Explanation:
To analyze the function and understand the path of the diver, we can observe the coefficients and the form of the equation.
The equation f(x) = -9x^2 + 9x + 1 represents a quadratic function, where the highest power of x is 2. This means that the path of the diver is a parabola.
The coefficient in front of x^2 is negative (-9), which indicates that the parabola opens downward. This suggests that the diver reaches a peak height and then descends back towards the water.
The equation also includes a linear term (9x), which means that the path of the diver is not a perfect parabola but is slightly influenced by a linear component.
Finally, the constant term (1) represents the initial height of the diver above the water. In this case, it is 1 meter.
To visualize the path of the diver, we can graph the function.
Sure! To graph the function f(x) = -9x^2 + 9x + 1, we can plot points and draw a smooth curve connecting them. Let's choose some x-values and calculate the corresponding f(x) values.
Let's start by evaluating the function for a few x-values:
For x = 0:
f(0) = -9(0)^2 + 9(0) + 1
= 1
So, when the diver starts from the end of the diving board (x = 0), the height above the water is 1 meter.
For x = 1:
f(1) = -9(1)^2 + 9(1) + 1
= 1
For x = 2:
f(2) = -9(2)^2 + 9(2) + 1
= -27 + 18 + 1
= -8
For x = -1:
f(-1) = -9(-1)^2 + 9(-1) + 1
= -9 + (-9) + 1
= -17
By calculating f(x) for more x-values, we can obtain more points to plot on the graph. Once we have several points, we can connect them to create a smooth curve that represents the path of the diver.
Let's plot these points and sketch the graph:
(x, f(x))
(0, 1)
(1, 1)
(2, -8)
(-1, -17)
By plotting these points and connecting them with a curve, we can visualize the path of the diver above the water.
Please note that the graph can be further refined with more points and by considering a broader range of x-values.