Exponential Logarithmic Transformations Worksheets Grade 12

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.
2. g(x) = 3·log₂(x + 9) − 4 from f(x) = log₂(x). Identify the transformations.
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.

…and 4 more problems

1. A solid is formed by rotating the region bounded by the curve y = e^x, the x-axis, the y-axis, and the vertical line x = 2 about the x-axis. Using calculus, determine the volume of this three-dimensional solid. (Express your answer in terms of π and e.)
2. Describe the transformations that map f(x) = log₁₀(x) to g(x) = -2 log₁₀(1/6 x + 1) + 1.
3. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is given by the function C(t) = 80e^(-0.15t) - 20e^(-0.4t), where t is time in hours. Determine the time at which the medication concentration reaches its maximum value, and find that maximum concentration.

…and 5 more problems

1. Mere is analyzing the transformations of the function f(x) = 2^x to obtain the function g(x) = 4 * 2^(x - 6) + 3. Describe in words all the transformations (including shifts, stretches/compressions, and reflections) that must be applied to the graph of f(x) to obtain the graph of g(x), and specify the order in which they are applied.
2. Mason is analyzing the graph of the logarithmic function f(x) = log_{12}(x). He then applies a sequence of transformations to obtain a new function g(x): first, a horizontal stretch by a factor of 9, then a reflection across the x-axis, and finally a vertical translation upward by 15 units. Write the equation of g(x) in terms of x and describe how the original graph of f(x) is transformed to produce g(x).
3. Consider the graph of the function f(x) = log₅(x). This graph is then transformed to produce g(x) = 2·log₅(5x + 25). Describe the sequence of transformations (in order) applied to f(x) to obtain g(x). Then, identify the new vertical asymptote and the coordinates of the point on g(x) that corresponds to the point (5, 1) on f(x).

…and 4 more problems