Questions and Answers
What transformation does the function f(x) = -(3)^x undergo?
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 transformation does f(x) = (1/2)^-x undergo?
Signup and view all the answers
What transformations does f(x) = -(2^x) - 7 have?
Signup and view all the answers
What is the horizontal shift for f(x) = 2^(x+2)?
Signup and view all the answers
F(x) = -(2^x) reflects across the x-axis.
Signup and view all the answers
What is the vertical shift for f(x) = 2^x + 1?
Signup and view all the answers
What transformations are applied to f(x) = 2^-x+3?
Signup and view all the answers
What is the effect of the transformation for f(x) = 3 ∙ 2^x?
Signup and view all the answers
What transformation does f(x) = 1/2 ∙ 2^x represent?
Signup and view all the answers
What transformations does f(x) = -(1/2)^x - 6 undergo?
Signup and view all the answers
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.
