![QUADRATIC FUNCTIONS
The equation 𝑓(𝑥) = 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐………………..(A)
Is called quadratic function. What is the difference between the
quadratic equation 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 and the quadratic function
𝑓(𝑥) = 𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐 ? The quadratic equation (A) is valid
for only two values of the variable x, the two roots of the equation 𝛼, 𝛽
for which the value of the expression is 0. But for other values of x,
𝑓(𝑥) = 𝑦 ≠ 0.The independent variable x can take any value from
-∞ to ∞ in this case. The interval ] − ∞, ∞[ is called domain of the
function and the range (or codomain of function) is the set of values y
can take, which depends upon the coefficients a, b and c; here it is a
smaller set ,not interval ] − ∞, ∞[ . Obviously the roots 𝛼, 𝛽 depend
upon the coefficients and vice versa.
We could write the above quadratic function like
𝑎𝑥2
+ 𝑏𝑥 + (𝑐 − 𝑦) = 0,or, 𝑎𝑥2
+ 𝑏𝑥 + 𝐶 = 0,where
𝐶 = 𝑐 − 𝑦; so that for each value of the dependent variable 𝒚 the
quadratic function is a new quadratic equation.
For a particular set of values of a, b and c ( say 5, – 6 and 1 for
example) we can write 𝑓(𝑥) = 𝑦 = 𝑎(𝑥 − 𝛼)(𝑥 − 𝛽)……..(B)
(in the particular case 𝑓(𝑥) = 𝑦 = 5 (𝑥 −
1
5
) (𝑥 − 1)).
The graph of quadratic function is a “curve” in the x-y Cartesian plane
whereas the quadratic equation stands for only two points in the
plane (𝛼, 0) and (𝛽, 0), when we put 𝑦 = 0 in the quadratic function.
These two points are the points where the graph of the quadratic
function cuts the X-axis (interception of the curve with X-axis).](https://image.slidesharecdn.com/lecture2quadraticfunctions-210527025230/85/Lecture-1-2-quadratic-functions-1-320.jpg)
Lecture 1.2 quadratic functions
- 1. QUADRATIC FUNCTIONS The equation 𝑓(𝑥) = 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐………………..(A) Is called quadratic function. What is the difference between the quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 and the quadratic function 𝑓(𝑥) = 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 ? The quadratic equation (A) is valid for only two values of the variable x, the two roots of the equation 𝛼, 𝛽 for which the value of the expression is 0. But for other values of x, 𝑓(𝑥) = 𝑦 ≠ 0.The independent variable x can take any value from -∞ to ∞ in this case. The interval ] − ∞, ∞[ is called domain of the function and the range (or codomain of function) is the set of values y can take, which depends upon the coefficients a, b and c; here it is a smaller set ,not interval ] − ∞, ∞[ . Obviously the roots 𝛼, 𝛽 depend upon the coefficients and vice versa. We could write the above quadratic function like 𝑎𝑥2 + 𝑏𝑥 + (𝑐 − 𝑦) = 0,or, 𝑎𝑥2 + 𝑏𝑥 + 𝐶 = 0,where 𝐶 = 𝑐 − 𝑦; so that for each value of the dependent variable 𝒚 the quadratic function is a new quadratic equation. For a particular set of values of a, b and c ( say 5, – 6 and 1 for example) we can write 𝑓(𝑥) = 𝑦 = 𝑎(𝑥 − 𝛼)(𝑥 − 𝛽)……..(B) (in the particular case 𝑓(𝑥) = 𝑦 = 5 (𝑥 − 1 5 ) (𝑥 − 1)). The graph of quadratic function is a “curve” in the x-y Cartesian plane whereas the quadratic equation stands for only two points in the plane (𝛼, 0) and (𝛽, 0), when we put 𝑦 = 0 in the quadratic function. These two points are the points where the graph of the quadratic function cuts the X-axis (interception of the curve with X-axis).
