph( if negative a, facing down, if postive a opens up).

 

Week7- Discriminants

During week 7 of pre-cal 11, we learned how to find the discriminant of a general form of a quadratic equation.

In this quadratic formula:

x = \frac {-b +/- \sqrt {b^2 - 4ac}}{2a}

The discriminant is the part under the root sign:

b^2 - 4ac

Finding the solution to the discriminant of a quadratic equation will provide you will the type of roots you will get from the aforementioned quadratic equation.

To find the discriminant of a quadratic equation you need to learn the meaning of a,b,c in this case:

ax^2+bx+c, is the general form of the equation, and when you are trying to find the discriminant just put b^2-4ac in, such as:

4x^2+2x+6, the discriminant will be 2^2-4(4×6)=-92

And the range the discriminant is in it will provide you with the number of roots you will get from this quadratic equation; If your answer is a positive number, there are 2 roots, If your answer is a negative number, there are no real roots, If your answer is a 0, there is 1 root.

 

Week 6-solving quadratic equation with quadratic formula

This week we learned a universal and fail-safe method of solving a quadratic equation, that is the quadratic formula: 

You might be wondering what do all the letters represent, well in any quadratic equations you will be able to rearrange it into the form of ax2+bx+c then you will be able to solve this equation with the quadratic formula. This is the easiest and the safest method of solving a quadratic equation in my personal opinion.

 

Week 5- factoring

This week we reviewed some of the basics of factoring from math 10 pre-cal, and we learned a phrase which will help us in using the optimized steps to determine the method of factoring that can be used for the expression. There are three big types of expression that we can factor which we are exposed to, the easy ones, the harder ones and the difference of Squares.

This week we reviewed the factoring of the binomials and the trinomials. In the case of the trinomials, a factorable expression must be written in the form of x2, x, n, there are two types of trinomial expression the easy ones: which the number before the term x2 is one, and the hard ones: the number before the term x2 is not 1. The methods of factoring these expressions use the first and the last term of the original expression as they are the result of only two number multiplying, and there are different method from there depends on the situations.

And there is the difference of squares which is arguably the easiest one, it is written in a binomial format with 2 square terms: x2- some square number. The method of factoring difference of square is extremely easy, you can find a pair of the conjugate with the second term of the conjugate equals the square root of the second term of the original expression and leave the first term as x.

Week 4- Radical

This week we learned the calculations that involve radicals such as add, subtract. In the case of add and subtract we need to convert the subjects that we are adding into mixed radicals with the same radican. EX: \sqrt{12}+\sqrt{27},2\sqrt{3}+3\sqrt{3},5\sqrt{3} when the radicals have the same radican you will have to add the exponent of the mixed radical and keep the radican the same. For multiplication and division you can just apply the rules of multiplication and division on entire radicals with one exception, that is you can’t write denominator as a radical.

Week 3- Absolute Value

In this week’s classes, we learned the concept of absolute value…

Absolute value expresses the distance between numbers on a number line, due to the fact that there can’t be a negative distance as it is a scalar value, everything coming out of the “||” are positive. and the expression within absolute value get prioritization in an equation, such as

3×2+|2-3|, 6+1=7.

Sum of Infinite Geometric Series

In this week’s Math 11 Pre-Cal I learned that you can actually determine the exact Sum of Infinite Geometric Series in certain situations, when they are converges.( when -1<r<1) as with a rate less than 1 and greater than -1 the next term will get closer and closer to zero, therefore there is a determinable  sum. In the case of a diverging series, the  sum will get infinity big and therefor we can’t determine the exact sum.

The equation for the sum of an regular Geometric Series is

Sn=\frac{a(1-r^n)}{1-r}

When -1<r<1 r^n approaches 0 as n increases indefinitely.

So, Sn approaches Sn= \frac{a(1-0)}{1-r}, therefor S\infty= \frac{a}{1-r}

EX:

A Infinite Geometric Series where r=0.5

8,4,2,1

S\infty= \frac{8}{1-0.5} S\infty= \frac{8}{0.5} S\infty= {16}

 

“Top of the World” English 11 Narrative Essay

Top of the World

-Jeremy Zhao

My feet hanging off the ledge, thousand feet of darkness lying beneath me; I feel like I’m really on top of the world.

It all started with a dream. Mountain biking has always been my passion. Although my hometown-shanghai, China is one of the biggest and most populated city on earth it is possible the most unlikely place for one to mountain bike, but I kept this hobby throughout my childhood.

Although there are a handful of trails a few hours’ drive from Shanghai, they are all poorly build, poorly marked and do not have any sort of challenge. But on the other side of the globe, it’s a completely different story…

The holy land and the founding place of mountain biking, British Columbia, Canada with more than 8000 marked trails and world’s best lifted accessed bike park -Whistler. Whistler has always been my dream destination for mountain biking, but it is halfway across the world.

Well, that’s not the case anymore. Now I live here, Coquitlam: five minutes’ drive from world class mountain bike trails and two hours drive from Whistler.

Whistler is such a majestic place, with the roaring rivers, the snow-capped mountains, it is truly the Garden of Eden for extreme sports and Heaven for sportsman alike, but one must admit the crown jewel of it all is the peak of Whistler. Sitting at 7000 feet, the peak chair of Whistler is one of the highest chairlift in BC. Although the peak zone is nothing of too much significance in winter, when the summer rolls around it becomes something special.  A trail like no other lies here, Top of the World: a trail covered with razor-sharp rocks and thousand feet clip to the side. Stretches over five kilometers and 2000 feet of elevation, a trail that’s limited to 100 riders daily, only open two months per year and a trail that cost extra 20 dollars per lap: this is not an easy trail to ride.

Even within so many restrictions, one must try this trail. And that’s exactly what I decide to do in the summer of 2016.

I sprint out of the bed; today is the day; I am going to ride Top of the World. But there is a catch… Anyone with common sense will ride whistler bike park with full face helmet and neck brace, unfortunately I broke my helmet the day before riding Top of the World. After debating with myself all night I decide YOLO and kept my plan of riding a few of the hardest trails with a half face trail helmet, I’m sure I was one of the handful maniacs who did that in 2016.

I walked into the bike park, my heart is pumping so hard, I’m not only riding one of the hardest and longest trail on the mountain that I have never ridden before, I’m also doing it without adequate protections, If I mess up and crash I will plow my face straight into those razor sharp rocks…

As those scary thoughts rushing through my mind, as time flew past. When I came out of my thoughts I was already at the roundhouse and about to load up the peak chair… On to the peak chair we go, I look down, the view is so different than winter. Without the soft puffy snow covering everything, the mountain doesn’t seem so kind with it’s scattered sharp rocks and hanging cliffs…

After a short ride, I’m at the peak of Whistler, it was a long journey but I made it… My feet hanging off the ledge, thousand feet of darkness lying beneath me. I felt like I’m really on top of the world. This is what I live for.

The ride was amazing and I didn’t plow my face into the ground as I thought I would…

Reflection:

I did well on:

Ok story

Ok organization

 

Need to improve:

Grammar

Sentence structure