Rational Functions Worksheets Grade 12

1. A civil engineer is designing a suspension bridge where the cable shape follows the rational function f(x) = (2x² - 8x + 6)/(x² - 9), where x represents the horizontal distance from the center of the bridge in meters and f(x) represents the cable height above the roadway. To ensure proper clearance and structural integrity, the engineer needs to determine the cable's long-term behavior and identify any positions where the height becomes undefined. Find all vertical and horizontal asymptotes of this bridge cable function.
2. Consider the rational function f(x) = (3x³ - 2x² + 5x - 1)/(2x³ + 4x - 7). Determine the horizontal asymptote of this function. If the asymptote is a horizontal line, provide the y-value of that line as your answer.
3. lim(x→∞) (5x⁴ - 2x³ + 7x - 1)/(3x⁴ + 4x² - 9) = ?

…and 3 more problems

1. lim(x→∞) (5x⁴ - 3x³ + 2x - 7)/(2x⁴ + 4x² - x + 9) = ?
2. lim(x→∞) (3x² - 2x + 1)/(x² + 5x - 3) = ?
3. Isabella is analyzing the efficiency of a chemical reaction in a new industrial process. The reaction rate, in moles per second, is modeled by the rational function R(x) = (3x² - 7x + 2) / (x² - 7x + 12), where x represents the temperature in hundreds of degrees Celsius. To ensure safe operation, Isabella must identify any temperatures where the reaction rate becomes undefined (vertical asymptotes) and determine the long-term reaction rate as temperature increases (horizontal asymptote). What are the vertical asymptotes and the horizontal asymptote of this reaction rate function? Additionally, describe the behavior of R(x) near the larger vertical asymptote from both sides.

…and 4 more problems

1. Hana is a marine biologist studying the feeding rate of a filter-feeding organism in a tidal estuary. She models the rate at which the organism filters water using the rational function F(t) = (4t² + 8t - 12)/(t² - 16), where F(t) represents the filtration rate in liters per hour and t represents the time in hours after low tide. Hana needs to understand the long-term filtration rate as time increases and identify any times when the model becomes undefined due to tidal changes. Determine all vertical asymptotes and the horizontal asymptote of this filtration rate function, and describe the behavior of F(t) near each vertical asymptote.
2. An environmental engineer is modeling the rate of pollutant absorption in a wetland system using the function R(x) = (2x² - 5x + 3)/(x² - 9), where x represents the concentration of pollutants in parts per million and R(x) is the absorption rate in grams per day. The engineering team needs to understand the system's maximum absorption capacity and identify any pollutant concentrations where the absorption model becomes undefined. Determine all vertical and horizontal asymptotes of this absorption rate function.
3. Consider the graph of the rational function f(x) = (3x^2 - 15) / (x^2 - 25). On a coordinate plane, describe the locations of the vertical asymptotes, the horizontal asymptote, and explain the behavior of the function near each vertical asymptote (including whether the function approaches positive or negative infinity from each side).

…and 2 more problems