- 2. A geogebra graph is given below for 𝑦 = 5𝑥2 − 6𝑥 + 1 = 5(𝑥 − 1 5 ) (𝑥 − 1). The graph cuts the X-axis where y = 0. The points are A(1/5,0) and B(1,0), precisely show the roots of the quadratic equation, 1/5 and 1. Only the two roots of the quadratic equation, 𝜶, 𝜷, determine its coefficients a, b and c and the cofficients determine the whole graph of the functon 𝒇(𝒙) = 𝒚 = 𝒂(𝒙 − 𝜶)(𝒙 − 𝜷). This is an example of ‘factor theorem’; if α is a root of the quadratic equation then as 𝑥 = 𝛼 or, 𝑥 − 𝛼 =0, So 𝑥 − 𝛼 is a factor of the quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 and hence it is a factor of the quadratic function 𝑓(𝑥) = 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. Similar statement may be made about the other root and the which gives the Graph of 𝑦 = 5𝑥2 − 6𝑥 + 1 Roots are at A(1/5,0), B(1,0)
- 3. other factor. The fact shall be further clarified from ‘remainder theorem’. The remainder theorem (Euclidean polynomial remainder theorem) says that if a polynomial function 𝑓(𝑥) is divided by a linear polynomial −𝑟 , the remainder is 𝑓(𝑟); i.e., the function value when x is replaced by r. Not only quadratic polynomials, this is valid for polynomials of any positive integer degree. Observe the following: 𝑓(𝑥) 𝑥 − 𝑟 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑥 − 𝑟 = 𝑎𝑥2 − 𝑎𝑟𝑥 + 𝑎𝑟𝑥 + 𝑏𝑥 + 𝑐 𝑥 − 𝑟 = 𝑎𝑥(𝑥−𝑟)+𝑥(𝑏+𝑎𝑟)+𝑐 𝑥−𝑟 = 𝑎𝑥 + (𝑥−𝑟)(𝑏+𝑎𝑟)+𝑟(𝑏+𝑎𝑟)+𝑐 𝑥−𝑟 =𝑎𝑥 + 𝑏 + 𝑎𝑟 + 𝑎𝑟2+𝑏𝑟+𝑐 𝑥−𝑟 = 𝑎𝑥 + 𝑏 + 𝑎𝑟 + 𝑓(𝑟) 𝑥−𝑟 Evidently 𝑎𝑟2 + 𝑏𝑟 + 𝑐 = 𝑓(𝑟), is the value of 𝒇(𝒙) when x is replaced by r. And 𝑎𝑟2 + 𝑏𝑟 + 𝑐 = 𝑓(𝑟), is also the remainder when 𝑓(𝑥) is divided by 𝑥 − 𝑟. The things will be more vivid if 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is divided by 𝑥 − 𝑟 in long division process. This is not only true of quadratic polynomials but all polynomials of higher integer degrees. Let 𝑓(𝑥) be a polynomial of any integer higher degree. Let the quotient be 𝑄(𝑥) when it is divided by 𝑥 − 𝑟 and R be the remainder. We can write 𝒇(𝒙) = 𝑸(𝒙)(𝒙 − 𝒓) + 𝑹. This is division algorithm.
- 4. The general statement of the division algorithm is Let 𝑓(𝑥) be a polynomial of any integer higher degree. Let the quotient be 𝑄(𝑥) when it is divided by 𝑔(𝑥), where degree of 𝑔(𝑥) is less than degree of 𝑓(𝑥) and 𝑅(𝑥) be the remainder and degree of 𝑅(𝑥) is less than that of 𝑔(𝑥) . We can write 𝒇(𝒙) = 𝑸(𝒙)𝒈(𝒙) + 𝑹(𝒙). If 𝑔(𝑥) divides (𝑥) , we et get 𝑓(𝑟) = 𝑄(𝑟). 0 + 𝑅 The factor theorem reduces to be a particular case of the remainder theorem; 𝑓(𝑥) = 0 if 𝑥 = 𝑟 and in any quadratic equation. This finally explains the difference and similarity between the quadratic function and quadratic equation. We can calculate the value of say𝑓(3), for 𝑥 = 3 without any tedious calculation or without undertaking the actual long division process. This would be 𝑓(3) = 𝑎. 32 + 𝑏. 3 + 𝑐 without actually calculating it. Roots of the quadratic function or quadratic equation The roots 𝛼, 𝛽 of the quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 are also called the roots of the quadratic function 𝑦 = 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. As per Fundamental theorem of algebra every polynomial equation has a root. If 𝛼 is a root of 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, then 𝑥 − 𝛼 is a factor of the function 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. Now divide this by 𝑥 − 𝛼 by long
- 5. division process and get 𝑓(𝑥) = (𝑎𝑥 + 𝑎𝛼 + 𝑏)(𝑥 − 𝛼) + 𝑅 , where 𝑓(𝛼) = 𝑅 = 0 by remainder theorem, putting 𝑥 = 𝛼 throughout. Then 𝑓(𝑥) = (𝑎𝑥 + 𝑎𝛼 + 𝑏)(𝑥 − 𝛼), which is also 0 when 𝑎𝑥 + 𝑎𝛼 + 𝑏 = 0. This gives 𝑥 − (−𝛼 − 𝑏 𝑎 ) = 0. Thus −𝛼 − 𝑏 𝑎 is another root of the function 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. So, observe that −𝛼 − 𝑏 𝑎 = 𝛽, the other root of the equation the quadratic equation must have two roots, for 𝛼 + 𝛽 = − 𝑏 𝑎 , as we already know. In some special case to be discussed later, both the roots may be equal or coincident. In this way it could be shown that a polynomial equation of degree n shall have n roots i.e. it may be decomposed into product of n linear factors What the fundamental theorem of algebra means is this. We may multiply any two factors 𝑥 − 𝛼, and 𝑥 − 𝛽 for any two arbitrary numbers 𝛼 and 𝛽 and their product (𝑥 − 𝛼)(𝑥 − 𝛽) = 𝑥2 − (𝛼 + 𝛽)𝑥 + 𝛼𝛽 is a quadratic expression. Fundamental theorem of algebra tells us just the reverse. Given any quadratic expression 𝑝𝑥2 + 𝑞𝑥 + 𝑟, p, q, r being numbers whatsoever, this expression can be expressed as multiplication of two linear 𝒙 − 𝜸 and 𝒙 − 𝜹such as 𝒂(𝒙 − 𝜸)(𝒙 − 𝜹), but the theorem does not tell us anything about how these factors may be found out, nor the theorem tells us about the nature of these roots; what kinds of numbers they may be. This theorem was first proved by Gauss three times in his life time in different ways, although it was known much earlier. This is one of the important proofs of the millennium.
