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The exercise involves determining the parameters \( a \) and \( b \) for the affine function \( f(x) = ax + b \), using the table of values for \( f(x) \).

The table provides:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\
f(x) & 2 & -8 & -18 & -28 & -38 & -48 & -58 \\
\hline
\end{array}
\]

### Step 1: Find the slope \( a \)
The slope \( a \) is the rate of change of \( f(x) \) as \( x \) increases. Since \( f(x) \) is linear, the difference between consecutive \( f(x) \) values divided by the difference in \( x \) values will give the slope:

\[
a = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Using \( (x_1, f(x_1)) = (2, 2) \) and \( (x_2, f(x_2)) = (4, -8) \):

\[
a = \frac{-8 - 2}{4 - 2} = \frac{-10}{2} = -5
\]

### Step 2: Find the intercept \( b \)
Now, using the formula \( f(x) = ax + b \), substitute one of the points, say \( (2, 2) \), and the calculated slope \( a = -5 \), to find \( b \):

\[
f(2) = -5(2) + b = 2
\]
\[
-10 + b = 2 \quad \Rightarrow \quad b = 12
\]

### Final Answer:
The equation of the function is:

\[
f(x) = -5x + 12
\]

Let me know if you would like further details or explanations. Here are 5 related questions to deepen your understanding:

1. How do you verify the equation \( f(x) = -5x + 12 \) using other points from the table?
2. What is the significance of the slope \( a = -5 \) in this context?
3. How would the graph of this function look, and how does it relate to the table?
4. How can you determine whether this function is increasing or decreasing?
5. How would you solve for \( x \) if \( f(x) = -38 \)?

**Tip:** In linear equations, the slope represents how much the function increases or decreases as \( x \) changes, while the intercept represents the function's value when \( x = 0 \).

On considère deux fonctions affines f et g. Pour f, on donne un tableau de valeurs obtenu à l'aide d'un tableur. La fonction f(x) = ax + b est à déterminer en calculant les coefficients a et b.

Learn how to solve for the affine function f(x) = ax + b using a table of values. This exercise demonstrates the calculation of the slope (a) and y-intercept (b) from a given set of points. Ideal for Grades 9-11 students exploring linear and affine functions.

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