Exponential Logarithmic Transformations

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Exponential Logarithmic Transformations · Grade 12 · Worksheet 3

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. Answer: The transformations are: a horizontal shift 6 units to the right, a vertical stretch by a factor of 4, and a vertical shift 3 units upward. There is no reflection. The order is: horizontal shift, then vertical stretch, then vertical shift. Solution: Write f(x) = 2^x and g(x) = 4 * 2^(x - 6) + 3. Identify the parameters: The exponent (x - 6) indicates a horizontal shift. Since it is x minus 6, the shift is 6 units to the right.
   Full step-by-step solution

   Step 1: Write f(x) = 2^x and g(x) = 4 * 2^(x - 6) + 3. Step 2: Identify the parameters: The exponent (x - 6) indicates a horizontal shift. Since it is x minus 6, the shift is 6 units to the right. Step 3: The coefficient 4 in front of the exponential indicates a vertical stretch by a factor of 4 (since 4 > 1, it stretches, not compresses). There is no negative sign, so no reflection. Step 4: The +3 at the end indicates a vertical shift upward by 3 units. Step 5: Determine the order: Start with f(x). First, apply the horizontal shift: shift right 6 units to get h(x) = 2^(x - 6). Step 6: Second, apply the vertical stretch: multiply by 4 to get k(x) = 4 * 2^(x - 6). Step 7: Third, apply the vertical shift: add 3 to get g(x) = 4 * 2^(x - 6) + 3. Step 8: The transformations are: horizontal shift 6 units right, vertical stretch by factor 4, vertical shift 3 units up. No reflection is present. The answer is: horizontal shift 6 units right, vertical stretch by factor 4, vertical shift 3 units up, applied in that order.

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). Answer: g(x) = -log_{12}(x/9) + 15; horizontal stretch by factor 9, reflection across x-axis, vertical shift up 15 units Solution: Start with the original function f(x) = log_{12}(x). This replaces x with x/9, giving f_1(x) = log_{12}(x/9). This multiplies the entire function by -1, giving f_2(x) = -log_{12}(x/9).
   Full step-by-step solution

   Step 1: Start with the original function f(x) = log_{12}(x). Step 2: Apply the horizontal stretch by a factor of 9. This replaces x with x/9, giving f_1(x) = log_{12}(x/9). Step 3: Apply the reflection across the x-axis. This multiplies the entire function by -1, giving f_2(x) = -log_{12}(x/9). Step 4: Apply the vertical translation upward by 15 units. This adds 15 to the function, giving g(x) = -log_{12}(x/9) + 15. Step 5: Describe the transformations: the graph of f(x) = log_{12}(x) is stretched horizontally by a factor of 9 (so it becomes wider), then reflected across the x-axis (flipped upside down), and finally shifted upward by 15 units. The answer is g(x) = -log_{12}(x/9) + 15.

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). Answer: Transformations: horizontal compression by factor 1/5, horizontal shift 5 units left, vertical stretch by factor 2. Vertical asymptote: x = -5. Corresponding point: (0, 2). Solution: Start with f(x) = log₅(x). Rewrite g(x) = 2·log₅(5x + 25). Factor inside the log: 5x + 25 = 5(x + 5).
   Full step-by-step solution

   Step 1: Start with f(x) = log₅(x). Step 2: Rewrite g(x) = 2·log₅(5x + 25). Factor inside the log: 5x + 25 = 5(x + 5). So g(x) = 2·log₅(5(x + 5)). Step 3: Using logarithm property: log₅(5(x+5)) = log₅(5) + log₅(x+5) = 1 + log₅(x+5). But to see transformations clearly, we keep it as is. Step 4: The transformations from f(x) = log₅(x) to g(x) = 2·log₅(5(x+5)) are: - Inside: replace x with 5(x+5). This means first a horizontal shift left by 5 units (x → x+5), then a horizontal compression by factor 1/5 (x → 5x). Order: shift then compression. - Outside: multiply by 2, which is a vertical stretch by factor 2. Step 5: The original vertical asymptote of f(x) is x = 0. After horizontal shift left 5: x = -5. Horizontal compression does not move the asymptote further. So new asymptote: x = -5. Step 6: Original point (5, 1) on f(x). Apply transformations to the x-coordinate: first shift left 5: 5 - 5 = 0. Then horizontal compression by factor 1/5: 0 × (1/5) = 0. So x-coordinate becomes 0. Step 7: Apply vertical stretch to y-coordinate: 1 × 2 = 2. So new point: (0, 2). Step 8: Final answer: Transformations: horizontal compression by factor 1/5, horizontal shift 5 units left, vertical stretch by factor 2. Vertical asymptote: x = -5. Corresponding point: (0, 2).

