Podcast
Questions and Answers
What is the base of an exponential function in the form f(x)= ab^cx?
What is the base of an exponential function in the form f(x)= ab^cx?
- x
- c
- b (correct)
- 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 ≠1
In an exponential function of the form f(x)= ab^cx, the number b is called the ____.
In an exponential function of the form f(x)= ab^cx, the number b is called the ____.
base
What are the restrictions on the value of b in an exponential function?
What are the restrictions on the value of b in an exponential function?
No matter what the base, an exponential function of the form f(x)=b^x always goes through the point ___.
No matter what the base, an exponential function of the form f(x)=b^x always goes through the point ___.
An exponential function of the form f(x)=b^x always goes through the point ( ___, b), where b is the base.
An exponential function of the form f(x)=b^x always goes through the point ( ___, b), where b is the base.
If f(x)=3^x and g(x)=7^x, then the graph of f(x) will be ___ the graph of g(x) when x > 0.
If f(x)=3^x and g(x)=7^x, then the graph of f(x) will be ___ the graph of g(x) when x > 0.
If f(x)=0.5^x and g(x)=0.3^x, then the graph of f(x) will be _____ the graph of g(x) when x < 0.
If f(x)=0.5^x and g(x)=0.3^x, then the graph of f(x) will be _____ the graph of g(x) when x < 0.
What is the mathematical constant e approximately equal to?
What is the mathematical constant e approximately equal to?
For an exponential function f(x)= ab^cx, changing the value for a will change the ____ to the value of a.
For an exponential function f(x)= ab^cx, changing the value for a will change the ____ to the value of a.
For an exponential function f(x)= ab^cx, changing the value for c will change the ____ b to b^c.
For an exponential function f(x)= ab^cx, changing the value for c will change the ____ b to b^c.
How can you find the inverse of a function?
How can you find the inverse of a function?
The graph of the inverse of a function may be found by ___ over the line y=x.
The graph of the inverse of a function may be found by ___ over the line y=x.
The inverse of an exponential function tells us the ___ to which the base of an exponential function is raised to give us the value x.
The inverse of an exponential function tells us the ___ to which the base of an exponential function is raised to give us the value x.
What is a common logarithm?
What is a common logarithm?
What is a logarithm?
What is a logarithm?
What is a logarithmic function?
What is a logarithmic function?
What is a natural logarithm?
What is a natural logarithm?
The number, y so that b^y=x is called the ____ of x.
The number, y so that b^y=x is called the ____ of x.
The base of the logarithm function cannot be equal to ________ and must be _____________.
The base of the logarithm function cannot be equal to ________ and must be _____________.
The greater the base, the closer the graph is to the ___ when x>1.
The greater the base, the closer the graph is to the ___ when x>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
- Exponential functions take the form f(x)= ab^cx, where a, b, and c are real numbers, with b being positive and not equal to 1.
- The base of an exponential function, denoted as b, is fundamental to the function's characteristics.
- Exponential functions always pass through the point (0, 1), regardless of the base.
- For any base b, an exponential function will pass through the point (1, b).
Characteristics of Exponential Graphs
- If two exponential functions, such as f(x)=3^x and g(x)=7^x, are compared, f(x) will be below g(x) for x > 0.
- Conversely, if f(x)=0.5^x and g(x)=0.3^x, then f(x) will be above g(x) when x < 0.
Important Constants
- The mathematical constant e is approximately equal to 2.71828 and is essential in exponential functions.
Function Modifications
- In the function f(x)= ab^cx:
- Changing the value of a affects the y-intercept, moving it to the value of a.
- Altering c changes the base to b^c.
Inverse Functions
- To find the inverse of a function, interchange the numbers in each ordered pair.
- The graph of an inverse function can be derived by reflecting the original graph over the line y=x.
- The inverse of an exponential function reveals the exponent needed to raise the base to achieve a given value, x.
Logarithmic Functions
- A common logarithm has a base of 10 and is denoted as log x.
- The logarithm logb x is defined such that b^y = x, where y represents the logarithm of x with base b.
- Logarithmic functions are expressed as f(x)= logb^x.
- The natural logarithm, noted as ln x, uses base e.
Restrictions and Graph Behavior
- The base of a logarithm cannot be 1 and must remain positive.
- When the base increases, the graph of the function approaches the x-axis as x > 1.
Graph Shifting and Stretching
- Horizontal shifts in the graph of the function f(x)=a+clogb(dx+g) occur when changing the parameter g.
- Vertical shifts are made by altering either parameter d or a.
- To stretch the graph vertically, change the parameter c.
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