Asymptotes

lessonbunny.com

Answer Key & Explanations

Asymptotes · Grade 12 · Worksheet 3

1. f(x) = (3x³ + 7x² - 5x + 1)/(x² - 9). Find all asymptotes. Answer: Vertical asymptotes: x = 3, x = -3; Slant asymptote: y = 3x + 7 Solution: Find vertical asymptotes by setting the denominator equal to zero. Denominator: x² - 9 = 0. Factor: (x - 3)(x + 3) = 0.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero. Denominator: x² - 9 = 0. Factor: (x - 3)(x + 3) = 0. So x = 3 and x = -3. Check that the numerator is not zero at these points: at x = 3, numerator = 3(27) + 7(9) - 15 + 1 = 81 + 63 - 15 + 1 = 130 (not zero); at x = -3, numerator = 3(-27) + 7(9) + 15 + 1 = -81 + 63 + 15 + 1 = -2 (not zero). So vertical asymptotes at x = 3 and x = -3. Step 2: Determine horizontal or slant asymptote. Degree of numerator = 3, degree of denominator = 2. Since numerator degree > denominator degree, there is no horizontal asymptote. Since numerator degree is exactly one more than denominator degree, there is a slant asymptote. Step 3: Perform polynomial long division to find the slant asymptote. Divide 3x³ + 7x² - 5x + 1 by x² - 9. - Divide leading terms: 3x³ / x² = 3x. Multiply: 3x(x² - 9) = 3x³ - 27x. Subtract: (3x³ + 7x² - 5x + 1) - (3x³ - 27x) = 7x² + 22x + 1. - Divide leading terms: 7x² / x² = 7. Multiply: 7(x² - 9) = 7x² - 63. Subtract: (7x² + 22x + 1) - (7x² - 63) = 22x + 64. The quotient is 3x + 7, and the remainder is 22x + 64. The slant asymptote is y = 3x + 7. Final answer: Vertical asymptotes: x = 3, x = -3; Slant asymptote: y = 3x + 7.

2. Emma is a civil engineer designing a new highway. The project's daily cost (in thousands of dollars) is modeled by the rational function C(t) = (3t³ - 5t + 1) / (t² - 9), where t represents the number of months since construction began. To understand the project's long-term financial trajectory and identify critical months when the cost model becomes undefined, Emma must find all vertical and horizontal or slant (oblique) asymptotes of this function. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 3 and t = -3; Slant asymptote: C = 3t Solution: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 9 = 0 Factor: (t - 3)(t + 3) = 0 Solve: t = 3 or t = -3 Check that numerator is not also zero at these points: At t = 3: numerator = 3(27) - 5(3) + 1 = 81 - 15 + 1 = 67 ≠ 0 At t = -3: numerator = 3(-27) - 5(-3)…
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero. Denominator: t² - 9 = 0 Factor: (t - 3)(t + 3) = 0 Solve: t = 3 or t = -3 Check that numerator is not also zero at these points: At t = 3: numerator = 3(27) - 5(3) + 1 = 81 - 15 + 1 = 67 ≠ 0 At t = -3: numerator = 3(-27) - 5(-3) + 1 = -81 + 15 + 1 = -65 ≠ 0 So vertical asymptotes are t = 3 and t = -3. Step 2: Determine the type of horizontal/slant asymptote by comparing degrees. Numerator degree: 3 (from 3t³) Denominator degree: 2 (from t²) Since numerator degree (3) is exactly one more than denominator degree (2), there is a slant (oblique) asymptote, not a horizontal asymptote. Step 3: Find the slant asymptote by polynomial long division. Divide 3t³ - 5t + 1 by t² - 9. First term: 3t³ ÷ t² = 3t Multiply: 3t × (t² - 9) = 3t³ - 27t Subtract: (3t³ - 5t + 1) - (3t³ - 27t) = 22t + 1 Second term: 22t ÷ t² = 0 (since degree of remainder is less than degree of divisor) The quotient is 3t with remainder 22t + 1. The slant asymptote is given by the quotient: C = 3t. Step 4: Final answer. Vertical asymptotes: t = 3 and t = -3 Slant asymptote: C = 3t

3. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (5t² + 3t - 2)/(t² - 4), where t represents hours after administration. To ensure safe dosage levels, she needs to identify all vertical and horizontal asymptotes of this concentration function. What are the equations of these asymptotes? Answer: Vertical asymptotes: x = 2 and x = -2; Horizontal asymptote: y = 5 Solution: C(t) = (5t² + 3t - 2) / (t² - 4) Vertical asymptotes occur where the denominator is zero (and numerator is not zero at the same point, otherwise it might be a hole).
   Full step-by-step solution

