Introduction

1. Introduction We will cover 3 topics today 1. Coordinates in a Plane 2. Graphs 3. Polar Coordinates

2. Coordinates in a Plane This equation gives two values of y for every value of x. This is called a one-to-many rule. Consider the function y = 2x This function gives one value of y for every value of x. This is called a one-to-one function. x is known as the independent variable and y is known as the dependent variable. A graph of the equation y = ±√x can be drawn by constructing a table of values. A function is defined as a rule that gives one value of y for each value of x. x 0 1 2 3 4 y 0 ±√1 ±√2 ±√3 ±√4

3. Coordinates in a Plane Is this a many-to-one or a one-to-many equation? Consider the function y = x2 This equation has two values of x for each value of y (except 0). This is called a many-to-one rule. This is a function. This is a many-to-one function.

4. Coordinates in a Plane We can deduce that The general formula for a straight line is given by A common formula describing a straight line is given by Where a, b and c are constants. y Determine m and c in terms of x1, x2, y1, and y2 x2, y2 x, y x1, y1 x

5. Graphs Find the radius and centre of the circle described by the following formula The general formula for a circle is given by y (a,b) r x If a curve is given by f(x,y) then f(x-a,y-b) represents the same curve translated by a distance ‘a’ on the x axis and ‘b’ on the y axis. Therefore the centre is at (1/2, -1) and the radius is 2.

6. Graphs The general formula for an ellipse is given by The general formula for a hyperbola is given by a > 0, b > 0 a > 0, b > 0 y y y=bx/a y=-bx/a b a -a x x -b

7. Graph Sketching • i) Zeros - obtain values of x where y = 0 • ii) Poles - obtain values of x that make the denominator equal to zero (called poles). iii) Asymptotes - these are straight lines that the curve approaches as either x → ±∞ or y → ±∞ iv) Special points - calculate values of f(x) explicitly at selected points to give further insight. e.g. at x = 0

8. Graph Sketching x=-1, y=∞ y (-1/2,2) (0,1) (1/2,0) x (2,-1) y=-2, x=∞

9. Graph Sketching x=-1, y=∞ y y=1, x=∞ (2,1/3) (1,0) x (0,-1) • Sketch the function (-1/2,3)

10. Polar Coordinates y • A spiral with an inner radius ‘a’ and an outer radius ‘b’ is described by r • Where θ runs from zero to 2πN. • N is the maximum number of revolutions and is given by x

11. Polar Coordinates • Using • Obtain the polar coordinates of the central ellipse • Using the trigonometric identities • (you need the following identities) • Simplifying • When θ runs from 0 to 2π the ellipse is traced out once.

12. Polar Coordinates • Given • Obtain the polar equation of the ellipse tilted through an angle α

13. Conclusion We have looked at 1. Coordinates in a Plane 2. Graphs 3. Polar Coordinates • Essential reading for next week • HELM Workbook 2.2: Graphs of Functions and Parametric Form • HELM Workbook 2.4: Characterising Functions • HELM Workbook 2.6: The Circle

14. COURSEWORK!!!!