Question:t\(\mathrm{h(t)}\)-2310152-1The table shows three values of t and the corresponding values for the linear function h. The function h is...

1. TRANSLATE the problem information

• Given information:
  • Linear function: \(\mathrm{h(t) = rt + s}\)
  • Table with three coordinate pairs: \(\mathrm{(-2, 31), (0, 15), (2, -1)}\)
  • Need to find: \(\mathrm{r + s}\)

• What this tells us: r is the slope and s is the y-intercept of this linear function

2. INFER the most efficient approach

• Since we need both r and s, let's find s first because it's easier
• The y-intercept s occurs when the input \(\mathrm{t = 0}\)
• Then we can use any two points to calculate the slope r

3. TRANSLATE the y-intercept information

• From the table: when \(\mathrm{t = 0}\), \(\mathrm{h(0) = 15}\)
• Substituting into \(\mathrm{h(t) = rt + s}\): \(\mathrm{h(0) = r(0) + s = s}\)
• Therefore: \(\mathrm{s = 15}\)

4. INFER and SIMPLIFY to find the slope r

• Using slope formula with points \(\mathrm{(0, 15)}\) and \(\mathrm{(2, -1)}\):
• \(\mathrm{r = \frac{h(t_2) - h(t_1)}{t_2 - t_1}}\)
  \(\mathrm{= \frac{-1 - 15}{2 - 0}}\)
  \(\mathrm{= \frac{-16}{2}}\)
  \(\mathrm{= -8}\)

5. SIMPLIFY the final calculation

• \(\mathrm{r + s = (-8) + 15 = 7}\)

Answer: C (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students confuse which variable represents the slope and which represents the y-intercept, or they don't recognize that when \(\mathrm{t = 0}\), the function value directly gives them s.

Some students might try to set up a system of equations using all three points instead of recognizing the direct path through the y-intercept. This leads to unnecessary complexity and potential calculation errors, possibly resulting in selection of an incorrect answer choice.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the approach but make arithmetic errors when calculating the slope \(\mathrm{(-16 \div 2)}\) or when adding the final values \(\mathrm{(-8 + 15)}\).

A common arithmetic mistake is getting the slope calculation wrong, perhaps calculating \(\mathrm{-16/2}\) as \(\mathrm{-7}\) instead of \(\mathrm{-8}\), which would give \(\mathrm{r + s = -7 + 15 = 8}\), leading to confusion since 8 isn't among the answer choices.

The Bottom Line:

This problem tests whether students can efficiently extract information from a table and connect it to the standard form of a linear function. The key insight is recognizing that the y-intercept is directly given in the table, making the solution much more straightforward than setting up complex equation systems.