   Let's go step-by-step. We have the function: C(t) = (5t² + 3t - 2) / (t² - 4) --- **Step 1: Identify vertical asymptotes** Vertical asymptotes occur where the denominator is zero (and numerator is not zero at the same point, otherwise it might be a hole). Denominator: t² - 4 = 0 t² = 4 t = 2 or t = -2 Check numerator at t = 2: 5(2)² + 3(2) - 2 = 5(4) + 6 - 2 = 20 + 6 - 2 = 24 ≠ 0 So t = 2 is a vertical asymptote. Check numerator at t = -2: 5(-2)² + 3(-2) - 2 = 5(4) - 6 - 2 = 20 - 8 = 12 ≠ 0 So t = -2 is a vertical asymptote. Vertical asymptotes: t = 2 and t = -2 In equation form: x = 2 and x = -2. --- **Step 2: Identify horizontal asymptote** For a rational function (polynomial / polynomial), compare degrees of numerator and denominator. Numerator degree: 2 (leading term 5t²) Denominator degree: 2 (leading term t²) When degrees are equal, horizontal asymptote is: y = (leading coefficient of numerator) / (leading coefficient of denominator) = 5 / 1 = 5 So horizontal asymptote: y = 5. --- **Final Answer:** Vertical asymptotes: x = 2 and x = -2 Horizontal asymptote: y = 5

4. Isabella is a materials scientist studying the heat dissipation of a new thermal coating. The temperature (in degrees Celsius) of a test panel over time (in minutes) is modeled by the rational function T(t) = (7t² + 15t + 8) / (t² - 11t + 28), where t represents time since the heat source was activated. To predict the panel's long-term temperature and identify critical times when the model fails, Isabella needs to find all vertical and horizontal asymptotes of this function. What are the equations of all asymptotes? Answer: Vertical asymptotes: t = 4, t = 7; Horizontal asymptote: T = 7 Solution: Find vertical asymptotes by setting the denominator equal to zero and solving for t. Denominator: t² - 11t + 28 = 0 Factor: (t - 4)(t - 7) = 0 So t = 4 and t = 7.
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting the denominator equal to zero and solving for t. Denominator: t² - 11t + 28 = 0 Factor: (t - 4)(t - 7) = 0 So t = 4 and t = 7. Check that the numerator is not zero at these t-values: At t = 4: numerator = 7(16) + 15(4) + 8 = 112 + 60 + 8 = 180 (not zero) At t = 7: numerator = 7(49) + 15(7) + 8 = 343 + 105 + 8 = 456 (not zero) Thus, vertical asymptotes at t = 4 and t = 7. Step 2: Find horizontal asymptote by comparing degrees. Numerator degree: 2 (highest power t²) Denominator degree: 2 (highest power t²) Since degrees are equal, the horizontal asymptote is the ratio of leading coefficients. Leading coefficient of numerator: 7 Leading coefficient of denominator: 1 Horizontal asymptote: T = 7/1 = 7 Final answer: Vertical asymptotes are t = 4 and t = 7; horizontal asymptote is T = 7.

5. 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.

6. A rational function is graphed on a coordinate plane with the equation f(x) = (2x² - 7x + 3)/(x² - 5x + 6). Describe all vertical and horizontal asymptotes of this function, providing their equations. Answer: x = 2, x = 3, y = 2 Solution: Find vertical asymptotes by setting denominator equal to zero x² - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2 or x = 3 These are vertical asymptotes since they don't cancel with numerator factors Check if numerator and denominator have common factors Numerator: 2x² - 7x + 3 = (2x - 1)(x - 3)…
   Full step-by-step solution

   Step 1: Find vertical asymptotes by setting denominator equal to zero x² - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2 or x = 3 These are vertical asymptotes since they don't cancel with numerator factors Step 2: Check if numerator and denominator have common factors Numerator: 2x² - 7x + 3 = (2x - 1)(x - 3) Denominator: (x - 2)(x - 3) The (x - 3) factor cancels, but x = 3 is still a vertical asymptote because the original function is undefined there Step 3: Find horizontal asymptote by comparing degrees Both numerator and denominator are degree 2 Horizontal asymptote = ratio of leading coefficients Leading coefficient of numerator: 2 Leading coefficient of denominator: 1 Horizontal asymptote: y = 2/1 = 2 Step 4: Final answer The vertical asymptotes are x = 2 and x = 3 The horizontal asymptote is y = 2