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The exercise gives us two affine functions, \( f \) and \( g \), and a table of values for \( f(x) \) based on different \( x \) values. We are also provided the formula \( f(x) = ax + b \), where \( a \) and \( b \) are constants, and we need to calculate their values.

### Given Table of Values for \( f(x) \):
- \( x = 2 \), \( f(x) = -8 \)
- \( x = 4 \), \( f(x) = -18 \)
- \( x = 6 \), \( f(x) = -28 \)
- \( x = 8 \), \( f(x) = -38 \)
- \( x = 10 \), \( f(x) = -48 \)
- \( x = 12 \), \( f(x) = -58 \)
- \( x = 14 \), \( f(x) = -68 \)

### Step 1: Finding the slope \( a \)
To find \( a \), we can use the slope formula for a linear function, which is:
\[
a = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Choosing two points from the table, such as \( (2, -8) \) and \( (4, -18) \), we calculate:
\[
a = \frac{-18 - (-8)}{4 - 2} = \frac{-18 + 8}{2} = \frac{-10}{2} = -5
\]

Thus, the slope \( a = -5 \).

### Step 2: Finding the y-intercept \( b \)
Now that we know \( a = -5 \), we can substitute one of the points into the equation \( f(x) = ax + b \) to find \( b \).

Using the point \( (2, -8) \):
\[
f(2) = -5(2) + b = -8
\]
\[
-10 + b = -8 \quad \Rightarrow \quad b = -8 + 10 = 2
\]

Thus, the y-intercept \( b = 2 \).

### Final Answer:
- \( a = -5 \)
- \( b = 2 \)

Let me know if you need further details or clarification.

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Here are five related questions to consider:
1. How would the values of \( a \) and \( b \) change if the table for \( f(x) \) had different points?
2. Can you verify the formula \( f(x) = -5x + 2 \) using another pair of points from the table?
3. How would you graph the function \( f(x) = -5x + 2 \)?
4. If a similar function \( g(x) = cx + d \) were given, how would you find \( c \) and \( d \)?
5. How does the slope \( a \) affect the behavior of the function \( f(x) \) as \( x \) increases?

**Tip:** When determining a linear function from a table of values, always use at least two points to calculate the slope and check the result with other points to ensure accuracy.

The exercise gives us two affine functions, f and g, and a table of values for f(x) based on different x values. We are provided the formula f(x) = ax + b, where a and b are constants, and we need to calculate their values.

This guide explains how to determine the slope and y-intercept of a linear function using a table of values. By analyzing points (2, -8) and (4, -18), we calculate the slope and use it to find the intercept, giving a comprehensive understanding of linear functions with real examples.

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