Let's analyze the given problem carefully.

Given Information:

• Student 1: Represents the linear function as an equation: $f(x) = 3x - 6$

• Student 2: Represents the function in a table with values:

  \hline x & f(x) \\ \hline -2 & -12 \\ -1 & -9 \\ 0 & -6 \\ 1 & -3 \\ 2 & 0 \\ \hline \end{array}$$

Solution Analysis:

1. Finding the Intercepts:

   • Equation from Student 1 ($f(x) = 3x - 6$):

     • Y-intercept:
       The y-intercept occurs when $x = 0$: $f(0) = 3(0) - 6 = -6$ So, the y-intercept is at $(0, -6)$.

     • X-intercept:
       The x-intercept occurs when $f(x) = 0$: $3x - 6 = 0$ $3x = 6$ $x = 2$ So, the x-intercept is at $(2, 0)$.

   • Table from Student 2:

     • From the table, when $x = 2$, $f(x) = 0$. This confirms that the x-intercept is at $(2, 0)$.
     • When $x = 0$, $f(x) = -6$. This confirms that the y-intercept is at $(0, -6)$.

2. Evaluating the Statements:

   • Option A:
     "The equation of $f(x)$ reveals an $x$-intercept at $(2, 0)$, while the table for $f(x)$ reveals a $y$-intercept at $(0, -6)$."

     This statement is true based on the calculations and values presented above.

   • Option B, C, and D:
     All these options are false because they either mix up the intercepts or give incorrect points for the intercepts.

Conclusion:

The correct answer is:

• Option A: The equation of $f(x)$ reveals an $x$-intercept at $(2, 0)$, while the table for $f(x)$ reveals a $y$-intercept at $(0, -6)$.

Would you like me to explain anything in more detail, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine the x-intercept of a linear equation given in standard form?
2. What is the slope of the function $f(x) = 3x - 6$, and how can it be interpreted?
3. How would the graph of $f(x) = 3x - 6$ look, and where would the intercepts be plotted?
4. What is the difference between x-intercepts and y-intercepts in any function?
5. How can you verify the intercepts of a function if only given its table of values?

Tip: To determine intercepts, set one variable to zero and solve for the other—this method works for both linear and non-linear functions.