ACT Preparation / Unit 11 ACT 1028 worked example, 60 to 90 seconds
Analyzing a Rational Function
The idea
A rational function is analyzed in four steps. First, factor the numerator and the denominator, and record the domain. Second, cancel common factors: each cancelled zero is a hole. Third, the zeros of the reduced denominator are vertical asymptotes, and the zeros of the reduced numerator are x-intercepts. Fourth, compare degrees for the horizontal asymptote, and find the y-intercept by setting x = 0.
In practice
Worked example: f(x) = (2x² − 2x)/(x² − 1). Step one: 2x(x − 1)/((x + 1)(x − 1)), so x cannot be 1 or −1. Step two: x − 1 cancels, leaving a hole at x = 1. Step three: the reduced form is 2x/(x + 1), with a vertical asymptote at x = −1 and an x-intercept at 0. Step four: equal degrees give y = 2/1 = 2, and the y-intercept is 0.
Where people slip
Find the hole's height from the reduced form: at x = 1, 2(1)/(1 + 1) = 1, so the hole is at (1, 1).
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Unlocks
- Nothing yet depends on this.
Something wrong in this lesson? Flag it · Sources for ACT Preparation