Exponential Logarithmic Graphs

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Answer Key & Explanations

Exponential Logarithmic Graphs · Grade 11 · Worksheet 2

1. Graph f(x) = 6^x and g(x) = log_6(x) on the same coordinate plane. Identify the asymptote of f(x) and the domain of g(x). Answer: Asymptote of f(x): y = 0; Domain of g(x): x > 0 Solution: Graph f(x) = 6^x. This is an exponential growth function. As x approaches negative infinity, 6^x approaches 0, so the horizontal asymptote is y = 0.
   Full step-by-step solution

   Step 1: Graph f(x) = 6^x. This is an exponential growth function. As x approaches negative infinity, 6^x approaches 0, so the horizontal asymptote is y = 0. The y-intercept is at (0, 1). Plot points: (-1, 1/6), (0, 1), (1, 6). Step 2: Graph g(x) = log_6(x). This is the inverse of f(x). The domain is x > 0 because the logarithm of a non-positive number is undefined. The vertical asymptote is x = 0. The x-intercept is at (1, 0). Plot points: (1/6, -1), (1, 0), (6, 1). Step 3: The graphs are reflections of each other across the line y = x. Final answer: The asymptote of f(x) is y = 0. The domain of g(x) is x > 0.

2. Aroha is a geochemist studying the groundwater contamination at a former industrial site. The concentration of a pollutant, in parts per billion (ppb), decreases exponentially over time according to the function C(t) = 1500 * (0.92)^t, where t is the number of years since the contamination source was removed. Aroha needs to determine when the concentration will drop to 30 ppb, the safe drinking water standard. Write an equation that models this situation and solve for the number of years required, expressing your answer in terms of logarithms. Answer: t = log(0.02) / log(0.92) years Solution: Set up the equation using the given function: 30 = 1500 * (0.92)^t Divide both sides by 1500 to isolate the exponential term: 30/1500 = (0.92)^t Simplify the fraction: 0.02 = (0.92)^t Take the logarithm of both sides (any base, but base 10 or natural log is common): log(0.02) = log((0.92)^t) Use…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 30 = 1500 * (0.92)^t Step 2: Divide both sides by 1500 to isolate the exponential term: 30/1500 = (0.92)^t Step 3: Simplify the fraction: 0.02 = (0.92)^t Step 4: Take the logarithm of both sides (any base, but base 10 or natural log is common): log(0.02) = log((0.92)^t) Step 5: Use the power rule of logarithms: log(0.02) = t * log(0.92) Step 6: Divide both sides by log(0.92) to solve for t: t = log(0.02) / log(0.92) Step 7: This is the exact expression in terms of logarithms. The answer is t = log(0.02) / log(0.92) years.

3. Mason is a computer scientist modeling the performance of a new cooling system for a high-performance processor. The temperature T (in degrees Celsius) of the processor after t minutes is modeled by the function T(t) = 27 + 72e^(-0.12t). Sketch the graph of this function, labeling the horizontal asymptote and the y-intercept. What is the long-term temperature of the processor as t approaches infinity? Answer: 27 degrees Celsius Solution: Identify the function: T(t) = 27 + 72e^(-0.12t). This is an exponential decay function shifted vertically. Find the horizontal asymptote.
   Full step-by-step solution

   Step 1: Identify the function: T(t) = 27 + 72e^(-0.12t). This is an exponential decay function shifted vertically. Step 2: Find the horizontal asymptote. As t → ∞, e^(-0.12t) → 0. So T(t) → 27 + 72(0) = 27. The horizontal asymptote is y = 27. Step 3: Find the y-intercept (when t = 0): T(0) = 27 + 72e^(0) = 27 + 72(1) = 99. The y-intercept is (0, 99). Step 4: The graph is a decreasing exponential curve starting at (0, 99) and approaching the asymptote y = 27 as t increases. Step 5: The long-term temperature as t approaches infinity is the horizontal asymptote value: 27 degrees Celsius.

