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)?"

If G(x) below represents the transformation of the graph of the main function y = (1/2)^x, what is the equation of G(x)?

Solution

To find the equation of G(x)G(x), which is a transformation of the function y=(12)xy = \left( \frac{1}{2} \right)^x, let's examine the transformation involved.

Step 1: Analyze the Function y=(12)xy = \left( \frac{1}{2} \right)^x

The base function y=(12)xy = \left( \frac{1}{2} \right)^x represents an exponential decay function. Its general shape starts high on the left (for x<0x < 0) and approaches zero as xx \to \infty.

Step 2: Analyze the Transformation

From the graph, it appears that the function has been reflected over the yy-axis, changing the growth direction. A reflection over the yy-axis can be applied by changing xx to x-x in the equation.

Step 3: Write the Transformed Equation

If we reflect y=(12)xy = \left( \frac{1}{2} \right)^x over the yy-axis, the new equation becomes: G(x)=(12)xG(x) = \left( \frac{1}{2} \right)^{-x}

Step 4: Simplify the Equation

Using the property ax=1axa^{-x} = \frac{1}{a^x}, we can rewrite this as: G(x)=2xG(x) = 2^x

Final Answer

The equation of G(x)G(x) is: G(x)=2xG(x) = 2^x

Would you like more details on transformations or the properties of exponential functions?

Further Questions

  1. What would the equation be if the graph was shifted up or down?
  2. How would a horizontal shift affect the equation?
  3. What is the impact of reflecting the graph over the xx-axis instead of the yy-axis?
  4. How would a vertical stretch or compression affect the equation?
  5. Can you identify the domain and range of G(x)=2xG(x) = 2^x?

Tip

When transforming functions, remember that changes inside the function argument xx (like x-x) 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