Questions and Answers
What is the base of an exponential function in the form f(x)= ab^cx?
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 ____.
base
What are the restrictions on the value of b in an exponential function?
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No matter what the base, an exponential function of the form f(x)=b^x always goes through the point ___.
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An exponential function of the form f(x)=b^x always goes through the point ( ___, b), where b is the base.
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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.
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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.
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What is the mathematical constant e approximately equal to?
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For an exponential function f(x)= ab^cx, changing the value for a will change the ____ to the value of a.
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For an exponential function f(x)= ab^cx, changing the value for c will change the ____ b to b^c.
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How can you find the inverse of a function?
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The graph of the inverse of a function may be found by ___ over the line y=x.
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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.
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What is a common logarithm?
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What is a logarithm?
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What is a logarithmic function?
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What is a natural logarithm?
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The number, y so that b^y=x is called the ____ of x.
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The base of the logarithm function cannot be equal to ________ and must be _____________.
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The greater the base, the closer the graph is to the ___ when x>1.
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To shift the graph of f(x)= a+clogb(dx+g) horizontally, change the parameter ____.
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To shift the graph of f(x)=a+clogb(dx+g) vertically, change parameters ___ or ___.
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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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Description
Test your knowledge of exponential functions, including their characteristics and graphical representations. Understand the significance of the constants involved, such as base 'b' and the constant 'e'. Prepare to explore how modifications to the function affect its graph and intercepts.
