The extended coordinate system

Now that we know the negative numbers, both axes can go on past the origin: the $\displaystyle x$ -axis goes on to the left to $\displaystyle -1$ , $\displaystyle -2$ …, and the $\displaystyle y$ -axis goes on down to $\displaystyle -1$ , $\displaystyle -2$ …

The coordinates are read as before, $\displaystyle x$ first and then $\displaystyle y$ . $\displaystyle {B}={(-2,3)}$ : two steps left of the origin and three steps up. $\displaystyle {D}={(2,-3)}$ : two steps right and three steps down. $\displaystyle {C}={(-4,-1)}$ : four steps left and one step down.

Four quadrants

A point on one of the axes is in no quadrant.

The axes cut the plane – the whole flat surface around them – into four parts, called quadrants. They are counted anticlockwise, starting from the one at the top right:

quadrant	first	second	third	fourth
$\displaystyle x$ -coordinate	positive	negative	negative	positive
$\displaystyle y$ -coordinate	positive	positive	negative	negative

In the picture: $\displaystyle A$ is in the first quadrant, $\displaystyle B$ in the second, $\displaystyle C$ in the third and $\displaystyle D$ in the fourth.

Exercise 1

Write the coordinates of each point, and which quadrant it is in.

Exercise 2

Draw a coordinate system and mark the points $\displaystyle (-3,2)$ , $\displaystyle (1,2)$ , $\displaystyle (1,-2)$ and $\displaystyle (-3,-2)$ .
Join them in order, and the last one to the first.
What shape do you get?
How long is its side?

Side length

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Without drawing: which quadrant is each point in? a. $\displaystyle (-7,12)$ b. $\displaystyle (15,-3)$ c. $\displaystyle (-0.5,-8)$ d. $\displaystyle (6,0)$

The length of a segment

The length is always the larger minus the smaller – and it is always positive.

The segment $\displaystyle PQ$ has the same $\displaystyle y$ -coordinate at both ends, and it crosses the $\displaystyle y$ -axis. From $\displaystyle -3$ to $\displaystyle 0$ there are three steps, and from $\displaystyle 0$ to $\displaystyle 4$ four more – seven together. As a calculation: $\displaystyle {4-(-3)}={7}$ .

And for the vertical segment $\displaystyle QR$ : $\displaystyle {2-(-3)}={5}$ .

Exercise 4

$\displaystyle A$ , $\displaystyle B$ and $\displaystyle C$ are three vertices of the rectangle $\displaystyle ABCD$ .
What are the coordinates of $\displaystyle D$ ?
Calculate the lengths of the sides and the area of the rectangle.

\displaystyle AB

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle BC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Area

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Mark the point $\displaystyle (2,3)$ .
Also mark the point whose $\displaystyle x$ -coordinate is the opposite, $\displaystyle (-2,3)$ ; the point whose $\displaystyle y$ -coordinate is the opposite; and the point whose two coordinates are both the opposites.
Join the four points.
What shape do you get?
Where do the axes cross it?

Formulas: algebra, grade 7

Open as a page

Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$