The value of a function of a single variable is written fx, where f is the function and x is the variable. We assume that both x and fx are real numbers and that for each value of x there is only one value fx. Technically speaking this makes f a ‘single real-valued function of a single real variable’, but we shall try to avoid such technical vocabulary where possible. Basically, it means that we can draw a graph of the function, with the values x along the horizontal axis and the corresponding values fx along the vertical axis, and that this graph

A linear function is one whose graph is a straight line. For instance, the function fx=3x+2 is linear because its graph is a straight line, shown in Figure I.1.1. A linear function defines a linear equation, i.e. 3x+2 =0 in this example. This has a root when x= −2/3. Readers may use the spreadsheet to graph other linear functions by changing the value of the coefficients a and b in the function fx = ax + b.

By contrast, the function fx = 4×2 + 3x + 2 shown in Figure I.1.2 defines an equation with no real roots because the function value never crosses the x-axis. The graph of a general quadratic function fx = ax2 + bx + c has a ‘∩’ or ‘∪’ shape that is called a parabola:

If the coefficient a > 0 then it has a ∪ shape, and if a < 0 then it has a ∩ shape. The size of a determines the steepness of the curve. • The coefficient b determines its horizontal location: for b > 0 the graph is shifted to the left of the vertical axis at x = 0, otherwise it is shifted to the right. The size of b determines the extent of the shift. • The coefficient c determines its vertical location: the greater the value of c the higher the graph is on the vertical scale.

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Readers may play with the values of a b and c in the spreadsheet for Figure I.1.2 to see the effect they have on the graph. At any point that the graph crosses or touches the x-axis we have a real root of the quadratic equation fx = 0. A well-known formula gives the roots of a quadratic equation ax2 + bx + c=0 where a b and c are real numbers.