Using a tableAt 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: 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, QuadraticsAll quadratics have this same symmetrical shape and for a general quadratic function in the form, f(x) = ax2 + bx + cWhere 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) ≡ -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 whenx = 1 (a = -1, b = 2, therefore -b/2a = 1) and y = 1. So, the graph can be sketched as: More Complex GraphsIf 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 graphIf 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.