Question

Explain why the graphs of reciprocals of linear functions (except horizontal ones) always have vertical asymptotes,

Answer 1

Answer:

The reason is because linear functions always have real solutions while some quadratic functions have only imaginary solutions

Step-by-step explanation:

An asymptote of a curve (function) is the line to which the curve is converging or to which the curve to line distance decreases progressively towards zero as the x and y coordinates of points on the line approaches infinity such that the line and its asymptote do not meet.

The reciprocals of linear function f(x) are the number 1 divided by function that is 1/f(x) such that there always exist a value of x for which the function f(x) which is the denominator of the reciprocal equals zero (f(x) = 0) and the value of the reciprocal of the function at that point (y' = 1/(f(x)=0) = 1/0 = ∞) is infinity.

Therefore, because a linear function always has a real solution there always exist a value of x for which the reciprocal of a linear function  approaches infinity that is have a vertical asymptote.

However a quadratic function does not always have a real solution as from the general formula of solving quadratic equations, which are put in the form, a·x² + b·x + c = 0 is  $\dfrac{-b \pm \sqrt{b^{2} - 4\cdot a\cdot c}}{2\cdot a}$, and when 4·a·c > b² we have;

b² - 4·a·c < 0 = -ve value hence;

√(-ve value) = Imaginary number

Hence the reciprocal of the quadratic function f(x) = a·x² + b·x + c = 0, where 4·a·c > b² does not have a real solution when the function is equal to zero hence the reciprocal of the quadratic function which is 1/(a·x² + b·x + c = 0) has imaginary values, and therefore does not have vertical asymptotes.