Match the function with a grab and explain the identifying factors that led you to your deduction. The functions: y=2^x
y= 3^-x
y=-3^-x
y=-0.5^-x
y =2^-x -2
y= 1.5^x -2

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.

Understand and analyze exponential functions including y = 2^x, y = 3^-x, y = -3^-x, y = -0.5^-x, y = 2^-x - 2, and y = 1.5^x - 2. Explore their graphs, identifying factors, and key characteristics such as growth, decay, shifts, and reflections. Suitable for high school students and anyone interested in advanced algebra and graph analysis.

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