The y-intercept of the graph of y = x^2 + 31 in the xy-plane is \(\mathrm{(0, y)}\). What is the...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = x^2 + 31}\)
  • We need to find the y-value of the y-intercept \(\mathrm{(0, y)}\)
• What this tells us: The y-intercept is where the graph crosses the y-axis

2. INFER the approach

• The y-intercept occurs when \(\mathrm{x = 0}\)
• To find the y-value, substitute \(\mathrm{x = 0}\) into the equation
• This will give us the exact coordinates of the y-intercept

3. SIMPLIFY by substitution

• Start with: \(\mathrm{y = x^2 + 31}\)
• Substitute \(\mathrm{x = 0}\): \(\mathrm{y = (0)^2 + 31}\)
• Evaluate: \(\mathrm{y = 0 + 31 = 31}\)

Answer: 31

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "y-intercept" means

Students might know it's a point on the graph but forget that the y-intercept specifically occurs when \(\mathrm{x = 0}\). They might try to set \(\mathrm{y = 0}\) instead (confusing it with x-intercept), leading to \(\mathrm{0 = x^2 + 31}\), which gives \(\mathrm{x^2 = -31}\). This leads to confusion since there's no real solution, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes in basic operations

Students correctly identify that they need to substitute \(\mathrm{x = 0}\), but make errors like thinking \(\mathrm{0^2 = 1}\) or incorrectly adding \(\mathrm{0 + 31}\). This might lead them to incorrect numerical answers if this were a multiple choice question.

The Bottom Line:

This problem tests whether students truly understand the geometric meaning of y-intercept and can translate that understanding into algebraic action. The computation is straightforward once the concept is clear.