function f is defined by f(x)=-a^x b where a and b are constants
Understanding the Function ( f(x) = -a^x b )
Answer: Let’s break down the function ( f(x) = -a^x b ) where ( a ) and ( b ) are constants.
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Identify the Components:
- ( a^x ): This is an exponential expression where the base ( a ) is raised to the power of ( x ).
- Negative Sign (-): The entire exponential expression is being multiplied by -1, which inverts its sign.
- Multiplication by ( b ): Finally, the expression is multiplied by the constant ( b ).
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Understanding the Behavior:
- Exponential Growth/Decay: If ( a > 1 ), the function represents exponential growth; if ( 0 < a < 1 ), it represents exponential decay.
- Effect of Constants:
- ( b ): Determines the vertical stretch or compression. If ( b > 0 ), the function is stretched upwards, while ( b < 0 ) stretches it downwards.
- Negative Sign (-): Flips the function vertically. If the function is originally increasing, it will decrease after applying the negative sign.
Example:
- If ( a = 2 ) and ( b = 3 ), then:
f(x) = -2^x \cdot 3 = -3 \cdot 2^x
Here, the function shows exponential growth inverted and stretched by 3.
Summary: The function ( f(x) = -a^x b ) combines exponential behavior with negative reflection and scaling by ( b ). Understanding each component helps in visualizing its overall shape and behavior.
Let me know if you need further clarification or additional examples! @anonymous