Finding an Exponential Function Through Two Points

Introduction

• Objective: Find an exponential function going through two given points (2, 6) and (5, 48).
• Model form: ( y = a \cdot b^x )

Setting Up the Equations

1. Point Substitution:
   • For point (2, 6):
     • Equation: ( 6 = a \cdot b^2 )
   • For point (5, 48):
     • Equation: ( 48 = a \cdot b^5 )
2. Aligning Equations:
   • Write equations on top of each other for easier manipulation.

Solving for ( b )

1. Elimination by Division:
   • Divide both sides:
     • ( \frac{6}{48} = \frac{a \cdot b^2}{a \cdot b^5} )
   • Simplifies to: ( \frac{1}{8} = b^{-3} )
2. Solving for ( b ):
   • Raise both sides to (-1/3) to solve for ( b ):
   • ( 8^{1/3} = 2 ), so ( b = 2 )

Solving for ( a )

1. Substitute & Solve:
   • Use any point, e.g., (2, 6):
   • Plug into model: ( 6 = a \cdot 2^2 )
   • Simplify: ( 6 = 4a )
   • Solve: ( a = \frac{6}{4} = \frac{3}{2} )

Final Exponential Model

• Equation: ( y = \frac{3}{2} \cdot 2^x )
• Verification:
  • Check using exponential regression or graphing to ensure it passes through both points.

Conclusion

• The exponential function ( y = \frac{3}{2} \cdot 2^x ) successfully models the data points provided.
• Process involved setting up equations, dividing to eliminate variables, and solving for unknowns ( a ) and ( b ).