Quadratic Blog Post

  1. I had trouble determining the x and y intercepts
  2. Find the x– and y-intercepts of 25x2 + 4y2 = 9
  3. https://www.youtube.com/watch?v=7reHz5OoulA
  4. x-intercepts:

    y = 0 for the x-intercepts, so:

    25x2 + 4y2 = 9
    25x2 + 4(0)2 = 9
    25x2 + 0 = 9
    x29/25
    x = ± ( 3/5 )

    Then the x-intercepts are the points ( 3/5, 0) and ( –3/5, 0)

    y-intercepts:

    x = 0 for the y-intercepts, so:

    25x2 + 4y2 = 9
    25(0)2 + 4y2 = 9
    0 + 4y2 = 9
    y29/4
    y = ± ( 3/2 )

    Then the y-intercepts are the points (0, 3/2 ) and (0, –3/2 )

    Whichever intercept you’re looking for, the other variable gets set to zero. (this was something I almost always forgot)

  5. I had trouble with this because I always forgot the step of making the other variable zero, know I know to remember to change the variable opposite to the one you are looking for. X-intercepts are always the same as ‘roots,’ ‘solutions,’ or ‘zeros.’ another key thing that messed me up. I think I got so stressed out while answering questions that I saw these other terms as something else completely.

Trig blog post

Strategy’s

Cosine law 

trig cos rule example

Cosine law is useful for finding the third side of a triangle when we know two sides and the angle between them (like the example above)

We know angle C = 37º, and sides a = 8 and b = 11

The law of cosine looks like: c2 = a2 + b2 − 2ab cos(C)

Step 1) Put in the known values – c2 = 82 + 112 − 2 × 8 × 11 × cos(37º)

Step 2) Do some calculations – c2 = 64 + 121 − 176 × 0.798…

Step 3) More calculations – c2 = 44.44…

Step 4) Take the square root – c = √44.44 = 6.67 to 2 decimal places

Answer)c = 6.67

You can also use Cosine Law to find the angles of a triangle when we know all three sides (example below)

trig cos rule example

The side of length “8” is opposite angle C, so it is side c. The other two sides are a and b.

Same as before, start with your basic equation – c2 = a2 + b2 − 2ab cos(C)

Step 1) put in the values – 82 = 92 + 52 − 2 × 9 × 5 × cos(C)

Step 2) calculate – 64 = 81 + 25 − 90 × cos(C)

Step 3) subtract the 25 from both sides – 39 = 81 − 90 × cos(C)

Step 4) Subtract 81 from both sides – −42 = −90 × cos(C)

Step 5) Swap sides – −90 × cos(C) = −42

Step 6) Divide both sides by −90 – cos(C) = 42/90

Step 7)Inverse cosine – C = cos-1(42/90)

Step 8) Calculator – C = 62.2°


Trig ratios 

Using a trig ratio is very helpful in solving for triangles. To solve a triangle means to find the length of all the sides and the measure of all the angles.

There are three steps:

Step 1) Choose which trig ratio to use.
– Choose either sin, cos, or tan by determining which side you know and which side you are looking for.

step 2) Substitute
– Substitute your information into the trig ratio.

Step 3) Solve
– Solve the resulting equation to find the length of the side.

1. Find b.

Step 1) Choose which trig ratio to use.

First, we know we must look at angle B because that is the angle we know the
measure of. So, looking at angle B, we want to identify which sides are involved. We know
one side is 8m, and that side is adjacent to angle B. The side we’re looking
for is opposite angle B. So we need to choose the trig ratio that has
opposite and adjacent. This of course is the tangent.

Step 2) Substitute

Next, we write our trig ratio: Tan B = opp/adj . Then, we substitute in the angle and the side we know: Tan 25 = b/8

Step 3) Solve

Now move the 8 to the other side by multiplying both sides by 8: 8 x tan 25 = b And use a calculator to find the answer. Well round to the nearest tenth: 3.7 m.

2. Find c.

Now that we know two sides, you could use the Pythagorean Theorem to find the third. But that’s less reliable because if you made a mistake on side b, then side c will also be wrong. So we are going to repeat the same process for side c.

Step 1) Choose the trig ratio to use.

We’re still using angle B. 8m is the adjacent and c is the hypotenuse. The trig ratio that uses the adjacent and hypotenuse is the cosine.

Step 2) Substitute

Write our trig ratio: Cos B = adj/hyp . Then, we substitute in the angle and the side we know: Cos 25 = 8/c

Step 3) Solve

Since our variable is on the bottom, we can start by cross multiplying: C x Cos 25 = 8 . Then we’ll divide both sides by cos 25°: C = 8/Cos 25 . And use a calculator to find the answer. Well round to the nearest tenth: 8.8 m.

