Asymptotes

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Asymptotes · Grade 12 · Worksheet 1

1. Find the horizontal asymptote of the function f(x) = (4x^3 - 2x^2 + 7)/(5x^3 + 3x - 1) as x approaches infinity. Enter the y-value only. Answer: 4/5 Solution: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials. Step 2: For rational functions where the degrees of numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 4 (from 4x^3). Step 4: The leading coefficient of the denominator is 5 (from 5x^3). Step 5: The horizontal asymptote is y = 4/5. The answer is 4/5.

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. f(x) = (7x² + 2x - 7)/(x² - 7x + 12). Find all asymptotes. Answer: Vertical asymptotes: x = 3, x = 4; Horizontal asymptote: y = 7 Solution: Find vertical asymptotes by setting denominator equal to zero: x² - 7x + 12 = 0. Factor: (x - 3)(x - 4) = 0. So x = 3 and x = 4.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting denominator equal to zero: x² - 7x + 12 = 0. Factor: (x - 3)(x - 4) = 0. So x = 3 and x = 4. These are vertical asymptotes because the numerator is not zero at these points (check: at x=3, numerator = 7(9)+2(3)-7 = 63+6-7 = 62, not zero; at x=4, numerator = 7(16)+2(4)-7 = 112+8-7 = 113, not zero). Step 2: Find horizontal asymptote. The degree of numerator is 2 (from 7x²). The degree of denominator is 2 (from x²). Since degrees are equal, the horizontal asymptote is the ratio of leading coefficients: 7/1 = 7. So y = 7. Step 3: Check for slant asymptote. Slant asymptotes occur only when numerator degree is exactly one more than denominator degree. Here degrees are equal, so no slant asymptote. The answer is: Vertical asymptotes: x = 3, x = 4; Horizontal asymptote: y = 7.

4. Aroha is a structural engineer analyzing the stress distribution in a bridge support beam. The stress (in megapascals) as a function of position x (in meters) along the beam is modeled by the rational function S(x) = (4x³ - 9x² + 6x - 1) / (x² - 7x + 12). To ensure the beam's safety, Aroha needs to identify all vertical asymptotes and any slant (oblique) asymptote of this stress function. Determine the equations of all asymptotes for S(x). Answer: Vertical asymptotes: x = 3, x = 4; Slant asymptote: y = 4x + 19 Solution: Find vertical asymptotes by setting the denominator equal to zero and solving for x. Denominator: x² - 7x + 12 = 0. Factor: (x - 3)(x - 4) = 0.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero and solving for x. Denominator: x² - 7x + 12 = 0. Factor: (x - 3)(x - 4) = 0. So x = 3 and x = 4 are candidates. Check if numerator is also zero at these points. For x = 3: 4(27) - 9(9) + 6(3) - 1 = 108 - 81 + 18 - 1 = 44 ≠ 0. For x = 4: 4(64) - 9(16) + 6(4) - 1 = 256 - 144 + 24 - 1 = 135 ≠ 0. Since numerator is non-zero at both, vertical asymptotes are x = 3 and x = 4. Step 2: Determine if there is a horizontal or slant asymptote. Degree of numerator = 3, degree of denominator = 2. Since degree of numerator is exactly one more than degree of denominator, there is a slant asymptote (no horizontal asymptote). Step 3: Find slant asymptote by polynomial long division. Divide 4x³ - 9x² + 6x - 1 by x² - 7x + 12. First term: (4x³)/(x²) = 4x. Multiply: 4x(x² - 7x + 12) = 4x³ - 28x² + 48x. Subtract from numerator: (4x³ - 9x² + 6x - 1) - (4x³ - 28x² + 48x) = 19x² - 42x - 1. Next term: (19x²)/(x²) = 19. Multiply: 19(x² - 7x + 12) = 19x² - 133x + 228. Subtract: (19x² - 42x - 1) - (19x² - 133x + 228) = 91x - 229. The quotient is 4x + 19, with remainder 91x - 229. The slant asymptote is y = 4x + 19. Step 4: Final answer: Vertical asymptotes: x = 3 and x = 4. Slant asymptote: y = 4x + 19.

5. Mere is a pharmaceutical researcher modeling the concentration of a new antibiotic in a patient's bloodstream. The concentration C(t) in milligrams per liter is given by the rational function C(t) = (4t² - 8t + 12) / (2t² - 18), where t is the time in hours after the medication is administered. To ensure patient safety and understand the drug's long-term behavior, Mere needs to identify all vertical and horizontal asymptotes of this function. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 3 and t = -3; Horizontal asymptote: C = 2 Solution: Factor the denominator: 2t² - 18 = 2(t² - 9) = 2(t - 3)(t + 3). Set denominator = 0: 2(t - 3)(t + 3) = 0, so t = 3 or t = -3. Check numerator at t = 3: 4(3)² - 8(3) + 12 = 4(9) - 24 + 12 = 36 - 24 + 12 = 24, not zero.
   Full step-by-step solution

   Step 1: Factor the denominator: 2t² - 18 = 2(t² - 9) = 2(t - 3)(t + 3). Set denominator = 0: 2(t - 3)(t + 3) = 0, so t = 3 or t = -3. Step 2: Check numerator at t = 3: 4(3)² - 8(3) + 12 = 4(9) - 24 + 12 = 36 - 24 + 12 = 24, not zero. Check at t = -3: 4(-3)² - 8(-3) + 12 = 4(9) + 24 + 12 = 36 + 24 + 12 = 72, not zero. So both are vertical asymptotes. Step 3: Compare degrees: numerator degree = 2, denominator degree = 2. They are equal, so horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 4 / 2 = 2. Thus horizontal asymptote is C = 2. Final answer: Vertical asymptotes: t = 3 and t = -3; Horizontal asymptote: C = 2.

6. lim(x→∞) (5x³ - 3x² + 2x - 7)/(2x³ + 4x² - x + 9) = ? Answer: 5/2 Solution: Identify the degrees of numerator and denominator. Both are degree 3.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 3. Step 2: For rational functions where numerator and denominator have the same degree, the limit as x approaches infinity is the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 5 (from 5x³). Step 4: The leading coefficient of the denominator is 2 (from 2x³). Step 5: The limit is 5/2. The answer is 5/2.

7. Liam is analyzing the population growth of a bacterial culture in his biology lab. The population size over time is modeled by the function P(t) = (3t² + 5t - 2)/(t² - 4), where t represents time in hours. As time approaches very large values, what horizontal line does the population approach, and what vertical lines represent time values where the population becomes undefined? Answer: Horizontal asymptote: y = 3; Vertical asymptotes: x = 2 and x = -2 Solution: For rational functions, horizontal asymptotes describe the end behavior as the input grows without bound.
   Full step-by-step solution

   For rational functions, horizontal asymptotes describe the end behavior as the input grows without bound. When the degrees of the numerator and denominator polynomials are equal, the horizontal asymptote is the ratio of the leading coefficients. Vertical asymptotes occur at values that make the denominator zero but not the numerator, representing points where the function is undefined. In real-world contexts like population modeling, these asymptotes provide important information about long-term behavior and limitations of the model.