https://file.aiquickdraw.com/mathbot/file/17255939080043wbilggt.jpg

Looking at the image, here is an analysis of the math problems and solutions shown:

### 1. Find the slope between the given points:
- **Problem 4**: Points (2,1) and (3,-2)
  - The formula for the slope is:  
    \[
    m = \frac{y_2 - y_1}{x_2 - x_1}
    \]
    Applying the coordinates:
    \[
    m = \frac{-2 - 1}{3 - 2} = \frac{-3}{1} = -3
    \]
  - Answer: \( \text{slope} = -3 \)

- **Problem 5**: Points (2,-2) and (4,8)
  - Applying the slope formula:
    \[
    m = \frac{8 - (-2)}{4 - 2} = \frac{8 + 2}{2} = \frac{10}{2} = 5
    \]
  - Answer: \( \text{slope} = 5 \)

### 2. Find the average rate of change in the interval [1,3] for the function \( y = 2x + 3 \)
- The rate of change is the slope of the line, which is the coefficient of \( x \) in the equation \( y = 2x + 3 \). Since the function is linear, the average rate of change is constant.
  - Answer: \( \text{rate of change} = 2 \)

### 3. For each table of values:
- **Problem 7**:
  - The values of \( q(x) \) increase by 3 for each unit increase in \( x \). This is a constant rate of change, meaning the function is linear.
  - Equation: \( q(x) = 3x - 16 \)
  - Answer: Linear

- **Problem 8**:
  - The first differences for \( h(x) \) are increasing by 2, indicating the second differences are constant. This is characteristic of a quadratic function.
  - Equation: The pattern suggests \( h(x) \) follows a quadratic form.
  - Answer: Quadratic

Let me know if you'd like to dive deeper into any part of these problems or have further questions!

### Follow-up Questions:
1. Can you explain how to derive the equation for a quadratic function based on a table of values?
2. What is the significance of the second differences in identifying quadratic functions?
3. How would you find the equation of a function if the table showed exponential growth instead?
4. Why does a linear function always have a constant slope?
5. Can you compare how the rate of change is computed for a quadratic versus an exponential function?

### Tip:
When dealing with tables of values, always check the first and second differences to identify whether the function is linear, quadratic, or something else.

The user submitted an image showing math problems involving slope calculation, average rate of change, and identifying function types based on given tables of values.

This webpage provides solutions to math problems involving slope calculations between points, average rate of change for linear functions, and identifying whether a function is linear or quadratic from a table of values. Suitable for students in Grades 8-10, it includes step-by-step solutions for understanding the key concepts.

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