Forms Of The Quadratic-part 1

To aid our understanding of this topic there are some key words that need to be defined. Throughout this article these key words are highlighted in bold italics.

. Base: A quantity that is raised to a power. For example, in , we say that 2 is raised to the power of 3. Therefore 2 is the base. Also in (read as 'x squared') x is raised to the power 2, so that x is the base. See power. . Coefficient: The prefixing number (part) of a term that is made up of the combination of a number and one or more letters/symbols (variables). The coefficient tells of the number of times the variable (s) was added together. Consider for example, the expression . In the term , 3 is the coefficient of . Therefore means . Also in the term 5xy, 5 is the coefficient of xy. Thus 5xy means . Therefore . Compare this definition with that of power. . Constant: A purely number term of an expression. Also called absolute term. Since constants are just numbers with no attached variables, they are said to be independent of variables. Thus constants are terms that are independent of variables.

. Degree:

1.(of terms) The sum of the powers of all variables in a term is the degree of that term. Consider, for example the term . There is only one variable, x, and its power is 2. Therefore the degree of is 2. Now consider the term . There are three variables: x, y and z with respective powers of 2, 1 and 4. Therefore the degree of is 2 + 1 + 4 = 7. Notice that 3 which is the power of 2 is not taken into the degree since the base 2 is not a variable. This shows that the degree of a term is defined in terms of powers of variables only. Finally, consider the term 17, which is only a number. Since there are no attached variables, the degree of 17 is 0. All terms that are numbers only, have degrees of 0. 2.(of expressions) If an expression is only one term, then the degree of that expression is the degree of that one term. If the expression has two or more terms, then its degree is same as that of the term with the greatest degree. Consider, for example, the expression . The terms are , and 2. Their respective degrees are 2, 1 and 0. Therefore the greatest degree is 2. Hence the degree of the expression is 2. See leading term. . Expression: A mathematical statement that is made up of one or more terms. The terms of an expression are separated one from another by either a plus sign (+) or a minus sign (-). For example, is an expression of three terms. . Function: A rule for assigning one set of numbers to another set of numbers. Generally the rule is stated in the form of an expression which has led to the interchangeably use of the two words. . Leading coefficient: The coefficient of the leading term. See coefficient and leading term, . Leading term: In an expression, the term with the greatest degree is called the leading term. See degree. . Power: Conventionally, the power of a quantity is written as a superscri pt immediately behind the quantity. The power tells of the number of times the quantity was multiplied together with itself. For example, (read as '2 cubed' or as '2 raised to the power of 3') tells us that the 2 was multiplied together with itself three times. Thus . Also in (read as 'x squared' or as 'x is raised to the power 2'), x is being multiplied together with itself two times. So that . Compare this definition with that of coefficient. See also base. . Term: The building block of expressions. A term may either be a number, a combination of one or more letters/symbols (variables) or a combination of numbers and letters/symbols (variables). There are two types of terms - (i) constant and (ii) variable. . Variable: A letter/symbol that can take on different values based on some predefined set of numbers. Usually variables are represented by the letters x, y or z.

A quadratic is an expression/function of the second degree (i.e. degree 2). The general form of the quadratic expression, in x (x is the variable), is ; where a, b and c are any numbers, with the restriction that . By definition, it is the presence of the term that makes the expression quadratic. So while b and c can be any numbers, including 0, a cannot be 0, since which destroys the term, thereby destroying the quadratic nature of . Consider now a particular case of the quadratic expression where a = 5, b = 3 and c = -2. That is, the particular quadratic expression . Let the variable x be a real number with values from -2 to 2. Mathematicians abbreviate the last sentence above by writing it as: , x ? R. Now for each of the predefined values of x, we can calculate a corresponding value for . So when x = -2, the corresponding calculated value for is 5(-2)2 + 3(-2) - 2 = 20 - 6 - 2 = 12. When x = -1, the corresponding calculated value for is 5(-1)2 + 3(-1) - 2 = 5 - 3 - 2 = 0. When x = 0, the corresponding calculated value for is 5(0)2 + 3(0) - 2 = 0 + 0 - 2 = -2 . and so on. A conveniently neat way of displaying values of the variable x, along with their corresponding calculated values of , is by using a table of values. Table 1 below is the table of values for the expression for x = -2, -1 0, 1, 2. Table 1 X -2 -1 0 1 2 5x2 20 5 0 5 20 +3x -6 -3 0 3 6 -2 -2 -2 -2 -2 -2 12 0 -2 6 24

It is clearly seen then, that different values of x produce correspondingly different calculated values of the expression . Therefore the value of varies in relation to the values of the variable x. Hence is a variable, the value of which is dependent on the variable x. For this reason is often called a dependent variable, while x is generally referred to as the independent variable. It is quite common (almost conventional) to represent the dependent variable, , by the letter y. With this convention, we invariably write . Another important concept that is commonly used by mathematicians, almost daily, is that of a function. To grasp something of this concept, let us return to our quadratic expression . We have seen that the expression is itself a variable whose value depends on x. An alternative way of saying this is that is a function of x; and we summarise this as . Viewed as a function, the expression is seen as a rule that tells us how values of x are to be treated in order to obtain corresponding final results. For example, the rule of our expression might be stated as: 'For every number, x, add 3 times the number to 5 times the square of the number itself; then subtract 2'. A diagrammatic representation of this rule is shown below as a flow diagram, commonly referred to as a function machine.

