\(\mathrm{y = 576^{(2x+2)}}\) The graph of the given equation in the xy-plane has a y-intercept of \(\mathrm{(r, s)}\). Which of...

1. TRANSLATE the problem information

• Given: \(\mathrm{y = 576^{2x+2}}\)
• Need to find: Which equation displays the s-value from y-intercept (r, s) as a constant, coefficient, or base

2. TRANSLATE what "y-intercept" means

• The y-intercept occurs where the graph crosses the y-axis
• This happens when \(\mathrm{x = 0}\)
• So substitute \(\mathrm{x = 0}\) into the original equation

3. SIMPLIFY to find the y-intercept

• \(\mathrm{y = 576^{2(0)+2}}\)
  \(\mathrm{= 576^{2}}\)
  \(\mathrm{= 576^2}\)
• Calculate: \(\mathrm{576^2 = 331{,}776}\) (use calculator)
• Therefore, the y-intercept is \(\mathrm{(0, 331{,}776)}\)
• This means \(\mathrm{r = 0}\) and \(\mathrm{s = 331{,}776}\)

4. INFER what the question is asking

• We need to find which equation shows \(\mathrm{s = 331{,}776}\) as a base, coefficient, or constant
• Look through the answer choices for 331,776

5. INFER and check equivalence

• Choice A: \(\mathrm{y = 331{,}776^{x+1}}\) shows 331,776 as the base
• SIMPLIFY to verify: Since \(\mathrm{331{,}776 = 576^2}\):
  \(\mathrm{y = (576^2)^{x+1}}\)
  \(\mathrm{= 576^{2(x+1)}}\)
  \(\mathrm{= 576^{2x+2}}\) ✓
• This matches our original equation!

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "y-intercept" means or how to find it

Students might think the y-intercept is some special property of the equation rather than simply the point where \(\mathrm{x = 0}\). Without setting \(\mathrm{x = 0}\), they can't find \(\mathrm{s = 331{,}776}\) and end up guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making calculation errors with \(\mathrm{576^2}\)

Students might incorrectly calculate \(\mathrm{576^2}\), getting a wrong value for s. This leads them away from recognizing that 331,776 appears in choice A, causing them to select Choice C or D based on seeing familiar numbers like 24 or 576.

The Bottom Line:

This problem tests whether students can connect the algebraic concept of y-intercept (substitute \(\mathrm{x = 0}\)) with recognizing equivalent exponential expressions. The key insight is that finding the y-intercept gives you the specific number to look for in the answer choices.