Quadratic equation is obtained by equating quadratic function to zero. General form of quadratic equation corresponding to quadratic function is :

a x 2 + b x + c = 0 ; a , b , c ∈ R , a > 0 a x 2 + b x + c = 0 ; a , b , c ∈ R , a > 0

Nature of a given quadratic function is best understood in terms of discriminant, D, of corresponding quadratic equation. This is given as :

D = b 2 − 4 a c D = b 2 − 4 a c

Quadratic equation is obtained by equating quadratic function to zero. Quadratic equation has at most two roots. The roots are given by :

α = - b − D 2 a = - b − b 2 − 4 a c 2 a α = - b − D 2 a = - b − b 2 − 4 a c 2 a

β = - b + D 2 a = - b + b 2 − 4 a c 2 a β = - b + D 2 a = - b + b 2 − 4 a c 2 a

The real roots of the quadratic equation are zeroes of quadratic function. The zeroes of quadratic function are real values of x for which value of quadratic function becomes zero. On graph, zeros are the points at which graph intersects y=0 i.e. x-axis.

Graph reveals important characteristics of quadratic function. The graph of quadratic function is a parabola. Working with the quadratic function, we have :

y = a x 2 + b x + c = a x 2 + b a x + c a y = a x 2 + b x + c = a x 2 + b a x + c a

In order to complete square, we add and subtract b 2 / 4 a 2 b 2 / 4 a 2 as :

⇒ y = a x 2 + b a x + b 2 4 a 2 + c a − b 2 4 a 2 ⇒ y = a x 2 + b a x + b 2 4 a 2 + c a − b 2 4 a 2

⇒ y = a { x + b 2 a 2 - b 2 − 4 a c 4 a } ⇒ y = a { x + b 2 a 2 - b 2 − 4 a c 4 a }

⇒ y + b 2 − 4 a c 4 a = a x + b 2 a 2 ⇒ y + b 2 − 4 a c 4 a = a x + b 2 a 2

⇒ y + D 4 a = a x + b 2 a 2 ⇒ y + D 4 a = a x + b 2 a 2

⇒ Y = a X 2 ⇒ Y = a X 2

Where,

X = x + b 2 a and Y = y + D 4 a X = x + b 2 a and Y = y + D 4 a

Figure 1: The graph is parabola.

Graph of quadratic function

Clearly, Y = a X 2 Y = a X 2 is an equation of parabola having its vertex given by (-b/2a, -D/4a). When a>0, parabola opens up and when a<0, parabola opens down. Further, parabola is symmetric about x=-b/2a.

Maximum and minimum values of quadratic function

The graph of quadratic function extends on either sides of x-axis. Its domain, therefore, is R. On the other hand, value of function extends from vertex to either positive or negative infinity, depending on whether “a” is positive or negative.

When a > 0, the graph of quadratic function is parabola opening up. The minimum and maximum values of the function are given by :

y min = - D 4 a at x = - b 2 a y min = - D 4 a at x = - b 2 a

y max ⇒ ∞ y max ⇒ ∞

Clearly, range of the function is [-D/4a, ∞).

When a < 0, the graph of quadratic function is parabola opening down. The maximum and minimum values of the function are given by :

Figure 2: The graph is parabola, which opens down.

Graph of quadratic function

y max = - D 4 a at x = - b 2 a y max = - D 4 a at x = - b 2 a

y min ⇒ − ∞ y min ⇒ − ∞

Clearly, range of the function is (-∞, -D/4a].

Example 1

Problem : Determine range of f x = - 3 x 2 + 2 x − 4 f x = - 3 x 2 + 2 x − 4

Solution : The determinant of corresponding quadratic equation is :

D = b 2 − 4 a c = 4 − 4 X − 3 X − 4 = 4 − 48 = - 44 ⇒ D < 0 D = b 2 − 4 a c = 4 − 4 X − 3 X − 4 = 4 − 48 = - 44 ⇒ D < 0

a = - 3 ⇒ a < 0 a = - 3 ⇒ a < 0

The graph of function is parabola opening down. Its vertex represents the maximum function value. The maximum and minimum values of function are given by :

⇒ y max = - D 4 a = - - 44 4 X - 3 = - 44 12 = - 11 3 ⇒ y max = - D 4 a = - - 44 4 X - 3 = - 44 12 = - 11 3

y min ⇒ − ∞ y min ⇒ − ∞

Range = (-∞, -11/3)

If D<0, then roots are complex conjugates. It means graph of function does not intersect x-axis. If a > 0, then parabola opens up. The value of quadratic function is positive for all values of x i.e.

