Podcast
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?
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz focuses on various transformations of exponential functions with distinct characteristics such as reflections, shifts, and decay. Each flashcard illustrates a function and its transformation properties. Test your understanding of how different parameters affect the graph's behavior.