Figure 1: The function machine for the expression .

Notice that the function effectively maps values of the independent variable, x, to corresponding values of the dependent variable, y. This mapping of values of x onto their corresponding y values under a function is mathematically depicted as . is read either as 'f at x is mapped unto y' or as 'f maps x unto y'.

We have now arrived at the place where we can represent the information in table 1 above in the language of function. So let f be the function that it is defined by . Since is a quadratic expression, then the function f is called a quadratic function. This point is worth generalising: functions are named based on the expressions on which they are defined. That being said, we now return to the place of representing the information in table 1 in functional notation. From the table we see that when x = -2, ; or when x = -2, when x = -1, ; or when x = -1, when x = 0, ;or when x = 0, ; etc..

Using functional notation the above information is presented as follows: ; or ; or ; or etc.

Orderly pairing each x value with its respective corresponding y value as (x, y), we have the following ordered pairs: (-2, 12), (-1, 0), (0, -2), (1, 6), (2, 24). Each of these ordered pairs identifies a unique point in the x-y plane and is referred to as the coordinates of the point. The first entry in each ordered pair is called the x-coordinate and the second entry, the y-coordinate. Plotting these points and connecting them for real values of x between -2 and 2, inclusive, produces the graph in figure 2 below.

Figure 2 We note several features of the graph: 1. The graph cuts the y-axis at the point (0, -2). This point is the y-intercept. Notice that the x-coordinate at the y-intercept is 0. This means that the y-coordinate of the y-intercept is the value of the expression/function when x = 0. This is always true at every y-intercept - y-coordinates of all y-intercepts are values of the expressions/functions when x = 0. Therefore the equation of the y-axis is x = 0. 2. The graph cuts the x-axis at points (-1, 0) and (0.4, 0). Therefore the points (-1, 0) and (0.4, 0) are called x-intercepts. Notice that the y-coordinates at the x-intercepts are all zeros. This is always true - all x-intercepts have corresponding y values of zero. For this reason, the equation of the x-axis is y = 0. The greater implication of this is that the expression/function has values of 0 at the x-coordinates of the x-intercepts. 3. The lowest point on the graph is (-0.3, -2.45). This lowest point is called the minimum turning point or minimum vertex of the graph of . The y-coordinate of this point is the smallest (minimum) value that the expression/function, , can have. That is, the smallest (minimum) value of is -2.45. The x-coordinate of the minimum turning point is the value of x which when input into the expression/function, , produces this corresponding y-coordinate. Therefore inputting x = -0.3 into gives . 4. The graph has an axis of symmetry which runs vertically through the vertex. For the graph of the expression 5x2 + 3x - 2, the equation of the axis of symmetry is x = -0.3. All graphs of the form are of similar shape (like the shape of your tongue) and are known as parabolas. They all have axes of symmetry running through their vertices, which can either be a minimum point, as in the above example, or a maximum point. However, not all parabolas will intercept the x-axis in two distinct places as in the example of the graph of y = 5x2 + 3x - 2. Some parabolas may touch the x-axis at only a single point while others may never at all intercept the x-axis. See figure 3. Figure 3 I. Graph of with minimum turning point. Parabola lies wholly above the x-axis so there are no x-intercepts. II. Graph of with minimum turning point. Parabola intercepts x-axis at one point only so there is only one x-intercept, (?, 0). III. Graph of with minimum turning point. Parabola intercepts x-axis at two points so there are two x-intercepts, (?, 0) and (?, 0). IV. Graph of with maximum turning point. Parabola lies wholly below the x-axis so there are no x-intercepts. V. Graph of with maximum turning point. Parabola intercepts x-axis at one point only so there is only one x-intercept, (?, 0). VI. Graph of with maximum turning point. Parabola intercepts x-axis at two points so there are two x-intercepts, (?, 0) and (?, 0).

These six diagrams of figure 3 exhaust all the possibilities of the position of the graph of in relation to the x-axis. Either the graph of will: i) have a maximum or a minimum turning point with no x-intercept; ii) have a maximum or a minimum turning point with only one x-intercept; iii) have a maximum or a minimum turning point with two x-intercepts. However, the diagrams raise pertinent questions. 1. Why is there no y-axis in each diagram, and what, if any, is the relation between the graph of and the y-axis? 2. How can we know whether the graph of will have a maximum turning point or a minimum turning point? 3. Can we by looking at the equation know whether or not the graph will intercept the x-axis? 4. If the graph does cross the x-axis, is there a simple way of determining the x-intercept(s) from its equation ? Let's start by answering question 2 first. By plotting and drawing a large number of graphs of the form , experience has revealed that these graphs will have minimum turning points whenever the coefficient of , a, is a positive number (such as, for example: 0.1, ½, 1 or 5). On the other hand, the graphs of will have maximum turning points whenever the coefficient of , a, is a negative number (such as, for example: -0.1, -½, -1 or -5). An analytical proof of this fact will be given later when we learn how to express the quadratic in another form. For now however, we take it as gospel that the graph of will have a minimum turning point if a is positive; but will have a maximum turning point if a is negative. Thus in the graph of will have a minimum turning point since the coefficient of is 5, which is a positive number. (See figure 2).

