PCH4: Real Functions of a Real Variable

Question 1

Domain

Answer 1

Set of all possible values of x

Question 2

Range

Answer 2

Set of all possible values of y

Question 3

Even and odd functions

Answer 3

A function is even if f(-x) = f(x)

A function is odd if f(-x) = - f (x)

Question 4

Equation of a circle with centre other than origin

Answer 4

centre (a, b), radius = r

(x-a)^2 + (y-b)^2 = r^2

Question 5

Equation of a circle with centre at origin

Answer 5

x^2 + y^2 = r^2

Question 6

What is a function?

Answer 6

Only has one value of y for any one value of x

Question 7

Function of a semi-circle whose centre is at the origin

Answer 7

y = √ r^2 - x^2 (concave down semi circle in postive y range)

y = - √ r^2 - x^2 (concave up semi circle in negative y range)

Question 8

Function of a semi-circle whose centre is at a point other than the origin

Answer 8

Centre = (a, b)

y = b + √ r^2 - (x-a)^2 (concave down semi circle)

         or

y = b - √ r^2 - (x - a)^2 (concave up semi circle)

Question 9

What is a locus?

Answer 9

A set of points obeying a given condition or conditions

Question 10

How do you find the locus of points when given two other points from which it is equidistant?

Answer 10

P(x,y) and Given points A(xa, ya) and B (xb, yb)

PA = PB
PA ^2 = PB ^2
Reduce to an equation in general form

Question 11

What is a parabola?

Answer 11

The locus of a point whose distance from a fixed point, called the focus, is equal to its distance from a fixed line, the directrix

Question 12

How do you find the locus of points when their distance from a fixed point is equal to their distance from a fixed line?
(Parabola)

Answer 12

Sub values of fixed point and locus point P (x, y) into equation for distance between points

Sub value of locus point and equation for fixed line (in general form) into formula for perpendicular distance

Distance between points equation = perpendicular distance equation

Square both sides and solve