Although the playing surface of a football or soccer field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side.
The cross section of a field with synthetic turf can be modeled by f(x) = -0.000234(x - 80)^2 + 1.5 where x and y are measured in feet.
Width = 160.064 feet wide
Height = 1.5 feet
how does the width and height relate to domain and range?
The domain and range can only be real life values, so the function can not have negative numbers in the solution of the sketch. Only the positive values would be measurements for a football field.
To understand how the width and height relate to the domain and range of the function f(x), let's break it down step by step.
Domain:
The domain of a function represents all the possible input values or x-coordinates for which the function is defined. In this case, the width of the field is given as 160.064 feet. Typically, the width of the playing surface encompasses the x-axis on a coordinate plane.
Thus, the domain of the function f(x) in this context would be the range of possible x-values within the width of the field. Given that the width of the field is 160.064 feet, the domain of the function f(x) would extend from the left end of the field, let's call it x = 0, to the right end, x = 160.064. Therefore, the domain is [0, 160.064], including both endpoints.
Range:
The range of a function represents all the possible output values or y-coordinates that can be obtained from the function for different x-values within the domain. In this case, the height of the field is given as 1.5 feet, and it corresponds to the y-coordinate.
Referring to the function f(x) = -0.000234(x - 80)^2 + 1.5, the highest point on the parabolic surface occurs when x = 80. Plugging this value into the function, we can calculate the highest point or maximum height.
f(80) = -0.000234(80 - 80)^2 + 1.5
= -0.000234(0)^2 + 1.5
= 1.5
Therefore, the maximum height or the highest point is 1.5 feet, as given. Since the height parameter remains constant, the range of the function f(x) would be [1.5, 1.5]. The range consists of a single value, representing the constant height of the field's surface.
In summary, the width of the field, which falls under the domain, determines the possible x-values within the coordinate plane, while the height of the field, which is fixed and constant, defines the range, which consists of a single value.