<< Chapter < Page Chapter >> Page >

Graph of quadratic function

The graph is parabola, which opens down.

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

y min

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

Problem : Determine range of 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

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 min

Range = (-∞, -11/3)

Nature of quadratic function

The discriminant of corresponding quadratic equation and coefficient of term “ x 2 ” of quadratic function together determine nature of quadratic function and hence its graph. Graphs of quadratic function is intuitive and helpful to remember results. As a matter of fact, we can interpret all properties of quadratic function, if we can draw its graph.

Case 1 : d<0

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

Graph of quadratic function

The discriminant is negative.

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

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

Case 2 : d=0

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

Graph of quadratic function

The discriminant is zero.

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

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.

Case 3 : d>0

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).

Graph of quadratic function

The discriminant is positive and a is positive.

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 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,∞).

Graph of quadratic function

The discriminant is positive and a is negative.

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 }

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask