Questions and Answers
What is the effect on the graph of an exponential function if the base, $b$, is greater than 1?
Which of the following statements correctly describes the range of an exponential function?
If you have the exponential equation $2^x = 32$, which of the following steps is the correct method to solve for $x$?
For an equation of the form $3^{2x} = 9$, what is the correct process to solve for $x$?
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What is the y-intercept of the exponential function defined by $f(x) = 5 imes 2^x$?
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What is the horizontal asymptote of any exponential function?
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An exponential decay function with base 0.5 has values that always increase as x increases.
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What form does an exponential function take?
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To solve for x in the equation $3^x = 27$, we should first rewrite it in terms of a common base, which is _____ in this case.
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Match the exponential functions with their corresponding behaviors:
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Study Notes
Exponential Function
Graphing Exponential Functions
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Definition: An exponential function is of the form ( f(x) = a \cdot b^x ), where:
- ( a ) is a non-zero constant (the initial value).
- ( b ) is a positive constant (the base), ( b \neq 1 ).
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Key Characteristics:
- Base ( b > 1 ): The graph increases rapidly as ( x ) increases (exponential growth).
- Base ( 0 < b < 1 ): The graph decreases as ( x ) increases (exponential decay).
- Y-intercept: Occurs at ( (0, a) ).
- Horizontal Asymptote: The line ( y = 0 ) serves as an asymptote.
- Domain: All real numbers ( (-\infty, \infty) ).
- Range: Positive real numbers ( (0, \infty) ).
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Graphing Steps:
- Identify ( a ) and ( b ).
- Plot the y-intercept at ( (0, a) ).
- Calculate additional points by substituting values of ( x ).
- Draw the curve approaching the horizontal asymptote ( y = 0 ).
Solving Exponential Equations
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General Form: An equation of the form ( a \cdot b^{kx} = c ).
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Methods:
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Equal Bases:
- Rewrite both sides to have the same base ( b ).
- Set the exponents equal: ( kx = \log_b(c) ).
- Solve for ( x ).
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Logarithmic Method:
- Take the logarithm of both sides: ( \log(a \cdot b^{kx}) = \log(c) ).
- Use properties of logarithms: ( \log(a) + kx \cdot \log(b) = \log(c) ).
- Rearrange to isolate ( x ):
- ( kx = \log(c) - \log(a) )
- ( x = \frac{\log(c) - \log(a)}{k \cdot \log(b)} )
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Examples:
- Example 1: ( 2^x = 16 ) → ( x = 4 ) (since ( 16 = 2^4 )).
- Example 2: ( 3^{2x} = 9 ) → ( 2x = 2 \rightarrow x = 1 ) (since ( 9 = 3^2 )).
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Special Cases:
- When the equation involves different bases, logarithmic methods are preferred.
- If ( b^x = 1 ), then ( x = 0 ) (regardless of base ( b ) as long as ( b \neq 0 )).
Exponential Functions Overview
- Exponential functions are defined as ( f(x) = a \cdot b^x ), where:
- ( a ) represents a non-zero constant (initial value).
- ( b ) is a positive constant with ( b \neq 1 ).
Graphing Exponential Functions
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Key Characteristics:
- Exponential Growth: Occurs when the base ( b > 1 ), causing the graph to rise rapidly as ( x ) increases.
- Exponential Decay: Occurs when ( 0 < b < 1 ), resulting in a graph that falls as ( x ) increases.
- Y-intercept: Located at the point ( (0, a) ).
- Horizontal Asymptote: The line ( y = 0 ) acts as an asymptote.
- Domain: All real numbers ( (-\infty, \infty) ).
- Range: Positive real numbers ( (0, \infty) ).
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Graphing Steps:
- Identify the values of ( a ) and ( b ).
- Plot the y-intercept at ( (0, a) ).
- Substitute various ( x ) values to find additional points.
- Sketch the curve that approaches the horizontal asymptote ( y = 0 ).
