Math Problem Statement
If G ( x ) G(x) below represents the transformation of the graph of the main function y
( 1 2 ) x y=( 2 1
) x , what is the equation of G ( x ) G(x)?"
Solution
To find the equation of , which is a transformation of the function , let's examine the transformation involved.
Step 1: Analyze the Function
The base function represents an exponential decay function. Its general shape starts high on the left (for ) and approaches zero as .
Step 2: Analyze the Transformation
From the graph, it appears that the function has been reflected over the -axis, changing the growth direction. A reflection over the -axis can be applied by changing to in the equation.
Step 3: Write the Transformed Equation
If we reflect over the -axis, the new equation becomes:
Step 4: Simplify the Equation
Using the property , we can rewrite this as:
Final Answer
The equation of is:
Would you like more details on transformations or the properties of exponential functions?
Further Questions
- What would the equation be if the graph was shifted up or down?
- How would a horizontal shift affect the equation?
- What is the impact of reflecting the graph over the -axis instead of the -axis?
- How would a vertical stretch or compression affect the equation?
- Can you identify the domain and range of ?
Tip
When transforming functions, remember that changes inside the function argument (like ) affect horizontal properties, while changes outside the function affect vertical properties.
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Function Transformations
Formulas
Reflection formula: f(-x) for y-axis reflection
Theorems
Exponential Growth and Decay
Suitable Grade Level
Grades 9-12
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