4. Graph f(x) = 3^(x-1) - 2. Identify the horizontal asymptote and the y-intercept. Answer: Horizontal asymptote: y = -2; y-intercept: (0, -5/3) Solution: Start with the parent function f(x) = 3^x. Its horizontal asymptote is y = 0 and y-intercept is (0, 1).
   Full step-by-step solution

   Step 1: Start with the parent function f(x) = 3^x. Its horizontal asymptote is y = 0 and y-intercept is (0, 1). Step 2: The transformation f(x) = 3^(x-1) - 2 involves a horizontal shift right by 1 unit and a vertical shift down by 2 units. Step 3: Horizontal asymptote: The vertical shift moves the asymptote from y = 0 to y = -2. Step 4: y-intercept: Set x = 0. f(0) = 3^(0-1) - 2 = 3^(-1) - 2 = 1/3 - 2 = 1/3 - 6/3 = -5/3. So the y-intercept is (0, -5/3). Step 5: Graph: Plot the y-intercept (0, -5/3), draw the horizontal asymptote y = -2 as a dashed line, and sketch the exponential curve increasing from left to right, approaching the asymptote as x goes to negative infinity. The answer is: Horizontal asymptote y = -2, y-intercept (0, -5/3).

5. Graph f(x) = 9^x and g(x) = log₉(x) on the same coordinate plane. Identify the asymptote and domain of each function. Answer: f(x): asymptote y=0, domain (-∞,∞); g(x): asymptote x=0, domain (0,∞) Solution: Graph f(x) = 9^x. This is an exponential function with base 9 > 1, so it increases rapidly. As x → -∞, 9^x → 0, so the horizontal asymptote is y = 0.
   Full step-by-step solution

   Step 1: Graph f(x) = 9^x. This is an exponential function with base 9 > 1, so it increases rapidly. As x → -∞, 9^x → 0, so the horizontal asymptote is y = 0. The domain is all real numbers (-∞, ∞). Step 2: Graph g(x) = log₉(x). This is the inverse of f(x). The domain is x > 0 because logarithms are only defined for positive arguments. As x → 0⁺, log₉(x) → -∞, so the vertical asymptote is x = 0. Step 3: Plot key points for f(x): (0,1), (1,9), (-1, 1/9). For g(x): (1,0), (9,1), (1/9, -1). The graphs are reflections of each other across the line y = x. The answer is: f(x) has asymptote y = 0 and domain (-∞, ∞); g(x) has asymptote x = 0 and domain (0, ∞).

6. Emma is a financial analyst studying the depreciation of a company's machinery. The value of a machine, V(t) in thousands of dollars, after t years is modeled by the exponential function V(t) = 80(0.75)^t. She needs to create a graph of this function for her report. What is the equation of the horizontal asymptote of V(t), and what does it represent in the context of the problem? Answer: y = 0 Solution: The function is V(t) = 80(0.75)^t, an exponential decay function. The horizontal asymptote is found by examining the behavior as t approaches infinity.
   Full step-by-step solution

   Step 1: The function is V(t) = 80(0.75)^t, an exponential decay function. The horizontal asymptote is found by examining the behavior as t approaches infinity. Step 2: As t increases, (0.75)^t approaches 0 because 0.75 is between 0 and 1. Step 3: Therefore, V(t) = 80 * (something approaching 0) approaches 0. Step 4: The horizontal asymptote is y = 0. Step 5: In context, this means that over a very long time, the machine's value will approach $0, but it will never become exactly $0 (in theory), representing the fact that machinery eventually becomes worthless but retains a nominal value.

7. Graph f(x) = 7^x and g(x) = log_7(x) on the same coordinate plane. Identify the asymptote of each function and state the domain and range of both. Answer: f(x) asymptote: y = 0; g(x) asymptote: x = 0; f(x) domain: (-∞, ∞), range: (0, ∞); g(x) domain: (0, ∞), range: (-∞, ∞) Solution: Graph f(x) = 7^x. This is an exponential function with base 7 > 1, so it increases rapidly. The horizontal asymptote is y = 0 because as x → -∞, 7^x → 0.
   Full step-by-step solution

   Step 1: Graph f(x) = 7^x. This is an exponential function with base 7 > 1, so it increases rapidly. The horizontal asymptote is y = 0 because as x → -∞, 7^x → 0. Domain: all real numbers (-∞, ∞). Range: (0, ∞). Step 2: Graph g(x) = log_7(x). Since logarithms are the inverse of exponentials, g(x) is the reflection of f(x) across the line y = x. The vertical asymptote is x = 0 because log_7(x) is undefined for x ≤ 0. Domain: (0, ∞). Range: all real numbers (-∞, ∞). Step 3: Key points for f(x): (0, 1), (1, 7), (-1, 1/7). Key points for g(x): (1, 0), (7, 1), (1/7, -1). The answer is: f(x) asymptote: y = 0; g(x) asymptote: x = 0; f(x) domain: (-∞, ∞), range: (0, ∞); g(x) domain: (0, ∞), range: (-∞, ∞).