Context: The function $f(x) = a(3.1^x + 3.1^b)$ has integer constants with $0 < a < b$, and forms I and II, $g(x) = a(3.1^x + k)$ and $h(x) = a(3.1)^x + m$, are equivalent to $f$. We must decide which form displays the $y$-coordinate of the $y$-intercept as a constant or coefficient.

Answer: B) Neither I nor II

Why: The $y$-intercept has $y$-coordinate $f(0) = a(3.1^0 + 3.1^b) = a(1 + 3.1^b)$. In form I, $k = 3.1^b$, so the displayed constants are $a$ and $k$, while the intercept is the combination $a(1 + k)$, which is not a number that appears in the equation. In form II, $m = a(3.1)^b$, and $h(0) = a(3.1)^0 + m = a + m$, again a sum rather than any displayed constant. Neither form shows $a(1 + 3.1^b)$ directly.

Traps:

• A) In form I the intercept still requires computing $a(1 + k)$; the coefficient $a$ alone is not the intercept.
• C) Both forms leave the intercept as a combination of two displayed numbers, never a single constant or coefficient.
• D) Treats $m$ like the constant term of a linear equation, but the exponential part still contributes $a(3.1)^0 = a$ at $x = 0$, so the intercept is $a + m$, not $m$.