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Questions and Answers
What is the base of an exponential function?
What is the base of an exponential function?
- b (correct)
- x
- c
- a
What is 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 ___.
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 ___.
The two restrictions on the value of b are that it must be a positive number and not equal ___.
No matter what the base, an exponential function of the form f(x)=bx always goes through the point ___.
No matter what the base, an exponential function of the form f(x)=bx always goes through the point ___.
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.
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.
If f(x)=3x and g(x)=7x, then the graph of f(x) will be ___ the graph of g(x) when x>0.
If f(x)=3x and g(x)=7x, then the graph of f(x) will be ___ the graph of g(x) when x>0.
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?
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?
To shift the graph of f(x)=a+clogb(dx+g) horizontally, change the parameter ___.
To shift the graph of f(x)=a+clogb(dx+g) horizontally, change the parameter ___.
To shift the graph of f(x)=a+clogb(dx+g) vertically, change parameters ___ or ___.
To shift the graph of f(x)=a+clogb(dx+g) vertically, change parameters ___ or ___.
To stretch the graph of f(x)=a+clogb(dx+g), change the parameter ___.
To stretch the graph of f(x)=a+clogb(dx+g), change the parameter ___.
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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.
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