Math Problem Statement

Graph each exponential function, write equations for the given data, and analyze the key features of the exponential function y = a·b^x.

Solution

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Question Breakdown:

1. Graph each exponential function.

  • f(x) = 3ᵡ:
    This represents exponential growth with a base of 3. As x increases, y grows rapidly.

  • f(x) = (1/4)ᵡ:
    This represents exponential decay since the base (1/4) is less than 1. As x increases, y approaches 0.

  • f(x) = 8·1.2ᵡ:
    This is another growth function with an initial value of 8 (y-intercept when x = 0) and a growth factor of 1.2.

4-6. Write an equation for each exponential function.

  • For each graph or table provided, analyze the growth/decay pattern to deduce the equation in the form y=abxy = a \cdot b^x, where aa is the initial value and bb is the base.

7. Key features of y=abxy = a \cdot b^x:

  • Include details about the horizontal asymptote, key points (such as y-intercept and another point), domain (xRx \in \mathbb{R}), and range (y>0y > 0 for growth and y<ay < a for decay if a<0a < 0).

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graphing Functions
Exponential Growth and Decay

Formulas

y = a·b^x (general exponential function)
Horizontal asymptote and domain/range relationships

Theorems

Properties of Exponential Functions
Key Points in Graphing Exponentials

Suitable Grade Level

Grades 8-10