A football is kicked from the ground and travels at a rate of 10 meters per second. The function 𝑓(𝑥)=10𝑥−5𝑥2 represents the ball's height above the ground at x seconds. Based on this function model, what is the football's highest point and how long will it take to hit the ground?
the max is on the axis of symmetry of the parabola
x = -b / 2 a = -10 / (2 * -5) = 1
1 second up, one second down ... 2 sec flight time
plug one into the original equation to find max height
F(x) = 10x-5x^2.
a. V^2 = Vo^2+2g*h = 0.
10^2+(-20)h = 0
h = 5 m. = max. ht.
b. 10x-5x^2 = 0 when the ball hits gnd.
Divide both sides by 5x:
2-x = 0
X = 2 s. to hit gnd.
a. Yes, the original Eq should be used to calculate max. ht. :
h max = 10x-5x^2 = 10*1-5*1^2 =
To find the highest point of the football and determine how long it will take to hit the ground, we need to find the vertex of the quadratic function 𝑓(𝑥) = 10𝑥 − 5𝑥^2.
The formula for the x-coordinate of the vertex of a quadratic function in the form 𝑓(𝑥) = 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 is given by 𝑥 = -𝑏 / (2𝑎).
Comparing 𝑓(𝑥) = 10𝑥 − 5𝑥^2 to 𝑎𝑥^2 + 𝑏𝑥 + 𝑐, we can see that 𝑎 = -5, 𝑏 = 10, and 𝑐 = 0.
Substituting these values into the formula, we get 𝑥 = -10 / (2 * (-5)) = 1.
Therefore, the football reaches its highest point at 𝑥 = 1 second.
To find the y-coordinate (height) at this point, substitute 𝑥 = 1 into the function 𝑓(𝑥):
𝑓(1) = 10(1) − 5(1)^2
= 10 - 5
= 5
Hence, the highest point of the football is at a height of 5 meters above the ground.
To determine how long it will take for the football to hit the ground, we need to find the value of 𝑥 when 𝑓(𝑥) = 0, since the height is zero at that time.
Set 𝑓(𝑥) = 0 and solve for 𝑥:
10𝑥 − 5𝑥^2 = 0
Dividing both sides by 5𝑥 gives:
2 − 𝑥 = 0
𝑥 = 2
Thus, it will take the football 2 seconds to hit the ground.