Questions and Answers
What is the base of an exponential function?
What is an exponential function?
a function of the form f(x)=ab^cx where a, b, and c are real numbers, b>0, b not = 0
In an exponential function of the form f(x)=abcx, the number b is called the ___.
base
The two restrictions on the value of b are that it must be a positive number and not equal ___.
Signup and view all the answers
No matter what the base, an exponential function of the form f(x)=bx always goes through the point ___.
Signup and view all the answers
No matter what the base, an exponential function of the form f(x)=bx always goes through the point ( ___, b) where b is the base.
Signup and view all the answers
If f(x)=3x and g(x)=7x, then the graph of f(x) will be ___ the graph of g(x) when x>0.
Signup and view all the answers
If f(x)=0.5x and g(x)=0.3x, what can be said about the graph of f(x) compared to g(x) when x is greater than 1?
Signup and view all the answers
To shift the graph of f(x)=a+clogb(dx+g) horizontally, change the parameter ___.
Signup and view all the answers
To shift the graph of f(x)=a+clogb(dx+g) vertically, change parameters ___ or ___.
Signup and view all the answers
To stretch the graph of f(x)=a+clogb(dx+g), change the parameter ___.
Signup and view all the answers
Study Notes
Exponential Functions
- The base of an exponential function, represented as ( f(x) = ab^c x ), is denoted by the letter ( b ).
- An exponential function follows the format ( f(x) = ab^c x ), where ( a, b, c ) are real numbers, ( b > 0 ), and ( b \neq 0 ).
- The value of ( b ) in the exponential function is critical as it determines the growth or decay rate of the function.
- The base ( b ) must be a positive number and cannot be equal to one, ensuring the function behaves correctly.
Graphical Characteristics
- For any exponential function of the form ( f(x) = b^x ), it consistently intersects the point ( (0, 1) ) on the graph.
- The graph also passes through the point ( (1, b) ), indicating that when ( x = 1 ), the function's output equals the base ( b ).
Comparative Analysis of Graphs
- If comparing ( f(x) = 3^x ) and ( g(x) = 7^x ), the graph of ( f(x) ) remains below ( g(x) ) when ( x > 0 ), illustrating that a smaller base yields smaller values in that interval.
- For functions ( f(x) = 0.5^x ) and ( g(x) = 0.3^x ), ( f(x) ) lies above ( g(x) ) when ( x < 1 ), indicating that a higher base results in higher values at lower inputs.
Transformations of Graphs
- To horizontally shift the graph of ( f(x) = a + c \log_b(dx + g) ), the parameter ( g ) should be modified.
- To achieve a vertical shift, the parameters ( a ) or ( d ) must be changed, impacting the overall positioning of the graph.
- The parameter ( c ) is crucial for stretching or compressing the graph vertically, affecting the steepness of the exponential curve.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on key concepts of Algebra II with these flashcards from Unit 8. This quiz covers definitions and essential terms related to exponential functions and their properties. Perfect for reinforcing your understanding and preparation for exams.
