This App will help you to avoid any unwanted slip-ups in the exam. Although most of the reminders are common sense, but from the evidence students still need reminding of them. Read through the tips and take note of the most relevant ones before tackling your exam.
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 → x2 + 3x − 2.
Select values of x and put the corresponding values of f(x) and into an organized table:
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).
All quadratics have this same symmetrical shape and for a general quadratic function in the form,
f(x) = ax2 + 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.
Sketch the curve that represents f(x) ≡ -x2 + 2x
When x = 0, y = 0.
Therefore it crosses the y-axis at (0,0)
f(x) = 0 when -x2 + 2x = 0, or x(2 - x) = 0.
For instance, when x = 0, or when x = 2.
It is a - x2 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.
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:
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.