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.