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:

  1. Select a few values of x (at least 5)
  2. Obtain the corresponding values of the function and enter them into a table
  3. 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, which you'll come across in later chapters). It's much easier than plotting points and more useful for later!

Example 1

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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.

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?

Example 3

Graph the function y = x x2

Example 4

Graph the function `y=1+1/x`

Example 5

Graph the function `y=sqrt(x+1)`

Example 6

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

`P=(100R)/(0.50+R)^2`

Plot the power as a function of the resistance.

Exercises

Graph the given functions

Q1 y = x3 x2

Q2 `y=sqrt(x)`

conical water storage
Conical water tank

Q3. (Application) Water flows out of a tank in the shape of an inverted cone (i.e. 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.

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