The function g is defined by \(\mathrm{g(x) = a(x+1)^2 + c}\), where a and c are constants. In the xy-plane,...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = a(x+1)^2 + c}\)
  • Graph passes through \(\mathrm{(-1, 5)}\) and \(\mathrm{(1, 21)}\)
  • Need to find: value of c

• What this tells us: If a point is on the graph, then substituting its coordinates into the function equation must make it true.

2. INFER the strategic approach

• Key insight: Look at the structure of \(\mathrm{g(x) = a(x+1)^2 + c}\)
• The term \(\mathrm{(x+1)^2}\) becomes zero when \(\mathrm{x = -1}\)
• This means using point \(\mathrm{(-1, 5)}\) will eliminate the unknown 'a' term, leaving only 'c' to solve for
• This is much simpler than trying to solve a system with both unknowns

3. TRANSLATE the point condition into an equation

• Since \(\mathrm{(-1, 5)}\) is on the graph: \(\mathrm{g(-1) = 5}\)
• Substitute \(\mathrm{x = -1}\) into the function: \(\mathrm{g(-1) = a(-1+1)^2 + c}\)

4. SIMPLIFY the substitution

• \(\mathrm{g(-1) = a(-1+1)^2 + c}\)
• \(\mathrm{g(-1) = a(0)^2 + c}\)
• \(\mathrm{g(-1) = a(0) + c}\)
• \(\mathrm{g(-1) = 0 + c = c}\)

5. Solve for c

• Since \(\mathrm{g(-1) = 5}\) and \(\mathrm{g(-1) = c}\)
• Therefore: \(\mathrm{c = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic advantage of using \(\mathrm{x = -1}\) to eliminate the 'a' term. Instead, they try to set up a system of equations using both points simultaneously:

• From \(\mathrm{(-1, 5)}\): \(\mathrm{a(0)^2 + c = 5}\) → \(\mathrm{c = 5}\)
• From \(\mathrm{(1, 21)}\): \(\mathrm{a(4) + c = 21}\) → \(\mathrm{4a + c = 21}\)

While this approach works, it's unnecessarily complex and increases chances for arithmetic errors. Students may get confused managing two equations with two unknowns when a direct solution exists.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic mistakes when evaluating \(\mathrm{(-1+1)^2}\), either forgetting that \(\mathrm{(-1+1) = 0}\), or incorrectly computing \(\mathrm{0^2 = 0}\). Some might mistakenly calculate \(\mathrm{(-1+1)^2}\) as \(\mathrm{(-1)^2 + 1^2 = 2}\), leading to incorrect equations and wrong values for c.

The Bottom Line:

This problem rewards students who can analyze the function structure before jumping into calculations. The key insight is recognizing when a substitution will strategically simplify the problem by eliminating unknowns.