4. The Graph of a Function

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).

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

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!

Example 1

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.

Answer

We start at t = 0 since negative values of time have no practical meaning here.

x	0	0.5	1	1.5	2
y	2	5.3	6.1	4.5	0.4

This shape is called a parabola and is common in applications of mathematics.

NOTE:

We could have written the function in this example as: h(t) = 2 + 9t - 4.9t².

Example 2

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?

Answer

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.

x	0	1	2
y	9	-0.8	-10.6

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.

Example 3

Graph the function y = x - x²

Answer

Example 4

Graph the function math expression

Let's get LiveMath to plot any of these functions in this section. My suggestion is to observe the general shape of each one and remember it!

LIVEMath

Answer

Example 5

Graph the function

Answer

Example 6

The electric power P (in watts) delivered by a battery as a function of the resistance R (in ohms) is :

Plot the power as a function of the resistance.

Answer

Exercises

Graph the given functions

Q1 y = x³ - x²

Answer

Conical water tank

Q3. (Application) Water flows out of an inverted cone (ie the water flows through the pointy end of the cone and the widest part of the cone is at the top). The volume of the water is decreasing at a constant rate.

Draw a sketch graph of the height of the water in the cone versus the time.

Answer