4. A pharmaceutical company is modeling the concentration of a new drug 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. The drug is considered effective when the concentration is at least 25 mg/L and safe when it is below 40 mg/L. During what time interval is the drug both effective and safe? Answer: 1.5 < t < 4.2 Solution: When analyzing drug concentration models, we often need to determine time intervals where the concentration stays within therapeutic ranges.
   Full step-by-step solution

   When analyzing drug concentration models, we often need to determine time intervals where the concentration stays within therapeutic ranges. This involves finding where an exponential function falls between two values, which typically requires solving equations using logarithmic properties. The intersection points define the boundaries of the safe and effective window.

5. Describe the transformations that map f(x) = 9^x to g(x) = -2 * 9^(x - 3) + 11. Answer: Horizontal shift right 3 units, vertical stretch by factor 2, reflection across the x-axis, vertical shift up 11 units. Solution: Start with f(x) = 9^x. The exponent (x - 3) indicates a horizontal shift: x is replaced by x - 3, so the graph shifts right by 3 units.
   Full step-by-step solution

   Step 1: Start with f(x) = 9^x. Step 2: The exponent (x - 3) indicates a horizontal shift: x is replaced by x - 3, so the graph shifts right by 3 units. Step 3: The factor -2 outside the exponential: the negative sign reflects the graph across the x-axis, and the 2 multiplies the output, giving a vertical stretch by a factor of 2. Step 4: The +11 at the end shifts the graph upward by 11 units. Step 5: Combining all: g(x) is obtained from f(x) by shifting right 3, reflecting across x-axis, stretching vertically by 2, and shifting up 11. The answer is: Horizontal shift right 3 units, vertical stretch by factor 2, reflection across the x-axis, vertical shift up 11 units.

6. Describe the transformations that map f(x) = log₆(x) to g(x) = -2log₆(x - 1) + 3. Answer: Reflection in the x-axis, vertical stretch by a factor of 2, horizontal shift right by 1 unit, vertical shift up by 3 units. Solution: Write g(x) in the general transformation form: g(x) = a·log₆(b(x - h)) + k. Here, g(x) = -2log₆(x - 1) + 3. Step 2: Identify a = -2.
   Full step-by-step solution

   Step 1: Write g(x) in the general transformation form: g(x) = a·log₆(b(x - h)) + k. Here, g(x) = -2log₆(x - 1) + 3. Step 2: Identify a = -2. Since a is negative, there is a reflection in the x-axis. The absolute value |a| = 2 indicates a vertical stretch by a factor of 2. Step 3: The argument is (x - 1), so h = 1. This means a horizontal shift to the right by 1 unit. Step 4: The constant term is k = 3, indicating a vertical shift upward by 3 units. Step 5: There is no coefficient b inside the logarithm (b = 1), so no horizontal stretch or compression. The transformations are: reflection in the x-axis, vertical stretch by a factor of 2, horizontal shift right by 1 unit, vertical shift up by 3 units.

7. Describe the transformations that map f(x) = 3^x to g(x) = -3^(x+5) - 7. Answer: Reflection across the x-axis, horizontal shift left 5 units, vertical shift down 7 units Solution: Start with f(x) = 3^x. The negative sign in front of 3^(x+5) indicates a reflection across the x-axis: h(x) = -3^x. The term (x+5) inside the exponent represents a horizontal shift.
   Full step-by-step solution

   Step 1: Start with f(x) = 3^x. Step 2: The negative sign in front of 3^(x+5) indicates a reflection across the x-axis: h(x) = -3^x. Step 3: The term (x+5) inside the exponent represents a horizontal shift. Since it is x+5, the graph shifts left by 5 units: k(x) = -3^(x+5). Step 4: The -7 outside the exponential function represents a vertical shift downward by 7 units: g(x) = -3^(x+5) - 7. Step 5: Combining all transformations: reflection across the x-axis, horizontal shift left 5 units, vertical shift down 7 units. The answer is: reflection across the x-axis, horizontal shift left 5 units, vertical shift down 7 units.