Sketching graphs
At times you will need to sketch a function to see what it looks like. An easy way of doing this is:
- Select values of x and then calculate the corresponding values of the function.
- Put these values in a table.
- Use this table to sketch the graph.
Using the above example, where f:x → x^{2} + 3x − 2.
Select values of x and put the corresponding values of f(x) and into an organized table:
x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
f(x) | 8 | 2 | -2 | -4 | -4 | -2 | 2 | 8 | 16 | 26 | 38 |
Now we can plot the values of f(x) on a graph, we can see a pattern in the values of f(x):
There are several important pieces of information about the function that need to be found. In particular where the graph crosses the x- and y-axes, and where the graph turns.
The graph shows us that:
a) The curve has a line of symmetry at the line
(because values of x that are symmetrical about the line x = -3/2, give the same value for f(x)).
b) The lowest value of y = -17/4 and this happens when
c) Using the quadratic formula,
...we can calculate the roots of this equation (where f(x) = 0).
So,
And,
All quadratics have this same symmetrical shape and for a general quadratic function in the form,
f(x) = ax^{2} + bx + c
Where a, b, and c are constants.
The main features we need to sketch a quadratic are:
- Where the graph crosses the y-axis. (At (0, c) as when x = 0, y = c).
- Where the graph crosses the x-axis. (Factorise or use the quadratic formula to solve f(x) = 0.)
- Where the graph turns. You can use differentiation, or completing the square (the quadratic formula), to find that:
Graphically, we see that this means:
Once you know this information you can sketch any quadratic function.
For example:
Sketch the curve that represents f(x) ≡ -x^{2} + 2x
When x = 0, y = 0.
Therefore it crosses the y-axis at (0,0)
f(x) = 0 when -x^{2} + 2x = 0, or x(2 - x) = 0.
For instance, when x = 0, or when x = 2.
It is a - x^{2} therefore it is a symmetrical ∩ shape, with its maximum value when
x = 1 (a = -1, b = 2, therefore -b/2a = 1) and y = 1.
So, the graph can be sketched as:
If we don't already know what a graph will look like we need to find its main features. These are:
- Where the graph crosses the y-axis, which is when x = 0. (i.e. at the constant).
- Where the graph crosses the x-axis. To find the roots (where the graph crosses the x-axis), we solve the equation y = 0
- Where the stationary points are. The stationary points occur when the gradient is 0. (i.e. differentiate.) Whether there are any discontinuities.
- Are there any discontinuities? A discontinuity occurs when the graph is undefined for a certain value of x. This occurs when x appears in the denominator of a fraction (you can't divide by zero).
- What happens as x approaches ±∞? When x becomes a large positive or a large negative number the graph will tend towards a certain value or pattern.
Now put all this information onto the graph and join up the points.
Example 1:
Sketch the graph
If x = -3 then the denominator is zero. As we cannot divide by zero the graph is undefined, and there is a discontinuity at x = 3.
As x → +∞, y → 2 (The -1 and +3 become insignificant.) As x → -∞, y → 2 as well. This means there is a horizontal asymptote (value that the graph tends towards) at x = 2.
So the final graph looks like this:
Example 2:
The graph of the function f(x) = 2/x looks like this:
The two asymptotes are the x-axis and y-axis.
This curve has a special discontinuity at x = 0 where f(0) is undefined.