Answer to Question #250131 in Calculus for shantel

Question

Can a quadratic function have a range of (- ∞, ∞)? Justify your answer.

b. Discuss the possibilities for the number of times the graphs of two different quadratic functions intersect?

 c. Discuss the circumstances under which the x-intercepts of the graph of a quadratic function are included in the solution set of a quadratic inequality and when they are not included.

Answer 1

a) General form of quadratic function is f(x)=ax²+bx+c, a≠0

Case-1: a>0

As x $\to$ ∞, f(x) $\to$ ∞

As x $\to$ $-$ ∞, f(x) $\to$ ∞

So both left and right sides upward

Case -2: a <0

As x $\to$ ∞, f(x) $\to$$-$ ∞

As x $\to$ $-$ ∞, f(x) $\to$ $-∞$

Again y = a(x + b/2a)²+(4ac-b²)/4a

So for finite values of a, b, c the value of y will extend infinitely in positive sense if a > 0 and in negative sense if a < 0 . As a is either +ve or -ve not both at a time y can not tend to both +∞ and -∞ simultaneously.

So range of a quadratic function can not be (-∞,∞)

b)

Let two quadratic function be

f(x) = ax²+bx+c and g(x)= px²+qx+r

For solving to find point of intersection ,

ax²+bx+c = px²+qx+r

=> (a-p)x² + (b-q)x + (c-r) = 0

Which is a quadratic equation and it can have 1, 2 or zero solutions.

So the graph of two quadratic function may intersect at one point or two points or may not intersect.

c)

Let f(x) = ax²+bx + c

So f(x) = a($x-$ $\alpha)(x-\beta)$

If the quadratic inequality be ax²+bx + c≥0 the

 the x-intercepts of the graph of a quadratic function are included in the solution set of a quadratic inequality if a < 0 and they are not included when a > 0

If the quadratic inequality be ax²+bx + c≤0 then

 the x-intercepts of the graph of a quadratic function are included in the solution set of a quadratic inequality if a > 0 and they are not included when a < 0