Podcast
Questions and Answers
What is the restriction for the value of b in an exponential function?
What is the restriction for the value of b in an exponential function?
In the exponential function $f(x) = a imes b^x$, what does the a-value represent?
In the exponential function $f(x) = a imes b^x$, what does the a-value represent?
Which of the following describes the transformation represented by the function $g(x) = 3^x - 7$?
Which of the following describes the transformation represented by the function $g(x) = 3^x - 7$?
What characteristic of the function $f(x) = 2^x$ indicates its behavior as x approaches negative infinity?
What characteristic of the function $f(x) = 2^x$ indicates its behavior as x approaches negative infinity?
Signup and view all the answers
For the function $f(x) = 5^x + 2$, what does the '+ 2' signify in terms of graph transformation?
For the function $f(x) = 5^x + 2$, what does the '+ 2' signify in terms of graph transformation?
Signup and view all the answers
Which property of exponents expresses the following: $x^{a+b} = x^a imes x^b$?
Which property of exponents expresses the following: $x^{a+b} = x^a imes x^b$?
Signup and view all the answers
In the case of the exponential function $f(x) = a imes b^{x}$, what does the graph typically show in relation to the horizontal asymptote?
In the case of the exponential function $f(x) = a imes b^{x}$, what does the graph typically show in relation to the horizontal asymptote?
Signup and view all the answers
What happens to the graph of an exponential function when it is multiplied by -1?
What happens to the graph of an exponential function when it is multiplied by -1?
Signup and view all the answers
Study Notes
Exponential Functions
- An exponential function is of the form f(x) = a • bx
- Restrictions: a ≠ 0, b > 0, b ≠ 1
- a-value represents the initial value or y-intercept (0, a)
- b-value represents the multiplier/factor or constant ratio
- Exponent properties: xa • xb = xa+b, (xa)b = xab, x-a = 1/xa
Exponential Function Characteristics
-
y-intercept: The point where the graph crosses the y-axis (x = 0)
- For f(x) = a • bx, the y-intercept is (0, a)
-
Horizontal Asymptote: A horizontal line that the graph approaches but never touches
- Equation of the horizontal asymptote: y = a
-
Domain: The set of all possible x-values for the function
- Usually all real numbers, written as (-∞, ∞)
-
Range: The set of all possible y-values for the function
- Usually (0, ∞) if a > 0
- Increasing/Decreasing: Determine if the function values increase or decrease as x increases
Transformations of Exponential Functions
-
Vertical Translations: Adding/subtracting a constant from the function.
- g(x) = f(x) + k (shifts up k units) or g(x) = f(x) - k (shifts down k units)
-
Horizontal Translations: Changing x.
- g(x) = f(x + h) (shifts left h units) or g(x) = f(x - h) (shifts right h units)
-
Vertical Stretches/Compressions: Multiplying the function by a constant (a factor)
- g(x) = a • f(x), if |a| > 1, it stretches vertically, if 0 < |a| < 1, it compresses vertically
-
Horizontal Stretches/Compressions: Changing x by a constant.
- g(x) = f(bx) compresses horizontally if |b| > 1, stretches horizontally if 0 < |b| < 1
-
Reflections: Multiplying the function by -1
- Reflections occur over the x-axis if g(x) = -f(x) or over the y-axis if g(x) = f(-x)
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Test your knowledge on exponential functions with this quiz. It covers the characteristics, properties, and restrictions of exponential functions. Understand concepts like y-intercepts, horizontal asymptotes, and growth behavior.
