Reciprocal of Linear and Quadratic Functions

Given \(y = \frac{2}{f(x)}\text{, 2 is actually an }a\text{ value}\) (mapping rule \((af(k(x - p)) + q\to(\frac{1}{k}x + p, ay + q)\))
Given a linear reciprocal \(\frac{1}{kx - c}\) the restriction of not dividing by zero will give \(x\neq\frac{c}{k}\) as a restriction. \(x = \frac{c}{k}\) also becomes the vertical asymptote.

Reciprocal Properties

• The closer a number gets to zero, the larger the reciprocal gets
• The closer a number gets to infinity, the closer to zero the reciprocal gets
• Reciprocating a number will not change its sign
• A reciprocal of a linear will always have a horizontal asmyptote of \(y = 0\)

  Key Features of a Reciprocal Function

• Domain (and sometimes range)
• Behavior near asymptotes
• End behaviors
• Intercepts (x and y)
• Asymptotes (horizontal and vertical)

  Testing Behaviors

  When testing vertical asymptotes, plug in numbers that are slightly larger than the value of the asymptote from both sides. Eg. for \(\frac{1}{x - 3}\) plug in 2.99 and 3.01 to see if the number is positive or negative. If the number is negative that side of the asmyptote shoots down. If the number is positive, that side of the asymptote shoots up.

From the negative side (plugging in 2.99 because \(2.99 < 3\) therefore is \(3^-\)) the function shoots down because plugging in a number from the negative side returns a negative value
From the positive side (plugging in 3.01 because \(3.01 > 3\) therefore is ##3^+$$) the function shoots up because plugging in a number from the positive side returns a positive

To test behaviors as the function approaches \(\pm\infty\) plug in super huge numbers like \(\pm 999\)
Plugging in -999 into the equation returns a number less than 0 (the horizontal asymptote is \(y = 0\)) and plugging in 999 returns a number larger than 0 (horizontal asymptote). Therefore when \(x\to -\infty\) it is below the horizontal asymptote and when \(x\to +\infty\) it is above the horizontal asymptote.

Quadratic Reciprocals

• Quadratics with two zeros will have a reciprocal with two vertical asymptotes splitting the function into three parts
• Number of zeros can be determined using the discriminant
• End behaviors and behaviors near asymptotes can be found in the same way as finding them for linears