Sketching graphs

At times you will need to sketch a function to see what it looks like. An easy way of doing this is:

  1. Select values of x and then calculate the corresponding values of the function.
  2. Put these values in a table.
  3. 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:

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):

Plot

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

The line of symmetry

(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

x

c) Using the quadratic formula,

Quadratic formula

...we can calculate the roots of this equation (where f(x) = 0).

So,

Roots

And,

Roots

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:

  1. Where the graph crosses the y-axis. (At (0, c) as when x = 0, y = c).
  2. Where the graph crosses the x-axis. (Factorise or use the quadratic formula to solve f(x) = 0.)
  3. Where the graph turns. You can use differentiation, or completing the square (the quadratic formula), to find that:

Main features

Graphically, we see that this means:

Graphs of quadratics

Once you know this information you can sketch any quadratic function.

For example:

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:

Example

If we don't already know what a graph will look like we need to find its main features. These are:

  1. Where the graph crosses the y-axis, which is when x = 0. (i.e. at the constant).
  2. Where the graph crosses the x-axis. To find the roots (where the graph crosses the x-axis), we solve the equation y = 0
  3. Where the stationary points are. The stationary points occur when the gradient is 0. (i.e. differentiate.) Whether there are any discontinuities.
  4. 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).
  5. 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

Example Example 1

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:

Graph

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.

Example 2

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