Sine law

The Sine law says when we divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C

5,8,9 Triangle

asin A = 8sin(62.2°) = 80.885… = 9.04…

bsin B = 5sin(33.5°) = 50.552… = 9.06…

csin C = 9sin(84.3°) = 90.995… = 9.05…

Example

triangle 35 degrees, 105 degrees, 7

 

Law of Sines: a/sin A = b/sin B = c/sin C

Step 1) Enter values – a/sin A = 7/sin(35°) = c/sin(105°)

Ignore a/sin A (not applicable in this example) – 7/sin(35°) = c/sin(105°)

Step 2) swap sides – c/sin(105°) = 7/sin(35°)

Step 3) Multiply both sides by sin(105°) – c = ( 7 / sin(35°) ) × sin(105°)

Step 4) Calculate – c = ( 7 / 0.574… ) × 0.966…

Step 5) some more calculating – c= 11.8

we can also use the Law of Sines to find an unknown angle. In this case it is best to turn the fractions upside down (sin A/a instead of a/sin A, etc)

triangle 63 degrees, 4.7, 5.5

Step 1) enter the values – sin A / a = sin B / 4.7 = sin(63°) / 5.5

Step 2) Multiply both sides by 4.7 – sin B = (sin(63°)/5.5) × 4.7

Step 3) Calculate – sin B = 0.7614…

Step 4) Inverse Sine – B = sin-1(0.7614…)

Answer) B = 49.6°

There is one tricky thing we have to look out for: Two possible answers. Imagine we know angle A, and sides a and b.

We can swing side a to left or right and come up with two possible results (a small triangle and a much wider triangle) Both answers are right!

 

 

Angles

Angle AOF = 28, because it is Vertically Opposite. Vertically Opposite angles are across from and equal each other.

Angle AOB = 152, because it is a Supplementary Angle. Supplementary Angles always add to 180 degrees and are on a line.

Angle FOC = 152, because it is Vertically opposite to Angles AOB.

Angle DOC = 76, because it is a Complimentary Angle. Complimentary Angles add to 90 degrees.

What Darwin Never Knew

Darwin’s Theory of Evolution is the widely held notion that all life is related and has descended from a common ancestor: the birds and the bananas, the fishes and the flowers- all related.

Charles Darwin was a biologist who visited the Galapagos Islands in 1931. While there he encountered many exotic and fascinating organisms and collected the bodies of birds during an expedition. On his way home he realized they were the same type of bird (finch), but they had adapted themselves for that island. Said events lead to his Theory of Evolution. Now that we have a much better understanding of DNA in the 21 century, we know that a difference in base pairings can create a large difference in a species growth. Then when we learned about “switches” that turn on and off specific genes which gave us an explanation for embryos, who all look similar in their early structures but change over periods of time. The discovery of DNA is important because now we recognize how even the smallest changes in DNA can have major effects in organisms. The discovery of DNA provides a scientific explanation between the development of species to the sophistication of cells that make up the blueprint for mankind.

bio bio

Water For Elephants – Inquiry

Why are humans so dependent on companionship?

Water for Elephants does a wonderful job of showcasing the best and worst of humanity through its characters, like August’s abuse and Jacobs kindness contrast beautifully. Marlena has spent her life being abused by August, so when she meets Jacob who is in stark contrast to him,  an example is when Jacob comforts Marlena when her horse dies, something she couldn’t do with August, “I’m marveling and not just a little unnerved at her stoic reaction when a strange noise rises from her throat. It’s followed by a moan, and next thing I know she’s bawling. She doesn’t even try to wipe the tears that slide down her cheeks, just stands hugging her arms with shoulders heaving, gasping for breath. She looks like she’s going to collapse in on herself.”  (7.131)

We need companionship to help us through the most difficult times in our lives. Someone to cry with and support you can be the best way to healthily deal with hardships.

 

We need companionship in a way of friendship as well, If it wasn’t for Camels compassion Jacob would have never found a place with the Circus. Camel shows courage and companionship to Jacob when he sticks up for his place on the train, proven when he says, “Just shut it. I don’t want to hear it. You’re a good kid, and I ain’t about to stand by and watch you mope off ‘cuz that fat old grouch don’t got time. I just ain’t. So have a little respect for your elders and don’t give me no trouble.” This momets creates a bond of loyalty between the two and from then on, they have each others back, which helps prolong Camels life.

Water For Elephants – Connections

Toxic Relationships

You say that you love me, but you act like you don’t

don’t take this the wrong way, but you put me in harms way 

A theme in Water for elephants is Toxic Relationships. It’s displayed in Marlena’s relationship with August, for example, her whole demeanor changes around him, she becomes submissive and ‘plays dumb,’ However when she is with Jacob she is almost guarded. In the circus where Jacob ends up being apart of, everyone in their own way is broken, and because most people have never been treated any better in their past lives, it’s hard to know any better. August is a dominant man who demands respect, undying loyalty, and almost ownership of the other patrons of the circus. In this sense August feels as if he has ownership over the performers, leading to their unavoidable mistreatment.