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Let's break down the problems from your image and tackle them step by step.

### Problems 10-12:
For these problems, you're tasked with finding the constant \( k \) for a linear function based on the given values.

1. **Problem 10 (Function \( g(x) \))**:
   - Given points: \( (1, -5), (3, -1), (5, k), (7, 7) \).
   - This is a linear function, so the slope \( m \) should be constant between any two points.
   - Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope between two known points (1, -5) and (3, -1).
   - Then, using the same slope, find the missing value of \( k \).

2. **Problem 11 (Function \( h(x) \))**:
   - Given points: \( (0, 7), (2, 3), (4, k), (8, -5) \).
   - Follow the same procedure as above: find the slope between two known points and solve for \( k \).

3. **Problem 12 (Function \( f(x) \))**:
   - Given points: \( (0, 1), (2, 4), (5, 16), (7, 22) \).
   - Again, find the slope between two known points and use it to solve for the missing constant.

### Problems 13-18:
For these problems, you're asked to determine whether the given function is concave up or concave down based on the data provided.

#### Concavity Test:
- If the second differences of the function's values (i.e., how the changes in the function values change) are positive, the function is concave up.
- If the second differences are negative, the function is concave down.

Let's check each table:

1. **Problem 13 (Function \( f(x) \))**:
   - Given points: \( (1, 0), (2.5, 3), (3, 4), (3.5, 13), (4, 23) \).
   - Find the first and second differences to determine concavity.

2. **Problem 14 (Function \( g(x) \))**:
   - Given points: \( (0, 8), (2, 8), (4, 8), (6, 15), (8, 35) \).
   - Similarly, calculate the second differences to check for concavity.

3. **Problem 15 (Function \( h(x) \))**:
   - Given points: \( (0, 5), (1, 4), (3, 9), (4, 12), (5, 15) \).
   - Use the same method to check for concavity.

4. **Problem 16 (Function \( k(x) \))**:
   - Given points: \( (0, 7), (2, 6), (4, 5), (6, 2), (8, -3) \).
   - Calculate the second differences to check for concavity.

5. **Problem 17 (Function \( p(x) \))**:
   - Given points: \( (0, 11), (2, 10), (4, 6), (6, 1), (8, -5) \).
   - Follow the same procedure as above.

6. **Problem 18 (Function \( f(x) \))**:
   - Given points: \( (0, 5), (2, 8), (4, 13), (6, 20), (8, 29) \).
   - Finally, check the second differences to determine concavity.

### Would you like me to walk you through solving one of these problems or provide more detailed steps for any specific table? Feel free to ask about any part you're unsure of!

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**Here are 5 related questions to extend your learning:**
1. How can we derive the general equation of a line given two points?
2. What is the significance of the slope in determining concavity?
3. How do second differences relate to the shape of a graph?
4. How do we interpret negative and positive concavity in real-world contexts?
5. What are some practical applications of concavity analysis in economics or physics?

**Tip:** Always remember to check the second differences when determining concavity from a table of values.

Find the value of the constant in the linear functions (Problems 10-12) and determine the concavity of given functions based on tables of values (Problems 13-18).

Solve linear functions and find constants by calculating slopes between points (Problems 10-12). Analyze concavity in tables of values for given functions by applying second differences (Problems 13-18). Suitable for students in Grades 9-11.

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