The table shows some values of t and the corresponding values \(\mathrm{V(t)}\) for a linear function V that models the...

1. TRANSLATE the problem information

• Given information:
  • Table of values for linear function V(t) modeling water volume
  • Need to find the t-intercept of y = V(t)
• What this tells us: The t-intercept is the point where the graph crosses the t-axis, meaning \(\mathrm{V(t) = 0}\)

2. INFER the approach

• Since we need the t-intercept but only have a table of values, we must find the equation of the linear function first
• For linear functions, we need slope and a point to write the equation

3. Calculate the slope using any two points from the table

• Using points \(\mathrm{(-2, 14)}\) and \(\mathrm{(1, 8)}\):
  \(\mathrm{slope = \frac{8 - 14}{1 - (-2)} = \frac{-6}{3} = -2}\)

4. SIMPLIFY to find the linear equation

• Using point-slope form with point \(\mathrm{(4, 2)}\):
  \(\mathrm{V(t) - 2 = -2(t - 4)}\)
  \(\mathrm{V(t) - 2 = -2t + 8}\)
  \(\mathrm{V(t) = -2t + 10}\)

5. SIMPLIFY to find the t-intercept

• Set \(\mathrm{V(t) = 0}\):
  \(\mathrm{0 = -2t + 10}\)
  \(\mathrm{2t = 10}\)
  \(\mathrm{t = 5}\)
• Therefore, the t-intercept is \(\mathrm{(5, 0)}\)

Answer: C. (5, 0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse t-intercept with y-intercept, looking for where \(\mathrm{t = 0}\) instead of where \(\mathrm{V(t) = 0}\).

Looking at the table, when moving toward \(\mathrm{t = 0}\), students might try to estimate or interpolate the value of \(\mathrm{V(0)}\) from the given points. They might calculate \(\mathrm{V(0) = 10}\) from their equation and incorrectly think the answer is \(\mathrm{(0, 10)}\) or get confused about which coordinate represents what. This leads to confusion and guessing among the available choices.

Second Most Common Error:

Poor INFER reasoning: Students attempt to find the t-intercept directly from the table without recognizing they need the complete linear equation first.

They might notice that \(\mathrm{V(t) = 2}\) when \(\mathrm{t = 4}\), and since 2 is close to 0, they incorrectly select \(\mathrm{(4, 0)}\) as the answer. They fail to understand that they need to find exactly where V(t) equals zero, not just where it's close to zero.

The Bottom Line:

This problem requires students to understand that intercepts are precise mathematical points, not approximations from nearby table values. The key insight is recognizing that finding any intercept of a function requires having the complete equation, not just discrete data points.