D < 0, a > 0 ⇒ f x > 0 for x ∈ R D < 0, a > 0 ⇒ f x > 0 for x ∈ R

Figure 3: The discriminant is negative.

Graph of quadratic function

If a < 0, then parabola opens down. The value of quadratic function is negative for all values of x i.e.

D < 0, a < 0 ⇒ f x < 0 for x ∈ R D < 0, a < 0 ⇒ f x < 0 for x ∈ R

Sign rule : If D<0, then sign of function is same as that of “a” for all values of x in R.

If D=0, then roots are equal and is given by –b/2a. It means graph of function just touches x-axis. If a > 0, then parabola opens up. The value of quadratic function is non-negative for all values of x i.e.

D = 0, a > 0 ⇒ f x ≥ 0 for x ∈ R D = 0, a > 0 ⇒ f x ≥ 0 for x ∈ R

Figure 4: The discriminant is zero.

Graph of quadratic function

If a < 0, then parabola opens down. The value of quadratic function is non-positive for all values of x i.e.

D = 0, a < 0 ⇒ f x ≤ 0 for x ∈ R D = 0, a < 0 ⇒ f x ≤ 0 for x ∈ R

Sign rule : If D=0, then sign of function is same as that of “a” for all values of x in R except at x=-b/2a, at which f(x)=0. We do not associate sign with zero.

If D>0, then roots are unequal and are given by (–b±D)/2a. It means graph of function intersects x-axis at α and β (β>α). If a > 0, then parabola opens up. The value of quadratic function is positive for all values of x in the interval (-∞,α) U (β,∞).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (α,β).

Figure 5: The discriminant is positive and a is positive.

Graph of quadratic function

D > 0, a > 0 ⇒ f x > 0 for x ∈ - ∞ , α ∪ β , ∞ ⇒ Sign of function same as that of “a” D > 0, a > 0 ⇒ f x > 0 for x ∈ - ∞ , α ∪ β , ∞ ⇒ Sign of function same as that of “a”

D > 0, a > 0 ⇒ f x = 0 for x ∈ { α , β } D > 0, a > 0 ⇒ f x = 0 for x ∈ { α , β }

D > 0, a > 0 ⇒ f x < 0 for x ∈ α , β ⇒ Sign of function opposite to that of “a” D > 0, a > 0 ⇒ f x < 0 for x ∈ α , β ⇒ Sign of function opposite to that of “a”

If a < 0, then parabola opens down. The value of quadratic function is positive for all values of x in the interval (α,β).The values of quadratic function are zero for values of x ∈{α,β}. The value of quadratic function is negative for all values of x in the interval (-∞,α) U (β,∞).

Figure 6: The discriminant is positive and a is negative.

Graph of quadratic function

D > 0, a < 0 ⇒ f x < 0 for x ∈ α , β ⇒ Sign of function same as that of “a” D > 0, a < 0 ⇒ f x < 0 for x ∈ α , β ⇒ Sign of function same as that of “a”

D > 0, a < 0 ⇒ f x = 0 for x ∈ { α , β } D > 0, a < 0 ⇒ f x = 0 for x ∈ { α , β }

D > 0, a < 0 ⇒ f x > 0 for x ∈ - ∞ , α ∪ β , ∞ ⇒ Sign of function opposite to that of “a” D > 0, a < 0 ⇒ f x > 0 for x ∈ - ∞ , α ∪ β , ∞ ⇒ Sign of function opposite to that of “a”

Sign rule : If D>0, then domain of function, which is R, is divided at root points in three intervals. The signs of function in side intervals are same as that of “a”, whereas sign of function in the middle interval is opposite to that of “a”.

Figure 7: Signs of function.

Sign rule of quadratic function