Math Revision 16.3 - Basic Functions

On the other hand, a function is decreasing if for any two points A and B (A is the leftmost, i.e. x_B > x_A), then f(

Quadratic functions

Quadratic functions have a general form

where a and b are coefficients and c is a constant (all of them are numbers), while x is the independent variable of the function.

The domain, codomain and range of a quadratic function are all unlimited unless the restrictions are explicitly stated in the clues. This is because this function has no square roots, logarithms, trigonometric elements, fractions, etc., which may bring restrictions in the allowed values.

The graph of a quadratic function is a parabola. The coefficient a of the quadratic function tells us what the orientation of the parabola is. Thus, if a > 0 the arms of the parabola are upwards; the graph of the quadratic function indicated by this parabola has a minimum at V(-b/2a, -Δ/4a), where Δ = b₂ - 4ac is the discriminant of the corresponding quadratic equation f(x) = 0.

On the other hand, if the coefficient a of the parabola is negative (a < 0), the arms of the parabola are downwards; the graph of the quadratic function indicated by this parabola has a maximum at V(-b/2a, -Δ/4a). The x-coordinate of the vertex is also the equation of the symmetry line that divides the quadratic function graph into two equal halves.

The sign of the discriminant Δ also tells us a lot of useful info about the number of x-intercepts of the graph. These intercepts are called roots when dealing with the corresponding quadratic equation ax^x + bx + c = 0 or zeroes when dealing only with the quadratic polynomial ax^x + bx + c. Thus, if Δ > 0, the graph of f(x) = ax^x + bx + c has two x-intercepts at

This means the graph intercepts the horizontal axis at two points: A(x¹, 0) and B(x^x, 0).

On the other hand, if the discriminant Δ is zero, the parabola has a single contact point with the horizontal axis, at the vertex V. If the discriminant is negative, the parabola graph has no intercept points with the horizontal axis.

Given that a parabola is a kind of curve, we need to know more than the above three points to sketch the graph. Another point we can easily find is the y-intercept, which has the x-coordinate equal to zero. The corresponding f(x) for x = 0 is

Therefore, the fourth point (say point C) identified on the parabola is at C(0, c).

We have also used the symmetrical point to C when plotting quadratic graphs given that the x-coordinate of the vertex acts as a midpoint for the x-coordinates of the points C and D. Thus, from the segment midpoint formula

or

The corresponding y-coordinate of point D is found by finding the value of the function for f(x) = f(xD). In other words, we substitute the x-coordinate of point D in the original function and find the corresponding y-value.

Another important element of quadratic functions is the gradient m, which - as explained in tutorial 15.8 - in quadratic functions has the formula m = 2ax + b. Since the gradient of a quadratic function varies depending on the x-value of the tangent point, the coefficient m of the tangent line to the quadratic graph y = mx + n is different at different points of the graph.

Cubic functions

Cubic functions have a general form of

where a, b and c are coefficients while d is a constant.

If factorization is possible, we can write the cubic function as a product of a linear and a quadratic function. This helps identify the zeroes of the cubic function (which is a polynomial function) by solving the equation f(x) = 0. If factorization is not possible, the roots (x-intercepts) are found through iterative methods.

Another thing that regards cubic functions is the gradient. The gradient m of the cubic function

is

Obviously, the gradient is different for different values of x. Since the gradient m gives the steepness of the tangent line to the graph at the given point, it is clear that there is an infinity of possible lines that can be tangent to the graph at different points. The general equation of all such tangents is

where n is a constant.

Another thing we can do with the gradient is to find any local minimum or maximum of the function. Thus, a tangent line y = m(x) must act as a kind of "floor" at the local minimum point of a graph or as a "roof" at the local maximum point, both of these tangent lines must be horizontal. Hence, the gradient must be zero at the local minimum or at the local maximum point of a function. This is true for all functions, including cubic ones.

We say "local" minimum or maximum because the graph may extend to infinity but we are interested in the wiggle formed when the graph line changes direction.

As for the domain and range of cubic functions, they are unlimited unless a restriction is explicitly given in the clues.

Last, we can find the y-intercept by substituting x = 0 in the original function, i.e. by finding f(0).

Exponential functions have the general form

where a and m are coefficients, b is the base and c is a constant.

The base b is often the Euler's number 'e'. In this way, we obtain a special exponential function of the general form

Euler number is an infinite series of fractions where the numerator of all fractions is 1, while the denominator is n! where n is the number a certain term occupies in the series.

