Chapter 4- Coordinate geometry, Graphs and Circles (Pure Maths)

Question 1

What are the three general equation of a straight line?

Answer 1

the general equation of a straight line is:

Y - Y1= m (X - X1)

Y=mX + c

aX + bY + c = 0

Question 2

whats the equation of midpoint?

Answer 2

midpoint = (x1+x2)/2 , (y1 +y2)/2

Question 3

whats the equation for the length of AB?

Answer 3

Length (AB) = √ (xb-xa)² + (yb-ya)²

Question 4

what’s the gradient of parallel lines have?

Answer 4

parallel lines have the same gradient

Question 5

what’s the gradient of perpendicular lines?

Answer 5

perpendicular lines multiply to -1

Question 6

what’s the equation for y being proportional to x?

Answer 6

y∝ x or Y = kX

Question 7

whats the equation for inversely proportional to?

Answer 7

y∝ 1/X or y=k/x

Question 8

what are cubics and quartics?

Answer 8

cubics and quartics are polynomials
cubics are x³
quartics are x⁴

Question 9

for the equation Y=kxⁿ what happens when n is EVEN?

Answer 9

Y=kxⁿ

when n is EVEN
you get an n or u shape

if k is positive you get a u shape

if k is negative you get an n shape

Question 10

for the equation Y=kxⁿ what happens when n is ODD?

Answer 10

Y=kxⁿ

when n is odd. you get a ‘corner to corner’ shape

if k is positive you get a ‘bottom left to top right shape’
the happy ending

if k is negative you get a ‘bottom left to top right shape’
the sad ending

Question 11

For the reciprocal function y=k/xᴺ or y=kx⁻ᴺ what does the curve look like when n is EVEN ?

Answer 11

when n is even get two bits next to each other

if k is positive both parts are above the x axis. the happy ending

if k is negative both parts are below the x axis. the sad ending

Question 12

For the reciprocal function y=k/xᴺ or y=kx⁻ᴺ what does the curve look like when n is ODD?

Answer 12

when n is odd you gets two bits opposite each other

if k is positive the graph starts at the top right and bottom left quadrants. the happy ending

if k is negative its in the top left and bottom right. the sad ending

Question 13

for y=f(x) + a, what does the ‘+a’ do?

Answer 13

y= f(x) + a

adding a number to the whole function translates the graph in the y direction

if a is positive the graph goes upwards

if a is negative it goes down

Question 14

for y=f(x+ a) , what does the ‘+a’ do?

Answer 14

y=f(x+ a)

the ‘+a’ translates the graph in the x direction

if a is positive it goes to the left

if a is negative the graph goes right

Question 15

what does the ‘a’ do in y=af(x)

Answer 15

the ‘a’ in y=af(x) stretches the whole function vertically by scale factor a

if a > 1 or a < -1, the graph is stretched
if -1 < a < 1, the graph is squashed

if a is negative the graphs reflected in the x axis

Question 16

what does the ‘a’ do in y=f(ax)

Answer 16

the ‘a’ in y=f(ax) stretches the whole function horizontally by scale factor 1/a

if a > 1 or a < -1, the graph is squashed
if -1 < a < 1, the graph is stretche

if a is negative the graphs reflected in the y axis

Question 17

what’s the general formula of a circle?

Answer 17

the general formula of a circle is

(x-a)² + (y-b)² = r²

Question 18

how to rearrange circle equations? e.g

x² + y² + 2fx + 2gy + c = 0

Answer 18

to rearrange x² + y² + 2fx + 2gy + c = 0

1) group y and xs together
2) complete the square for the x and y terms
3) rearrange in correct form

Question 19

what’s a circumcircle?

Answer 19

a circumcircle is when a circle passes through all vertices of a triangle

Question 20

what are the three useful properties of a circle?

Answer 20

1) the triangle formed from the ends of a diameter has a right angle triangle
2) a perpendicular line form the centre to a chord bisects the cord
3) a tangent to the cirlcle meets a radius at a right angle