A soccer ball is kicked from the ground at an upward velocity of 25 feet per second. This situation can be represented by the function f (x) = -16x2 + 25x. Graph the function. Identify the x-intercept(s) and interpret its meaning. (1 point)
Item 4
(1.25,0); The soccer ball will return to the ground 1.25 seconds after it has been kicked.
Item 5
(0,1.563); The soccer ball will return to the ground 1.563 seconds after it has been kicked.
Item 6
(0.781,9.766); The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
(1.563,0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.
To graph the function f(x) = -16x^2 + 25x, we can plot points on a coordinate plane using different values of x and finding the corresponding y-values.
For the x-intercepts, we need to find the values of x when y = 0. Setting f(x) = 0, we have:
-16x^2 + 25x = 0
Factoring out x:
x(-16x + 25) = 0
From this equation, we can see that x = 0 or -16x + 25 = 0. Solving the second equation:
-16x + 25 = 0
-16x = -25
x = 25/16 ≈ 1.563
So the x-intercept is approximately (1.563, 0). This means that the soccer ball will return to the ground approximately 1.563 seconds after it has been kicked. Therefore, Item 5 is the correct interpretation of the x-intercept.
The graph of the function will also have a vertex, which represents the maximum height of the soccer ball. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -16 and b = 25 in our function.
x = -25 / (2 * -16)
x = 25/32 ≈ 0.781
Plugging this value back into the function, we can find the y-coordinate of the vertex:
f(25/32) = -16(25/32)^2 + 25(25/32)
≈ -16(625/1024) + (625/32)
≈ -625/64 + 625/32
≈ (625/32) - (625/64)
≈ (625 - 625/2) / 64
≈ 625/64
So the vertex of the graph is approximately (0.781, 9.766). This means that the soccer ball will reach a maximum height of 9.766 feet approximately 0.781 seconds after being kicked. Therefore, Item 6 is the correct interpretation of the vertex.