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Questions and Answers
What transformation does the function f(x) = -(3)^x undergo?
What transformation does the function f(x) = -(3)^x undergo?
- Vertical shift down 7
- Reflection across the x-axis (correct)
- Growth (correct)
- Decay
What is the horizontal shift of g(x) = 4^(x-2)?
What is the horizontal shift of g(x) = 4^(x-2)?
Right 2 units
What kind of transformation does h(x) = (1/2)^x+3 represent?
What kind of transformation does h(x) = (1/2)^x+3 represent?
- Reflection across the x-axis
- Decay (correct)
- Horizontal shift
- Vertical shift up 3 units (correct)
What transformation does f(x) = (1/2)^-x undergo?
What transformation does f(x) = (1/2)^-x undergo?
What transformations does f(x) = -(2^x) - 7 have?
What transformations does f(x) = -(2^x) - 7 have?
What is the horizontal shift for f(x) = 2^(x+2)?
What is the horizontal shift for f(x) = 2^(x+2)?
F(x) = -(2^x) reflects across the x-axis.
F(x) = -(2^x) reflects across the x-axis.
What is the vertical shift for f(x) = 2^x + 1?
What is the vertical shift for f(x) = 2^x + 1?
What transformations are applied to f(x) = 2^-x+3?
What transformations are applied to f(x) = 2^-x+3?
What is the effect of the transformation for f(x) = 3 ∙ 2^x?
What is the effect of the transformation for f(x) = 3 ∙ 2^x?
What transformation does f(x) = 1/2 ∙ 2^x represent?
What transformation does f(x) = 1/2 ∙ 2^x represent?
What transformations does f(x) = -(1/2)^x - 6 undergo?
What transformations does f(x) = -(1/2)^x - 6 undergo?
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Study Notes
Exponential Functions Transformation Overview
- Exponential functions can undergo various transformations such as reflections, shifts, and stretches.
- General form of exponential functions: f(x) = a * b^(x-h) + k, where:
- a: vertical stretch or compression
- b: base of exponential
- (h, k): horizontal and vertical shifts, respectively
Reflections Across Axes
- Reflection across the x-axis indicates a negative coefficient for the exponential term.
- Example: f(x) = -(3^x) shows reflection across x-axis with base 3, exhibiting growth behavior.
- Reflection across the y-axis occurs when the variable x is negated, as in f(x) = (1/2)^-x, indicating decay.
Vertical and Horizontal Shifts
- A vertical shift moves the graph up or down based on the constant added or subtracted.
- Example: h(x) = (1/2)^x + 3 shifts the graph up 3 units, with a decay base of ½.
- A horizontal shift moves the graph left or right determined by the value within the parentheses.
- Example: g(x) = 4^(x-2) indicates a shift to the right by 2 units, maintaining growth with base 4.
Stretches and Compressions
- Vertical stretch is represented by a coefficient greater than 1 multiplying the function.
- Example: f(x) = 3 * 2^x stretches the graph vertically by a factor of 3.
- Vertical compression occurs when the coefficient is between 0 and 1.
- Example: f(x) = 1/2 * 2^x compresses the graph vertically by a factor of ½.
Combined Transformations
- Transformations can be combined in a single function.
- Example: f(x) = -(2^x) - 7 reflects across the x-axis and shifts down 7 units while exhibiting growth.
- Another example is f(x) = -(1/2)^x - 6, which reflects across the x-axis and shifts down 6 units, implying decay.
Notable Transformations
- g(x) = 4^(x-2):
- Horizontal shift to the right 2 units, base 4, maintains growth.
- f(x) = 2^-x + 3:
- Reflection across y-axis, vertical shift up 3 units showcasing decay.
- f(x) = 2^x + 1:
- A straightforward vertical shift upward by 1 unit with base 2 and growth.
Summary
- Understanding the transformations of exponential functions increases the ability to graph and interpret various forms efficiently.
- Each transformation affects the function's direction, growth/decay behavior, and position in the coordinate system.
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