Given a modified exponential curve defined as \( x=a+b c^{t} \). Show all the steps

The modified exponential curve \( x = a + b c^t \) has a horizontal asymptote at \( a \). Its behavior depends on the value of \( c \): if \( c > 1 \), the curve increases from \( a \) to \( \infty \) as \( t \) increases; if \( 0 < c < 1 \), the curve decreases from \( \infty \) to \( a \) as \( t \) increases. The growth rate of \( x \) with respect to \( t \) is determined by the derivative \( \frac{dx}{dt} = b c^t \ln(c) \), which is positive for \( c > 1 \) and negative for \( 0 < c < 1 \).