- 6. In this manner we may show that a polynomial of degree n must have at least one root, by fundamental theorem of algebra, and eventually must have n roots, whether distinct or not. We doubly emphasize this point, a quadratic equation cannot have more than two distinct roots, and in the same vein, a polynomial of degree n cannot have more than n distinct roots. If not, see the consequence. If possible, let the quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 have 3 distinct roots, 𝛼, 𝛽, 𝛾 such that 𝛼 ≠ 𝛽 ≠ 𝛾. So we have, 𝑎𝛼2 + 𝑏𝛼 + 𝑐 = 0………………………………..(a) 𝑎𝛽2 + 𝑏𝛽 + 𝑐 = 0………………………………..(b) 𝑎𝛾2 + 𝑏𝛾 + 𝑐 = 0………………………………..(c) Subtracting (b) from (a), 𝑎(𝛼2 − 𝛽2) + 𝑏(𝛼 − 𝛽) = 0 Or, (𝛼 − 𝛽)(𝑎𝛼 + 𝑎𝛽 + 𝑏) = 0 Or, 𝑎𝛼 + 𝑎𝛽 + 𝑏 = 0 since 𝛼 ≠ 𝛽…………………(d) Subtracting (c) from (a), 𝑎(𝛼2 − 𝛾2) + 𝑏(𝛼 − 𝛾) = 0 In the same way we get 𝑎𝛼 + 𝑎𝛾 + 𝑏 = 0 since 𝛼 ≠ 𝛾………(e) Subtracting (e) from (d), we get, 𝑎(𝛽 − 𝛾) = 0………………..(f) If 𝛽 ≠ 𝛾, we have to accept that a = 0, Similarly we can show that b =0 and c = 0. Thus 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 becomes 0.𝑥2 + 0𝑥 + 0 = 0, which is no more a quadratic equation, even it is not an equation. It is true for any value of x, 𝛼, 𝛽, 𝛾, 𝛿, 𝜀, …, anything . We call this is an identity, which is true for every value of the variable. In other words, if a quadratic equation has three distinct roots, it has infinite number of different roots.
- 7. Example 1 Show that 𝑎2−𝑥2 (𝑎−𝑏)(𝑎−𝑐) + 𝑏2−𝑥2 (𝑏−𝑐)(𝑏−𝑎) + 𝑐2−𝑥2 (𝑐−𝑎)(𝑐−𝑏) − 1 = 0 is an identity. Solution:, Put x = a in the expression and see that it reduces to 0. So a is a root of the equation. Similarly show that b and c are also roots of the equation. Since we have not assumed 𝑎 = 𝑏 = 𝑐, they are different from each other and the quadratic equation has 3 different roots. So it must be identically 0. Example 2 Show that 𝑎2(𝑥−𝑏)(𝑥−𝑐) (𝑎−𝑏)(𝑎−𝑐) + 𝑏2(𝑥−𝑐)(𝑥−𝑎) (𝑏−𝑐)(𝑏−𝑎) + 𝑐2(𝑥−𝑎)(𝑥−𝑏) (𝑐−𝑎)(𝑐−𝑏) − 𝑥2 = 0 is an identity. Solution:, Put x = a in the expression and see that it reduces to 0. So a is a root of the equation. Similarly show that b and c are also roots of the equation. Since we have not assumed 𝑎 = 𝑏 = 𝑐, they are different from each other and the quadratic equation has 3 different roots. So it must be identically 0.