The equation for the area of a square, \( A = s^2 \), is a nonlinear function. This is because the area \( A \) is proportional to the square of the side length \( s \). In a nonlinear function, the change in the output (area) is not proportional to the change in the input (side length). If you were to graph this equation, you would see a parabolic curve, which is characteristic of nonlinear functions.
The equation for the area of a square could be written: A=s^2
Is this a linear or nonlinear function?
7 answers
What are some of the values in the table where s is the side length and A is the area?
Here are some values for the side length \( s \) and the corresponding area \( A \) based on the equation \( A = s^2 \):
| Side Length \( s \) | Area \( A \) | |----------------------|---------------| | 0 | 0 | | 1 | 1 | | 2 | 4 | | 3 | 9 | | 4 | 16 | | 5 | 25 | | 6 | 36 | | 7 | 49 | | 8 | 64 | | 9 | 81 | | 10 | 100 |
This table shows the relationship between the side length and the area for squares with side lengths from 0 to 10. As you can see, the area increases at a rate that becomes faster as \( s \) increases, which is characteristic of a nonlinear function.
When you plot these points, is the graph increasing or decreasing?
When you plot the points from the table where the side length \( s \) is on the x-axis and the area \( A \) is on the y-axis (using the equation \( A = s^2 \)), the graph will show an increasing trend.
Specifically, as the side length \( s \) increases, the area \( A \) also increases. The graph will appear as a parabola opening upwards, showing that the area grows more rapidly as \( s \) increases. Thus, the relationship is one of increasing values, making it a nonlinear increasing function.
What numbers would be included in the domain?
The domain of the function \( A = s^2 \) (where \( s \) is the side length of the square and \( A \) is the area) consists of all possible values for \( s \) that make sense in the context of squares.
In this case:
- Non-negative Values: The side length \( s \) cannot be negative, as a square cannot have a negative length. Therefore, \( s \) must be greater than or equal to 0.
- Real Numbers: Any real non-negative number is valid for \( s \). This includes whole numbers, fractions, and irrational numbers.
Thus, the domain can be expressed as:
- In interval notation: \( [0, \infty) \)
- In set notation: \( { s \in \mathbb{R} ,|, s \geq 0 } \)
So the domain consists of all non-negative real numbers.