https://file.aiquickdraw.com/mathbot/file/1726241258576ofvdmu6d.jpg

The table in the image requires identifying the type and behavior of given functions as either linear or exponential. Here's how to determine the type and behavior for each function:

### Types of Functions:
1. **Linear Functions** are in the form \(y = mx + b\), where \(m\) and \(b\) are constants.
2. **Exponential Functions** are in the form \(y = a \cdot b^x\), where \(a\) is a constant and \(b\) is the base of the exponential.

### Behavior of Functions:
- **Linear Growth/Decay**: Linear functions have a constant rate of change (slope).
  - Positive slope \(m > 0\): Linear Growth.
  - Negative slope \(m < 0\): Linear Decay.
- **Exponential Growth/Decay**: Exponential functions have a rate of change proportional to their value.
  - Base \(b > 1\): Exponential Growth.
  - Base \(0 < b < 1\): Exponential Decay.

### Analyzing Each Function:
1. **\(f(x) = 45x - 80\)**
   - Type: Linear (since it's in the form \(mx + b\))
   - Behavior: Growth (since the slope \(45 > 0\))

2. **\(d(y) = 8(4)^y\)**
   - Type: Exponential (since it's in the form \(a \cdot b^x\))
   - Behavior: Growth (since the base \(4 > 1\))

3. **\(p(t) = 190(1.2)^t\)**
   - Type: Exponential
   - Behavior: Growth (since the base \(1.2 > 1\))

4. **\(g(x) = -18x + 250\)**
   - Type: Linear
   - Behavior: Decay (since the slope \(-18 < 0\))

5. **\(h(x) = 200(0.9)^x\)**
   - Type: Exponential
   - Behavior: Decay (since the base \(0.9 < 1\))

### Summary Table:

| Function               | Type      | Behavior       |
|------------------------|-----------|----------------|
| \(f(x) = 45x - 80\)    | Linear    | Growth         |
| \(d(y) = 8(4)^y\)      | Exponential | Growth       |
| \(p(t) = 190(1.2)^t\)  | Exponential | Growth       |
| \(g(x) = -18x + 250\)  | Linear    | Decay          |
| \(h(x) = 200(0.9)^x\)  | Exponential | Decay        |

Would you like more details or have any questions about the functions?

### Related Questions:
1. How do you identify the slope in a linear function?
2. What are some real-world examples of exponential growth and decay?
3. How do you determine the y-intercept of a linear function?
4. What is the difference between exponential growth and linear growth?
5. Can an exponential function have a negative base?

### Tip:
When analyzing functions, always look for the form of the equation first; it helps in identifying the type and behavior.

Identifying the Characteristics of Linear and Exponential Functions

Learn how to distinguish between linear and exponential functions with this example table. The problem includes various functions and their behaviors, categorized as either growth or decay. Perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation