FUNCTIONS : FUNCTIONS Rectangular Coordinates Slide 2: A good way of presenting a function is by graphical representation.
Graphs give us a visual picture of the function. Slide 3: The rectangular co-ordinate system consists of:
the origin (0,0)
the four quadrants Slide 4: Normally, the values of the independent variable (generally the x-values) are placed on the horizontal axis, while the values of the dependent variable (generally the y-values) are placed on the vertical axis. The x-value, called the abscissa, is the perpendicular distance of P from the y-axis.
The y-value, called the ordinate, is the perpendicular distance of P from the x-axis.
The values of x and y together, written as (x, y) are called the co-ordinates of the point P. Ex:Three vertices of a rectangle are A(-3 , -2), B(4 , -2) and C(4,1). Where is the fourth vertex D? : Ex:Three vertices of a rectangle are A(-3 , -2), B(4 , -2) and C(4,1). Where is the fourth vertex D? Since the opposite sides of a rectangle are equal and parallel, we can see that:
the y co-ordinate of D is 1
the x co-ordinate of D is -3
Hence, the co-ordinates of D are (-3, 1). Ex: Where are all points (x , y) for which x < 0 and y < 0? : Ex: Where are all points (x , y) for which x < 0 and y < 0? Since
x < 0 means that x is negative,
and y < 0 also means that y is negative,
then the only region where both co-ordinates for all points are negative is the "third quadrant (III)". Ex: Where are all the points whose abscissas equal their ordinates? : Ex: Where are all the points whose abscissas equal their ordinates? "Abscissas" means x-values, while "ordinates" means y-values.
So the question means "where on the rectangular system do we have x = y for all points (x, y)?"
In other words, we want a line connecting points like (-3, -3) and (0, 0) and (5, 5) and (700, 700).
The line we want cuts the first and third quadrants in half at 45°. We can write this line as y = x. Ex: Where are all the points (x, y) for which x = 0 and y < 0? : Ex: Where are all the points (x, y) for which x = 0 and y < 0? They are on the negative part of the y-axis. The Graph of a Function : The Graph of a Function Slide 10: The graph of a function is the set of all points whose co-ordinates (x, y) satisfy the function y = f(x). This means that for each x-value there is a corresponding y-value which is obtained when we substitute into the expression for f(x). Slide 11: Since there is no limit to the possible number of points for the graph of the function, we will follow this procedure at first:
- select a few values of x
- obtain the corresponding values of the function
- plot these points by joining them with a smooth curve Slide 12: However, you are encouraged to learn the general shapes of certain common curves (like straight line, parabola, trigonometric and exponential curves) - it's much easier than plotting points and more useful for later! A man who is 2 m tall throws a ball straight up and its height at time t (in s) is given by h = 2 + 9t - 4.9t2 m. Graph the function. : A man who is 2 m tall throws a ball straight up and its height at time t (in s) is given by h = 2 + 9t - 4.9t2 m. Graph the function. We start at t = 0 since negative values of time have no practical meaning here. This shape is called a parabola and is common in applications of mathematics. Slide 14: The velocity (in m/s) of the ball in Example 1 at time t (in s) is given by
v = 9 - 9.8t
Draw the v-t graph. What is the velocity when the ball hits the ground? EXAMPLE: Slide 15: Since we recognise it is a straight line, we only need to plot 2 points and join them. But we find 3 ponts, just to check. Slide 16: The ball hits the ground at approx t = 2.05 s (we can see this from Example 1). The velocity when the ball hits the ground from the graph we just drew is about -11 m/s.
Normally, we take velocity in the up direction to be positive. EX: Graph the function y = x - x2 : EX: Graph the function y = x - x2 (a) Determine the values in the table Slide 18: (b) Plot these points (c) Indicate that the curve continues. EX: Graph the function : EX: Graph the function (a) Note : y is not defined for x = 0, due to division by 0
Hence, x = 0 is not in the domain
(b) determine the values in the table Slide 20: c) Check what happens between x = -1 and x = 1: when x = , y = -1 when x = , y = 3 (d) As x gets closer to 0, the points get closer to the y-axis, although they do not touch it. The y-axis is called an asymptote of the curve. Ex: Graph the function : Ex: Graph the function (a) Note: y is not defined for values of x less than -1
Hence, x < -1 is not in the domain (b) determine the values in the table: Slide 22: (c) The positive square root is indicated, hence, the range consists of all positive values of y, including 0 (ie. y ≥ 0) Graph the given function(1) y = x3 - x2 : Graph the given function(1) y = x3 - x2 (2) : (2) We can only take the square root of a positive number so x ≥ 0. The square root of a number can only be positive, so y ≥ 0.
This graph is actually one half of a parabola, with horizontal axis. THANK YOU : THANK YOU