Question

Write two different rational functions whose graphs have the same end behaviour as the graph of y=3x^2

Answer 1

Answer:

               $y=x^2+5x+20\\ \\ y=8x^2+35$

Explanation:

The end behavior of a rational function is the limit of the function as x approaches negative infinity and infinity.

Note that the the values of even functions are the same for ± x. That implies that their limits for ± ∞ are equal.

The limits of the quadratic function of general form $y=ax^2+bx+c$ as x approaches negative infinity or infinity, when $a$  is positive, are infinity.

That is because as the absolute value of x gets bigger y becomes bigger too.

In mathematical symbols, that is:

$\lim_{x \to -\infty}3x^2=\infty\\ \\ \lim_{x \to \infty}3x^2=\infty$

Hence, the graphs of any quadratic function with positive coefficient of the quadratic term will have the same end behavior as the graph of y = 3x².

Two examples are:

         $y=x^2+5x+20\\ \\ y=8x^2+35$