Solving Exponential Equations
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General Form: Given by ( a \cdot b^{kx} = c ).
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Solution Methods:
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Equal Bases Method:
- Convert both sides to a common base ( b ).
- Equate the exponents: ( kx = \log_b(c) ).
- Solve for ( x ).
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Logarithmic Method:
- Apply logarithm to both sides: ( \log(a \cdot b^{kx}) = \log(c) ).
- Utilize properties of logarithms to simplify:
- ( \log(a) + kx \cdot \log(b) = \log(c) ).
- Rearrange to isolate ( x ):
- ( kx = \log(c) - \log(a) )
- ( x = \frac{\log(c) - \log(a)}{k \cdot \log(b)} ).
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Examples:
- Example 1: For the equation ( 2^x = 16 ), ( x = 4 ) (since ( 16 = 2^4 )).
- Example 2: In the equation ( 3^{2x} = 9 ), it simplifies to ( 2x = 2 ), yielding ( x = 1 ) (as ( 9 = 3^2 )).
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Special Cases:
- Different bases should be handled with logarithmic methods.
- If ( b^x = 1 ), then it follows that ( x = 0 ), given ( b \neq 0 ).
Definition Of Exponential Functions
- Exponential function form is expressed as ( f(x) = a \cdot b^x ) where:
- ( a ) is the initial value, a non-zero constant.
- ( b ) is a positive constant representing the base, where ( b \neq 1 ).
- The variable ( x ) appears in the exponent, indicating its key role in growth/decay.
- Common bases used in exponential functions include:
- Base ( e ): Represents the natural exponential function ( f(x) = e^x ).
- Base 10: Represented as ( f(x) = 10^x ).
Properties Of Exponential Functions
- Growth and Decay Characteristics:
- Exponential growth occurs when ( b > 1 ).
- Exponential decay occurs when ( 0 < b < 1 ).
- Domain and Range:
- The domain of exponential functions is all real numbers, denoted as ( (-\infty, \infty) ).
- The range consists of positive real numbers, represented as ( (0, \infty) ).
- Asymptotic Behavior:
- Exhibits a horizontal asymptote at ( y = 0 ).
- Graphical Behavior:
- Function curves are continuous and smooth, passing through the y-intercept at ( (0, a) ).
Solving Exponential Equations
- To solve equations in the form ( a \cdot b^x = c ):
- Rearrange to isolate the exponential part: ( b^x = \frac{c}{a} ).
- Apply logarithms on both sides to find ( x ):
- ( x = \log_b\left(\frac{c}{a}\right) ).
- Alternatively, use natural logarithms for calculation:
- ( x = \frac{\ln(\frac{c}{a})}{\ln(b)} ).
- In cases where the base is not apparent:
- Rewriting the equation with a common base can simplify the solution.
- Utilize logarithmic properties for further simplification.
Graphing Exponential Functions
- General Shape:
- Functions that exhibit exponential growth curve upward; decay functions curve downward.
- Important Points for Graphing:
- Plot the y-intercept at ( (0, a) ).
- For growth or decay, plot the point ( (1, a \cdot b) ).
- Display behavior in negative ( x ) by plotting ( (-1, \frac{a}{b}) ).
- Asymptotic Nature:
- Exponential graphs approach but never meet the x-axis (at ( y = 0 )).
- Increasing and Decreasing Trends:
- Exponential growth functions consistently increase for all ( x ).
- Exponential decay functions consistently decrease for all ( x ).
- Transformation Effects:
- Adjustments to ( a ) result in vertical stretching or shrinking.
- Modifications to ( b ) influence the rate of growth or decay.
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Description
This quiz covers graphing and solving exponential functions, defined in the form f(x) = a · b^x. Key concepts include characteristics of exponential growth and decay, identifying y-intercepts, and understanding horizontal asymptotes. Test your knowledge on the steps necessary to graph these functions accurately.
