Category: D W Math tools

Transformation Matrices

Rotate a figure 180 degrees around the origin:

From your original figure, just reverse the polarity of the ordered pairs.

Point A(x,y) will be Point a(-x, -y)

(-6,3) (-3,3) (-2,5) (-2,1) (-6,1) original
(6,-3) (3,-3) (2,-5) (2,-1) (6,-1) rotated 180 degrees around origin.

 

Reflect over y-axis

From the original, reverse the polarity of the x value. Point A(x,y) will become point a(-x,y)

(-6,3) (-3,3) (-2,5) (-2,1) (-6,1) original
(6,3) (3,3) (2,5) (2,1) (6,1) reflected over y axis

 

Rotate 90 degrees counter clockwise around origin

The new position of point M (h, k) will become M’ (k, -h)

 

 

Working with numbers in JavaScript

Make a string into a number for use in calculations : Number(string);

Round a number to certain place after making calulations:
The Math.round() function returns the value of a number rounded to the nearest integer.

Math.round(x)

The Math.abs() function returns the absolute value of a number.

var abs = Math.abs(x);

The Math.floor() function returns the largest integer less than or equal to a number. This is basically to round DOWN:

Math.floor(x)

The Math.ceil() function returns the smallest integer greater than or equal to a number. This is to round UP:

Math.ceil(x)

Direct, Inverse and Joint Variation Equations

Direct variation is a variation equation , relating one variable to one or more others using multiplication and or division.

We can use this to solve for d in a distance rate time problem. d=rt

if the rate is going to be a constant when the other variables are changing, then :
distance varies directly as time and
the distance is directly proportional to time.

Examples: page 321 in text has lots more
In electronics, Resistance of wire varies directly as its length R=kL
In medicine, dosage is directly proportionate to weight of person d=kw

 

Joint Variation is when one quantity may vary directly as a product of two or more other quantities.

If a variable y varies jointly with variable x and z, then y=kxz where k is the constant of proportionality.

In geometry the area A of a triangle varies jointly as its base b and height h. A=kbh
If F varies jointly as H and G then  F=kHG

Z varies jointly as W and Y . If Z=330 when W=11 and Y=10, find Z when W=60 and Y=10
First find the variation: The variation is Z=kWY.

Second find k using the first set of values 330=k(11)(10) | 330=k(110) | k=330/110 | k=3
Third solve for Z using the second set of values and the k you just found… Z=(3)(60)(10) | z=1800

Inverse variation is when one quantity increases as the other decreases and vice versa. y=k/x

We would use this in a distance rate time equation (d=r/t | t=d/r) when we were solving for t
T=k/r

For the weight of an astronaut problem, the weight W of an object varies inversely as the square of the distance d from the center of the earth.When w=111, d=3978 find k. It really helped me to leave the numbers unmultiplied

W=k/d^2

111=k/3978^2 | k=111(3978^2)

now find w when d=434+3978
w=111(3978^2)/(434+3978)^2
w=90.24

 

Combined variation is awesome..

T varies direclty as the square of D and inversly as F. The variation is : (T=kD^2)/F

The number of phone calls between two cities, N, varies directly as the populations (p1+p2) and inversly as distance between, d

The variation is N=kp1p2/d
If 69000 is N number of calls and d distance between cities  is 340 and p1 p2=(40800)(230000) then find k

69000= k((40800)(230000))/340
69,000=27,600,000k
k=69000/27600000
k=.0025

how many calls, N, are made when d=455and p1p2=(70,000)(160,000) and k=.0025
N=kp1p2/d
N=.0025(70,000)(160,000)/455
N=612538 rounded to nearest integer

F=kq1q2/d^2
F=6 when q1=9 and q2=3 and d=3
6=k(9)(3)/3^2
3k=6
k=2

find f when q1=6 q2=4 and d=2
F=(2)(6)(4)/2^2
f=12