Exponential Logarithmic Transformations

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Exponential Logarithmic Transformations · Grade 12 · Worksheet 1

1. Dr. Chen is studying the growth of a bacteria culture in her lab. The population P(t) after t hours is modeled by the function P(t) = 500e^(0.15t). Due to limited resources, the culture can only sustain a maximum population of 10,000 bacteria. Determine how many hours it will take for the population to reach this maximum sustainable level, rounding your answer to the nearest tenth of an hour. Answer: 20.0 Solution: P(t) = 500 * e^(0.15t) Maximum sustainable population = 10,000. Set up the equation for when the population reaches 10,000. 10,000 = 500 * e^(0.15t) Divide both sides by 500 to isolate the exponential term.
   Full step-by-step solution

   We are given the population model: P(t) = 500 * e^(0.15t) Maximum sustainable population = 10,000. Step 1: Set up the equation for when the population reaches 10,000. 10,000 = 500 * e^(0.15t) Step 2: Divide both sides by 500 to isolate the exponential term. 10,000 / 500 = e^(0.15t) 20 = e^(0.15t) Step 3: Take the natural logarithm of both sides to solve for t. ln(20) = ln(e^(0.15t)) ln(20) = 0.15t * ln(e) Since ln(e) = 1, we have: ln(20) = 0.15t Step 4: Solve for t. t = ln(20) / 0.15 Step 5: Calculate ln(20). ln(20) ≈ 2.995732274 Step 6: Divide by 0.15. t ≈ 2.995732274 / 0.15 t ≈ 19.97154849 Step 7: Round to the nearest tenth of an hour. t ≈ 20.0 Final answer: 20.0 hours.

2. g(x) = 3·log₂(x + 9) − 4 from f(x) = log₂(x). Identify the transformations. Answer: Vertical stretch by factor 3, horizontal shift left 9 units, vertical shift down 4 units Solution: Start with parent function f(x) = log₂(x). The coefficient 3 outside the log multiplies the output: g(x) = 3·log₂(x + 9) − 4. This is a vertical stretch by a factor of 3.
   Full step-by-step solution

   Step 1: Start with parent function f(x) = log₂(x). Step 2: The coefficient 3 outside the log multiplies the output: g(x) = 3·log₂(x + 9) − 4. This is a vertical stretch by a factor of 3. Step 3: Inside the log, x is replaced by (x + 9). This shifts the graph horizontally. Since it is x + 9, the shift is left by 9 units. Step 4: The constant −4 outside the log subtracts 4 from the output. This is a vertical shift downward by 4 units. Step 5: There are no reflections (no negative sign in front of the log or inside the argument). The answer is: vertical stretch by factor 3, horizontal shift left 9 units, vertical shift down 4 units.

3. Aroha is a sound engineer analyzing the intensity of a sound wave as it passes through a specialized filter. The output intensity I(x) in decibels after the sound has traveled x meters through the filter is modeled by the function I(x) = 12 + 3 log_2(4x - 8). She needs to describe all transformations applied to the parent function f(x) = log_2(x) to obtain I(x). Identify each transformation (horizontal shift, vertical shift, stretch/compression, reflection) and state the values. Answer: Horizontal shift right by 2 units, horizontal compression by factor 1/4, vertical stretch by factor 3, vertical shift up by 12 units. Solution: Start with the parent function f(x) = log_2(x). The given function is I(x) = 12 + 3 log_2(4x - 8). Factor inside the logarithm: 4x - 8 = 4(x - 2).
   Full step-by-step solution

   Step 1: Start with the parent function f(x) = log_2(x). Step 2: The given function is I(x) = 12 + 3 log_2(4x - 8). Step 3: Factor inside the logarithm: 4x - 8 = 4(x - 2). So I(x) = 3 log_2(4(x - 2)) + 12. Step 4: Rewrite in transformation form: I(x) = 3 log_2(4(x - 2)) + 12. Step 5: Identify transformations from f(x) = log_2(x): - The term (x - 2) inside the log means a horizontal shift right by 2 units. - The factor 4 inside the log means a horizontal compression by a factor of 1/4 (since the graph is squeezed horizontally). - The factor 3 outside the log means a vertical stretch by a factor of 3. - The +12 outside means a vertical shift up by 12 units. Step 6: There is no reflection (no negative sign inside or outside the log). The answer is: Horizontal shift right by 2 units, horizontal compression by factor 1/4, vertical stretch by factor 3, vertical shift up by 12 units.

4. Noah is studying the transformation of an exponential function. The base function is f(x) = 7^x. He then considers a new function g(x) = -2 * 7^(x - 8) + 12. Describe the sequence of transformations applied to the graph of f(x) to obtain the graph of g(x), and identify the coordinates of the point on g(x) that corresponds to the point (0, 1) on f(x). Answer: Reflection across x-axis, vertical stretch by factor 2, horizontal shift 8 units right, vertical shift 12 units up; corresponding point is (8, 10). Solution: Identify the transformations. g(x) = -2 * 7^(x - 8) + 12 is of the form a * f(x - h) + k, where a = -2, h = 8, k = 12.
   Full step-by-step solution