Again, in such functions, the domain is unlimited, as the independent variable x can take any value. The range, however, here is limited as, for example, if the base is positive the exponential part cannot be negative. Therefore, the function cannot be smaller than the constant c. Therefore, such functions have a minimum value of c. On the other hand, if the base is negative, the function f(x) can take a negative value for odd values of the index and a positive value for even indices, so the graph is a set of individual points not connected through a solid line.

As for the gradient, minimum, maximum, etc., the procedure is similar as in the functions discussed earlier. The gradient of an exponential function f(x) = ax is m = ax · ln a.

Logarithmic functions

Logarithmic functions f(x) = loga x are the inverse of exponential functions with base a, where the variable x in logarithmic functions is in the part known as an argument. When the base of an exponential function is the Euler number 'e', the corresponding logarithmic equation is called the natural logarithm (in short 'ln') function. Hence, the above pairs of functions are the inverse of each other. The inverse of a function f(x) is denoted as f - 1(x). Using this notation, we can express the relationship between exponential and logarithmic functions as:

The graph of a logarithmic function (but not only; this rule is available for all functions) is symmetrical to its corresponding exponential function with respect to the line y = x.

When dealing with logarithmic functions, we consider only functions with a positive base because this corresponds to exponential functions with a positive base. If the base of an exponential function is positive, the value of the function switches between positive and negative depending on whether the exponent x is even or odd. Hence, we are not able to obtain a graph for this function.

Logarithmic functions have no y-intercepts as they do not touch the Y-axis. Logarithmic functions have only x-intercepts, which for logarithmic functions of the form f(x) = logax is always at (1, 0). Therefore, to plot the graph of a logarithmic function we give other 4-5 values to the variable x, where some are between 0 and 1 and the rest are greater than 1. Then, we connect these points smoothly.

The gradient of the logarithmic function f(x) = loga x is

In this way, we can obtain the equation of the tangent line at any point of the graph as in the previous functions.

As for the domain and range, it is clear that the domain of logarithmic functions includes only the positive numbers. Hence, we have D = (0, + ∞). On the other hand, the range of a logarithmic function has no restrictions; it can be positive, negative or zero depending on the values of the independent variable x.

Square root function

Square root function f(x) = √x is the inverse of the quadratic function f(x) = x^x in the positive section of the y-axis. However, the square root function is not defined in the negative part of the horizontal axis since the square root of negative numbers does not exist in the set of real numbers. If we consider the parent function of each of the two above types, we have

Thus, it is clear that the domain of the parent square root function is D = [0, + ∞).

Despite the fact that a square root contains two values, one positive and one negative, in the square root functions, we accept only the positive values because if we accept both, it would not be a function anymore, given that in functions, an x-value cannot have in correspondence two y-values. Therefore, the range of square root functions is also limited only to the positive region of the Y-axis.

It is clear that any constant added to the parent square root function above makes the graph shift vertically, while any coefficient preceding x when inserted inside the root or preceding the square root when inserted outside the root, stretches or compresses the graph.

The gradient m of the parent square root function f(x) = √x is

When the square root function is given in the form

the corresponding gradient m is

Absolute value function

The absolute value of a number (denoted by two vertical lines aside the number) represents the distance of a number from the origin, no matter in which direction it lies.

From the above description, it is clear that the absolute value function f(x) is identical to the corresponding function without the absolute value symbols g(x) only for f(x) ≥ 0, where the two graphs of f(x) and g(x) are identical. The rest of the function f(x) (where without the symbols of absolute value it would normally give a negative output) has a graph that is obtained by mirroring g(x) on the upper part of the coordinates system.

Despite the fact that the domain of an absolute value function is not restricted, the restriction occurs at the range, as it cannot be negative.

The absolute value function may contain other types of expressions besides linear ones. The procedure, however, is the same as described above.

Reciprocal functions

Reciprocal functions have the variable at the denominator of a fraction. It is known that the general formula of reciprocal functions is

where a, h and k are all numbers. The simplest form of a reciprocal function occurs when h = 0, a = 1 and k = 0. This is called the parent reciprocal function and has the form

The graph of reciprocal functions is called a hyperbola. A hyperbola extends in two quadrants and has two asymptotes: one horizontal and one vertical. They represent the limit values of the function.

The constant h makes the graph shift horizontally by h units while the constant k makes it shift vertically by k units in respect to the reciprocal function y = a/x. The coefficient a instead, makes the graph more distant from the axes when it is greater than 1, although the overall shape of the graph does not change.

The gradient of the parent reciprocal function f(x) = 1/x is

As for the general reciprocal function

its gradient is