Coordinate Geometry, Graphs and Circles

Question 1

What do parallel lines have in common?

Answer 1

Gradient

Question 2

What is of not about the gradients of perpendicular lines?

Answer 2

They multiply to give -1.

Question 3

What is a perpendicular line called with regards to another?

Answer 3

The normal.

Question 4

What does it mean if two variables are in direct proportion?

Answer 4

Changing one variable will change the other by the same scale factor so multiplying by any constant will have the same effect on both.

Question 5

How is “y is directly proportional to x” written?

Answer 5

“y = kx” where k is the constant or constant of proportionality or “y (stretched lower case alpha) x”

Question 6

What does it mean if two variable are inversely proportional?

Answer 6

Doubling one halves the other, so on.

Question 7

How is “y is inversely proportional to x” written?

Answer 7

“y = k(1/x)” where k is the constant or constant of proportionality or “y (stretched lower case alpha) 1/x”

Question 8

What are the three equations that can be used to describe a straight line?

Answer 8

• y-b = m(x-a) for point (a,b) on the line
• y = mx+c
• ax+by+c = 0 where a, b and c are integers

Question 9

What is the equation for the midpoint of a line segment?

Answer 9

Midpoint of the line (ab) = ((xa+xb)/2 , (ya+yb)/2)

Question 10

How do you find the length of a line segment?

Answer 10

Pythagoras’ Theorem: length of line (ab)^2 = (xb-xa)^2 + (yb - ya)^2

Question 11

What are the two general curves for any graph in the form y = kx^n (where n is positive?

Answer 11

• When n is even you get an u/n shape when k is positive/negative.
• When n is odd you get a corner to corner shape with a stationary point in the middle.

Question 12

What information do you need to sketch a graph and how do you find it?

Answer 12

You need to know where the graph crosses the axes:

• To find where a graph crosses the y-axes, just set x = 0 and find the value of y.
• To find where it crosses the x-axis, factorise the polynomial - it crosses the x-axis when each bracket is set to equal to 0.

Question 13

If you factorise a polynomial and get a squared bracket what does this mean?

Answer 13

This is a double root, and the graph will only touch the x-axis, not cross it.

Question 14

If you factorise a polynomial and get a cubed bracket what does this mean?

Answer 14

This is a triple root, which still crosses the x-axis, but flattens out as it does so.

Question 15

What is a reciprocal function?

Answer 15

Reciprocal functions are those of the form y = k/x^n or y = kx^-n where k is a constant.

Question 16

What is the general shape of any reciprocal graph when n is even?

Answer 16

An L shape which appears also reflected in the y axis. If k is negative then the whole graph is reflected in the x-axis.

Question 17

What is the general shape of any reciprocal graph when n is odd?

Answer 17

An L shape which appears also, reflected in the line y = -x. If k is negative the whole thing is reflected in the y-axis.

Question 18

What is a translation?

Answer 18

Moving the graph of a function either horizontally or vertically. The shape of the graph itself doesn’t change, it just moves.

Question 19

What kind of translation is y = f(x) + a?

Answer 19

Adding a number to the whole function which translates the graph in the y-direction.
-If a > 0, the graph goes upwards.
-If a < 0, the graph goes downwards.
This can be described by a column vector:
(0)
(a)

Question 20

What kind of translation is y = f(x + a)?

Answer 20

Writing 'x + a' instead of 'x' translates the graph in the x-direction.
-If a > 0, the graph goes to the left.
-If a < 0, the graph goes to the right
As a column vector, this would be:
(-a )
( 0 )

Question 21

What are stretches and reflections?

Answer 21

The graph of a functions can be stretched, squashed or reflected by multiplying the whole function or the x’s in the function by a number. The result you get depends on what you multiply and whether the number is positive or negative.

Question 22

What are the characteristics of stretching a graph like: y = af(x)

Answer 22

Multiplying the whole function by a stretches the graph vertically by a scale factor of a.
-If a > 1 or a < -1, the graph is stretched.
-If a < 1 or a > -1, the graph is squashed.
-If a is negative, the graph is also reflected about the x-axis.
For every point on the graph, the x-coordinate stays the same and the y-coordinate is multiplied by a.

Question 23

What are the characteristics of stretching a graph like: y = f(ax)?

Answer 23

Writing ‘ax’ instead of ‘x’ stretches the graph horizontally by a scale factor of 1/a.
-If a > 1 or a < -1, the graph is squashed.
-If -1 < a < 1, the graph is stretched.
-If a is negative, the graph is also reflected about the y-axis.
For every point on the graph, the y-coordinate stays the same and the x-coordinate is multiplied by 1/a.

Question 24

What is the equation of a circle?

Answer 24

(x - a)^2 + (y - b)^2 = r^2, where r is the radius and the centre is (a, b).

Question 25

How do you put an un-factorised equation of a circle into the correct form?

Answer 25

Complete the square.

Question 26

What are the important properties of a circle?

Answer 26

• The triangle formed from the ends of a diameter has a right angle.
• The perpendicular from the centre to a chord bisects the chord.
• A tangent to the circle meets a radius at a right angle.

Question 27

What is are circumcircles?

Answer 27

Circles that pass through all three vertices of a triangle.