Now to answer question 3. By simply looking at the equation one will not get any clue as to whether or not its graph will intercept the x-axis. However, there is a simple calculation involving the numbers a (the coefficient of ), b (the coefficient of ) and c (the constant term) that does provide such a clue. The calculation is an analytical conclusion reached by writing the quadratic in another form. This calculation is so important that it is given the name discriminant and is represented by the symbol ? (delta, the Greek capital letter D). It will be later proven that the discriminant, ?, is given by . . If has a negative value then the graph of will not intersect the x-axis (figures 3I and 3IV). . If is equal to zero then the graph of will intersect the x-axis at a single point only (figures 3II and 3V). . If has a positive value then the graph of will intersect the x-axis at two distinct points (figures 3III and 3VI). Returning to our example and comparing it to the general form of the quadratic, , we have , and . Hence , which is a positive value. Therefore the graph of will have two distinct x-intercept. (See figure 2).

Question 4 asks if there is a simple way of finding the point(s) where the graph of the parabola intersects the x-axis, if such a point does exist. There are two methods for finding exactly where the graph of intersects the x-axis; but whether any of them is considered simple is left for the reader to later decide. These two methods are based on expressing in the two other forms of the quadratic. (We will study these two forms of the quadratic later). From one of these forms we obtain a formula into which we can directly input the numbers a, b and c and come up exactly with the x-intercept. The formula is: or , since . The ± sign in the formula tells us that there should be two x-intercepts, at most, which are separately given as and . Notice the square root of the discriminant in these formulae. In answering question 3 above, it was stated that: . If is negative, then the graph of will not intersect the x-axis. This is because if is negative then its square root is undefined. . If is zero, then its square root is also zero. Hence the two x-intercepts and coincide to become one and equal, . . If is positive, then the x-intercepts are distinctly given by and . From figure 2 we saw that the graph of has x-intercepts (-1, 0) and (0.4, 0). We will now use the formulae to arrive at these same results. Recall that in we have , and , so that , which is positive and therefore there are two distinct x-intercepts given by and . Hence the x-intercepts are and ; That is, and ; That is, = (-1, 0) and = (0.4, 0) ? Now let's answer question 1: "Why are there no y-axis in each of the graphs of figure 3?" The simple, yet truthful answer is, "Primarily to focus our attention on the relationship between the graph of and the x-axis". "Is there any relation between the graph of and the y-axis?" As stated before, the graph of has a turning point. In relation to the y-axis the turning point may be to its left (where x is negative), to its right (where x is positive) or on it (where x = 0). It can be shown (and later will be) that the x-coordinate of the turning point is given by . Hence in the graph of if: . a and b are of the same sign (that is, either both negative or both positive) then the turning point will be on the left of the y-axis. . a and b are of opposite signs then the turning point will be on the right of the y-axis. . b = 0 then the turning point will be on the y-axis. Thus the y-axis will be the axis of symmetry. Consider again the equation with , and . Since a and b are both of the same sign then the turning point of the graph of will be on the left of the y-axis (that is, the turning point is on the negative side of the x-axis). More precisely, the x-coordinate of the turning point as given by is (see figure 2). Once the x-coordinate of the turning point is obtained, its corresponding y-coordinate can be calculated by substituting it (the x-coordinate) into the equation . Thus substituting x = -0.3 into the equation gives the corresponding y-coordinate of the turning point as . Hence the turning point of the graph of is (-0.3, -2.45). Another point of interest between the graph of and the y-axis is the y-intercept. It is easy to deduce this point by simply looking at the graph's equation in the general form . We know that the y-intercept has its x-coordinate equal to zero, then its y-coordinate is simply equal to c, the constant term in the equation . Hence all graph of have y-intercept of (0, c). In our example of the graph of , the y-intercept is (0, -2). In closing we summarise the main points in table 2 below. Table 2 General form Example

. Has maximum/minimum turning point if a is negative/positive. . Has a minimum turning point since a = 5 is positive. . Has: i. no x-intercept if is negative. ii. only one x-intercept if . iii. two distinct x-intercept if is positive. . Has two distinct x-intercept since is a positive value.

. If x-intercepts exist, they are given by . . Has x-intercepts and .

. Has y-intercept at (0, c). . Has y-intercept at (0, -2). . The x-coordinate of the turning point is . . The x-coordinate of the turning point is .

. The corresponding y-coordinate of the turning point is found by substituting into . . The corresponding y-coordinate of the turning point is found by substituting into . So that