To analyze Jamaal's height over time and determine the nature of the function [tex]\( y = -16x^2 + 20x + 4 \)[/tex], let's carefully examine the mathematical form of the equation.
### Step-by-Step Solution:
1. Identify the Function Type:
- The given height function is [tex]\( y = -16x^2 + 20x + 4 \)[/tex].
- This is a quadratic function, characterized by the presence of the [tex]\( x^2 \)[/tex] term.
2. Quadratic Functions:
- A quadratic function is any function that can be written in the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( a \neq 0 \)[/tex].
- In this particular function [tex]\( y = -16x^2 + 20x + 4 \)[/tex], the coefficient [tex]\( a \)[/tex] is [tex]\(-16\)[/tex], [tex]\( b \)[/tex] is 20, and [tex]\( c \)[/tex] is 4.
3. Characteristics of Quadratic Functions:
- Quadratic functions are non-linear because they form a parabolic shape when graphed. This is due to the [tex]\( x^2 \)[/tex] term, which causes the rate of change to vary.
- Unlike linear functions (which can be written in the form [tex]\( y = mx + b \)[/tex] and have a constant rate of change), quadratic functions do not have a constant rate of change.
4. Conclusion:
- Since [tex]\( y = -16x^2 + 20x + 4 \)[/tex] fits the standard form of a quadratic function, and quadratic functions are always nonlinear, we can conclude that this function is nonlinear.
Thus, the best description of the function is:
D. The function is nonlinear.
