To determine the nature of the function [tex]\( y = 3 \cdot 2^z + 12 \)[/tex], let's analyze its form and structure.
1. Identify Components: The function is given by [tex]\( y = 3 \cdot 2^z + 12 \)[/tex].
- The term [tex]\( 3 \cdot 2^z \)[/tex] is crucial here. Notice that [tex]\( 2^z \)[/tex] is an exponential component where the base (2) is raised to the power of the variable [tex]\( z \)[/tex].
- The coefficient 3 is simply a constant multiplier.
- The term 12 is an added constant which doesn't change the type of the function but shifts it vertically.
2. Check for Linear Form: A linear function has the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants.
- In our case, there isn't an expression of the form [tex]\( mz + b \)[/tex], as the primary term involves an exponent on [tex]\( z \)[/tex]. Hence, this is not a linear function.
3. Check for Quadratic Form: A quadratic function has the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
- Our expression doesn't include a squared term ([tex]\( z^2 \)[/tex]); instead, the variable [tex]\( z \)[/tex] is in the exponent, ruling this out as a quadratic function.
4. Check for Exponential Form: An exponential function typically has the form [tex]\( y = a \cdot b^z + c \)[/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and importantly, [tex]\( b \)[/tex] (the base) is raised to the power of [tex]\( z \)[/tex], the variable.
- Here, [tex]\( y = 3 \cdot 2^z + 12 \)[/tex] fits this form perfectly with [tex]\( a = 3 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 12 \)[/tex].
Given that the primary term includes [tex]\( 2^z \)[/tex] which characterizes the function as exponential, we conclude that:
[tex]\[ y = 3 \cdot 2^z + 12 \][/tex] is an exponential function.
Thus, the correct classification is:
Exponential