   Step 1: Identify the transformations. g(x) = -2 * 7^(x - 8) + 12 is of the form a * f(x - h) + k, where a = -2, h = 8, k = 12. Step 2: The factor a = -2 indicates a vertical stretch by factor 2 and a reflection across the x-axis (because of the negative sign). Step 3: The term (x - 8) indicates a horizontal shift 8 units to the right. Step 4: The +12 indicates a vertical shift 12 units upward. Step 5: To find the corresponding point, start with (0, 1) on f(x). First, apply the horizontal shift: x becomes 0 + 8 = 8, y stays 1. Step 6: Apply the vertical stretch and reflection: multiply y by |a| = 2, then reflect: 1 * 2 = 2, then 2 * (-1) = -2. Step 7: Apply the vertical shift: add 12 to y: -2 + 12 = 10. Step 8: The corresponding point on g(x) is (8, 10). The answer is: Reflection across x-axis, vertical stretch by factor 2, horizontal shift 8 units right, vertical shift 12 units up; corresponding point is (8, 10).

5. Tane is studying the transformation of a logarithmic function. The function f(x) = log_3(x) is transformed to g(x) = -3 log_3(x - 5) + 7. Describe all transformations applied to f(x) to obtain g(x), including the order in which they occur. Answer: Reflection across the x-axis, vertical stretch by a factor of 3, horizontal shift 5 units to the right, vertical shift 7 units up Solution: Start with f(x) = log_3(x). Identify the transformations from g(x) = -3 log_3(x - 5) + 7. The factor of 3 in front of the logarithm indicates a vertical stretch by a factor of 3.
   Full step-by-step solution

   Step 1: Start with f(x) = log_3(x). Step 2: Identify the transformations from g(x) = -3 log_3(x - 5) + 7. Step 3: The factor of 3 in front of the logarithm indicates a vertical stretch by a factor of 3. Step 4: The negative sign in front of the 3 indicates a reflection across the x-axis. Step 5: The (x - 5) inside the logarithm indicates a horizontal shift 5 units to the right. Step 6: The +7 outside the logarithm indicates a vertical shift 7 units up. Step 7: Order of transformations: First, horizontal shift right 5 units. Second, vertical stretch by a factor of 3. Third, reflection across the x-axis. Fourth, vertical shift up 7 units. The answer is: Reflection across the x-axis, vertical stretch by a factor of 3, horizontal shift 5 units to the right, vertical shift 7 units up.

6. Mere is studying the transformation of an exponential function. The parent function is f(x) = 4^x. The transformed function is g(x) = -2 * 4^(x - 5) + 7. Describe all transformations applied to f(x) to obtain g(x), including the order of transformations. Answer: Reflection over the x-axis, vertical stretch by a factor of 2, horizontal shift 5 units to the right, vertical shift 7 units upward. Solution: Identify the parent function: f(x) = 4^x. Write the transformed function: g(x) = -2 * 4^(x - 5) + 7. Compare to the general form: a * f(x - h) + k, where a = -2, h = 5, k = 7.
   Full step-by-step solution

   Step 1: Identify the parent function: f(x) = 4^x. Step 2: Write the transformed function: g(x) = -2 * 4^(x - 5) + 7. Step 3: Compare to the general form: a * f(x - h) + k, where a = -2, h = 5, k = 7. Step 4: Since a = -2, first apply a vertical stretch by a factor of 2 (because |a| = 2), then reflect over the x-axis (because a is negative). These two can be combined as a reflection and stretch. Step 5: The term (x - 5) means a horizontal shift 5 units to the right (since x - h with h = 5). Step 6: The +7 means a vertical shift 7 units upward. Step 7: Order of transformations from parent to g: start with f(x) = 4^x. First, apply the vertical stretch by a factor of 2 and reflection over the x-axis: h(x) = -2 * 4^x. Then, shift horizontally 5 units right: p(x) = -2 * 4^(x - 5). Finally, shift vertically 7 units up: g(x) = -2 * 4^(x - 5) + 7. The answer is: reflection over the x-axis, vertical stretch by a factor of 2, horizontal shift 5 units to the right, vertical shift 7 units upward.

7. Noah is analyzing the transformation of a logarithmic function used to model the brightness of a star as seen through a telescope filter. The original brightness function is f(x) = log₂(x). After calibration, the observed brightness is given by g(x) = -3 log₂(x - 9) + 7. Describe all transformations (shifts, stretches, reflections) that map f(x) onto g(x), and identify the domain of g(x). Answer: Reflection across the x-axis, vertical stretch by a factor of 3, horizontal shift right by 9 units, vertical shift up by 7 units; domain: x > 9 Solution: Identify the parent function: f(x) = log₂(x).
   Full step-by-step solution

   Step 1: Identify the parent function: f(x) = log₂(x). Step 2: The coefficient -3 indicates two transformations: a reflection across the x-axis (due to the negative sign) and a vertical stretch by a factor of 3 (due to the absolute value 3). Step 3: The term (x - 9) inside the logarithm indicates a horizontal shift to the right by 9 units. Step 4: The constant +7 outside the logarithm indicates a vertical shift upward by 7 units. Step 5: To find the domain of g(x), set the argument of the logarithm greater than 0: x - 9 > 0 => x > 9. Step 6: Thus, the transformations are: reflection across the x-axis, vertical stretch by factor 3, horizontal shift right 9, vertical shift up 7; domain: x > 9.