To match the given functions with their corresponding graphs, let's analyze the behavior of each function. I'll outline key identifying factors for each:

1. $y = 2^x$

• Identifying Factors:
  • Exponential growth function.
  • The graph passes through the point (0,1) since $2^0 = 1$.
  • As $x$ increases, $y$ increases rapidly.
  • As $x$ decreases, the graph approaches the x-axis but never touches it (asymptotic to the x-axis).

2. $y = 3^{-x}$

• Identifying Factors:
  • Exponential decay function.
  • Equivalent to $y = \frac{1}{3^x}$.
  • The graph passes through the point (0,1).
  • As $x$ increases, $y$ approaches 0.
  • As $x$ decreases, $y$ increases rapidly.

3. $y = -3^{-x}$

• Identifying Factors:
  • Reflects the exponential decay function $y = 3^{-x}$ across the x-axis.
  • The graph passes through the point (0,-1).
  • As $x$ increases, $y$ approaches 0 from below.
  • As $x$ decreases, $y$ decreases rapidly in the negative direction.

4. $y = -0.5^{-x}$

• Identifying Factors:
  • This is also an exponential decay function, but with a negative sign.
  • Reflects $y = 0.5^{-x}$ (which is equivalent to $y = 2^x$) across the x-axis.
  • The graph passes through the point (0,-1).
  • As $x$ increases, $y$ decreases rapidly in the negative direction.
  • As $x$ decreases, $y$ approaches 0 from below.

5. $y = 2^{-x} - 2$

• Identifying Factors:
  • Exponential decay function, shifted down by 2 units.
  • Equivalent to $y = \frac{1}{2^x} - 2$.
  • The graph passes through the point (0, -1) (because $2^{-0} - 2 = 1 - 2 = -1$).
  • As $x$ increases, $y$ approaches -2, but never touches it (asymptotic to $y = -2$).

6. $y = 1.5^x - 2$

• Identifying Factors:
  • Exponential growth function, shifted down by 2 units.
  • The graph passes through the point (0,-1) (because $1.5^0 - 2 = 1 - 2 = -1$).
  • As $x$ increases, $y$ increases but starts from -2.

Matching Process:

1. Identify asymptotes: Graphs with horizontal asymptotes like $y = -2$ correspond to shifted functions.
2. Sign analysis: Positive growth or decay will have curves above the x-axis (unless shifted down).
3. Exponential base comparison: The rate of increase/decrease helps to distinguish between bases like 2, 1.5, and 3.

Would you like to discuss these in relation to specific graphs or need further clarification?

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Relative Questions:

1. How does shifting a function vertically affect its graph?
2. What is the impact of a negative exponent in exponential functions?
3. How can the base of an exponential function change the steepness of the graph?
4. Why does a function with a negative sign reflect over the x-axis?
5. What role do asymptotes play in identifying exponential functions?

Tip: When analyzing functions, always start by considering their basic form before applying transformations like shifts, reflections, or scaling.