Asymptotes

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Answer Key & Explanations

Asymptotes · Grade 12 · Worksheet 2

1. Emma is an environmental engineer analyzing the concentration of a contaminant in a river over time. She models the concentration, C(t), in parts per million, using the rational function C(t) = (5t² + 20t - 15) / (t² - 25), where t represents the time in days since the start of monitoring. To understand the long-term behavior of the contaminant and identify days when the model breaks down, Emma needs to find all vertical and horizontal asymptotes of this function. What are the equations of these asymptotes? Answer: Vertical asymptotes: t = 5 and t = -5; Horizontal asymptote: C = 5 Solution: Find vertical asymptotes by setting the denominator equal to zero: 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: t² - 25 = 0. Factor: (t - 5)(t + 5) = 0. So t = 5 and t = -5. Check that the numerator is not zero at these points: at t = 5, numerator = 5(25) + 20(5) - 15 = 125 + 100 - 15 = 210 ≠ 0; at t = -5, numerator = 5(25) + 20(-5) - 15 = 125 - 100 - 15 = 10 ≠ 0. Both are vertical asymptotes. Step 2: Find horizontal asymptote by comparing degrees. Numerator degree: 2 (leading term 5t²). Denominator degree: 2 (leading term t²). Since degrees are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator = 5/1 = 5. Step 3: Final answer: Vertical asymptotes: t = 5 and t = -5; Horizontal asymptote: C = 5.

2. lim(x→∞) (5x³ - 3x² + 2x - 7)/(2x³ + 4x² - x + 1) = ? Answer: 5/2 Solution: Identify the degrees of numerator and denominator. Both are degree 3. For rational functions where degrees are equal, the limit at infinity equals the ratio of the leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 3. Step 2: For rational functions where degrees are equal, the limit at infinity equals the ratio of the leading coefficients. Step 3: Leading coefficient of numerator is 5, leading coefficient of denominator is 2. Step 4: The limit equals 5/2. The answer is 5/2.

3. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration function is given by C(t) = (3t² + 5t - 2)/(t² - 4), where t represents hours after administration. To determine the long-term behavior of the medication concentration, the engineer needs to find all vertical and horizontal asymptotes of this function. What are the equations of these asymptotes? Answer: Vertical asymptotes: x = 2 and x = -2; Horizontal asymptote: y = 3 Solution: Horizontal asymptotes represent the long-term behavior or equilibrium state of a system, while vertical asymptotes indicate values where the model breaks down or becomes undefined.
   Full step-by-step solution

   Rational functions often model real-world scenarios where quantities approach limiting values. Horizontal asymptotes represent the long-term behavior or equilibrium state of a system, while vertical asymptotes indicate values where the model breaks down or becomes undefined. To find horizontal asymptotes, compare the degrees of the numerator and denominator polynomials. For vertical asymptotes, identify values that make the denominator zero but not the numerator, unless there are common factors that cancel.

4. f(x) = (3x² - 2x + 1)/(x² - 4) = ? Answer: y = 3 Solution: To find the horizontal asymptote of the rational function f(x) = (3x² - 2x + 1)/(x² - 4), we compare the degrees of the numerator and denominator and examine the leading coefficients.
   Full step-by-step solution

   To find the horizontal asymptote of the rational function f(x) = (3x² - 2x + 1)/(x² - 4), we compare the degrees of the numerator and denominator and examine the leading coefficients. Step 1: Identify the degrees of the numerator and denominator. - The numerator is 3x² - 2x + 1. The highest power of x is 2, so the degree is 2. - The denominator is x² - 4. The highest power of x is also 2, so the degree is 2. Step 2: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients. - The leading coefficient of the numerator is 3 (from the term 3x²). - The leading coefficient of the denominator is 1 (from the term x²). Step 3: Calculate the ratio of the leading coefficients. - Ratio = (leading coefficient of numerator) / (leading coefficient of denominator) = 3 / 1 = 3. Step 4: Therefore, the horizontal asymptote is the horizontal line y = 3. This means that as x approaches positive infinity or negative infinity, the value of the function f(x) approaches 3. ANSWER: y = 3

5. Ava, an environmental engineer, is analyzing the rate of water flow through a filtration system. The flow rate (in liters per minute) is modeled by the rational function R(t) = (3t^3 - 2t + 7)/(t^2 - 10t + 21), where t represents time in minutes since the system was activated. To determine the long-term behavior of the flow rate and identify critical moments when the system might fail, Ava needs to find all vertical and slant (oblique) asymptotes of this function. What are the equations of these asymptotes? Answer: Vertical asymptotes: t = 3 and t = 7; Slant asymptote: R = 3t + 30 Solution: Find vertical asymptotes by setting denominator = 0: t^2 - 10t + 21 = 0. Factor: (t - 3)(t - 7) = 0. So t = 3 and t = 7.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting denominator = 0: t^2 - 10t + 21 = 0. Factor: (t - 3)(t - 7) = 0. So t = 3 and t = 7. Check that numerator is not zero at these points: at t=3, numerator = 3(27) - 2(3) + 7 = 81 - 6 + 7 = 82 ≠ 0; at t=7, numerator = 3(343) - 2(7) + 7 = 1029 - 14 + 7 = 1022 ≠ 0. Both are vertical asymptotes. Step 2: Since numerator degree (3) > denominator degree (2), there is a slant asymptote. Perform polynomial long division: (3t^3 + 0t^2 - 2t + 7) divided by (t^2 - 10t + 21). Divide leading terms: 3t^3 / t^2 = 3t. Multiply divisor by 3t: 3t(t^2 - 10t + 21) = 3t^3 - 30t^2 + 63t. Subtract from numerator: (3t^3 + 0t^2 - 2t + 7) - (3t^3 - 30t^2 + 63t) = 30t^2 - 65t + 7. Divide new leading term: 30t^2 / t^2 = 30. Multiply divisor by 30: 30(t^2 - 10t + 21) = 30t^2 - 300t + 630. Subtract: (30t^2 - 65t + 7) - (30t^2 - 300t + 630) = 235t - 623. Quotient is 3t + 30, remainder is 235t - 623. The slant asymptote is the quotient: R = 3t + 30. Final answer: Vertical asymptotes at t = 3 and t = 7; slant asymptote at R = 3t + 30.

6. Noah is a civil engineer studying the structural load distribution on a new bridge design. The load distribution over time is modeled by the rational function L(t) = (6t² + 11t - 1) / (t² - 1), where L(t) represents the load in kilonewtons and t represents time in hours after construction begins. To ensure the bridge's safety, Noah must identify all vertical and horizontal asymptotes of this function, which will help him understand critical time points when the model becomes undefined and the long-term load behavior. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 1 and t = -1; Horizontal asymptote: L = 6 Solution: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 1 = 0. Factor: (t - 1)(t + 1) = 0.
   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. So t = 1 and t = -1. Step 2: Check if the numerator is also zero at these points. For t = 1: numerator = 6(1)² + 11(1) - 1 = 6 + 11 - 1 = 16 ≠ 0. For t = -1: numerator = 6(-1)² + 11(-1) - 1 = 6 - 11 - 1 = -6 ≠ 0. Since the numerator is non-zero at both points, both t = 1 and t = -1 are vertical asymptotes. Step 3: Find horizontal asymptote by comparing degrees. Both numerator and denominator are degree 2 (highest power is t²). When degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator). Leading coefficient of numerator: 6. Leading coefficient of denominator: 1. So horizontal asymptote is L = 6/1 = 6. Step 4: Final answer: Vertical asymptotes at t = 1 and t = -1; Horizontal asymptote at L = 6.