Asymptotes

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Answer Key & Explanations

Asymptotes · Grade 12 · Worksheet 3

1. Sophia is a pharmacokineticist modeling the concentration of a new antibiotic in a patient's bloodstream. The concentration function is C(t) = (4t² - 7t + 2) / (t² - 11t + 28), where C(t) is in micrograms per milliliter and t is time in hours after the first dose. Determine all vertical and horizontal asymptotes of this function to understand critical time points and long-term drug behavior. Answer: Vertical asymptotes: t = 4 and t = 7; Horizontal asymptote: C = 4 Solution: Find vertical asymptotes by setting the denominator equal to zero: t² - 11t + 28 = 0. Factor: (t - 4)(t - 7) = 0. So t = 4 and t = 7 are potential vertical asymptotes.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero: t² - 11t + 28 = 0. Factor: (t - 4)(t - 7) = 0. So t = 4 and t = 7 are potential vertical asymptotes. Step 2: Check if the numerator is zero at these points. At t = 4: 4(16) - 7(4) + 2 = 64 - 28 + 2 = 38 (not zero). At t = 7: 4(49) - 7(7) + 2 = 196 - 49 + 2 = 149 (not zero). Since the numerator is not zero at either point, both are vertical asymptotes. Step 3: Find horizontal asymptote by comparing degrees. Numerator degree = 2, denominator degree = 2. Since degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator) = 4/1 = 4. Step 4: Final answer: Vertical asymptotes are t = 4 and t = 7; horizontal asymptote is C = 4.

2. Mere is an astrophysicist analyzing the trajectory of a comet passing near a star. The comet's distance from the star (in millions of kilometers) over time t (in days) is modeled by the rational function D(t) = (4t³ - 11t + 6) / (t² - 25). To predict when the comet might approach dangerously close or escape the star's gravitational influence, Mere needs to find all vertical, horizontal, and slant asymptotes of this function. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 5 and t = -5; Slant asymptote: D = 4t Solution: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 25 = 0 Factor: (t - 5)(t + 5) = 0 So t = 5 and t = -5.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 25 = 0 Factor: (t - 5)(t + 5) = 0 So t = 5 and t = -5. Check that numerator is not zero at these points: At t = 5: 4(125) - 11(5) + 6 = 500 - 55 + 6 = 451 ≠ 0 At t = -5: 4(-125) - 11(-5) + 6 = -500 + 55 + 6 = -439 ≠ 0 Thus, vertical asymptotes at t = 5 and t = -5. Step 2: Determine horizontal or slant asymptote by comparing degrees. Numerator degree: 3 Denominator degree: 2 Since numerator degree (3) is exactly one more than denominator degree (2), there is a slant asymptote (no horizontal asymptote). Step 3: Perform polynomial long division to find the slant asymptote. Divide numerator 4t³ - 11t + 6 by denominator t² - 25. First term: (4t³) / (t²) = 4t Multiply: 4t * (t² - 25) = 4t³ - 100t Subtract: (4t³ - 11t + 6) - (4t³ - 100t) = 89t + 6 Since the remainder (89t + 6) has degree 1, which is less than denominator degree 2, the quotient is 4t. Thus, the slant asymptote is D = 4t. Step 4: Final answer. Vertical asymptotes: t = 5 and t = -5 Slant asymptote: D = 4t

3. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the function C(t) = (3t² + 5t - 2)/(t² - 4), where C(t) represents the concentration in milligrams per liter and t represents time in hours after administration. Determine all vertical and horizontal asymptotes of this function to understand the long-term behavior and critical time points of the medication concentration. Answer: Vertical asymptotes: t = 2, t = -2; Horizontal asymptote: y = 3 Solution: Rational functions can have vertical asymptotes where the denominator equals zero, indicating points where the function value becomes unbounded.
   Full step-by-step solution

   Rational functions can have vertical asymptotes where the denominator equals zero, indicating points where the function value becomes unbounded. Horizontal asymptotes describe the function's end behavior as the input approaches positive or negative infinity. These asymptotes are determined by comparing the degrees of the numerator and denominator polynomials and their leading coefficients, which helps predict long-term trends in mathematical models.

4. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (3t² + 5t - 2)/(t² - 4), where t represents hours after administration. To ensure proper dosage timing, she needs to identify all vertical and horizontal asymptotes of this concentration function. What are the equations of these asymptotes? Answer: x = 2, x = -2, y = 3 Solution: When analyzing rational functions for asymptotes, vertical asymptotes are found by setting the denominator equal to zero and solving, excluding any points that are also zeros of the numerator (which would indicate removable discontinuities instead).
   Full step-by-step solution

   When analyzing rational functions for asymptotes, vertical asymptotes are found by setting the denominator equal to zero and solving, excluding any points that are also zeros of the numerator (which would indicate removable discontinuities instead). Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials - if they have the same degree, the horizontal asymptote is the ratio of the leading coefficients.

5. Aroha is an environmental engineer studying the rate at which a wetland filters pollutants. She models the rate of filtration (in liters per hour) using the rational function R(t) = (5t³ - 3t + 7) / (t² - 1), where t represents time in days since monitoring began. To predict long-term behavior and identify times when the model breaks down, Aroha needs to find all vertical and slant (oblique) asymptotes of this function. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 1 and t = -1; Slant asymptote: R = 5t Solution: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 1 = 0 Factor: (t - 1)(t + 1) = 0 Solutions: t = 1 and t = -1 Check that the numerator is not zero at these points.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 1 = 0 Factor: (t - 1)(t + 1) = 0 Solutions: t = 1 and t = -1 Step 2: Check that the numerator is not zero at these points. At t = 1: numerator = 5(1)³ - 3(1) + 7 = 5 - 3 + 7 = 9 ≠ 0 At t = -1: numerator = 5(-1)³ - 3(-1) + 7 = -5 + 3 + 7 = 5 ≠ 0 So vertical asymptotes exist at t = 1 and t = -1. Step 3: Determine if there is a slant asymptote. Degree of numerator: 3 Degree of denominator: 2 Since 3 = 2 + 1, there is a slant asymptote. Step 4: Perform polynomial long division of (5t³ - 3t + 7) by (t² - 1). Divide the leading term: 5t³ ÷ t² = 5t Multiply: 5t * (t² - 1) = 5t³ - 5t Subtract: (5t³ - 3t + 7) - (5t³ - 5t) = 2t + 7 Since the remainder (2t + 7) has degree 1, which is less than degree 2 of the divisor, the division stops. Step 5: The quotient is 5t, so the slant asymptote is R = 5t. Final answer: Vertical asymptotes at t = 1 and t = -1; Slant asymptote at R = 5t.

6. A rational function is graphed on a coordinate plane. The function is f(x) = (2x² - 5x - 3) / (x² - 9). Describe all vertical and horizontal asymptotes of this function using proper mathematical notation. Answer: Vertical asymptotes: x = 3, x = -3; Horizontal asymptote: y = 2 Solution: When analyzing rational functions, vertical asymptotes are found by setting the denominator equal to zero and solving for x, excluding any values that also make the numerator zero (which would indicate removable discontinuities). If the numerator's degree is less than the denominator's, the…
   Full step-by-step solution

   When analyzing rational functions, vertical asymptotes are found by setting the denominator equal to zero and solving for x, excluding any values that also make the numerator zero (which would indicate removable discontinuities). Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the numerator's degree is less than the denominator's, the horizontal asymptote is y=0. If the numerator's degree is greater, there is no horizontal asymptote (though there may be an oblique asymptote).