A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball in meters t seconds after it is hit is given by a quadratic function h(t) = −5t^2 + 10t + 1. What is the maximum height of the baseball?

Method 1: if you know Calculus

h'(t) = -10t + 10 = 0 for a max of h(t)
10t = 10
t = 1
h(1) = -5(1) + 10 + 1 = 6
max height is 6 m when t= 1 second

Method 2: complete the square of the quadratic
h(t) = -5(t^2 - 2t + ....) + 1
= -5(t^2 - 2t + 1 - 1 ) + 1
= -5((t-1)^2 - 1) + 1
= -5(t-1)^2 + 6
vertex is (1,6), which tells me the max is 6 when t = 1

method 3:
the x of the vertex is -b/(2a) for the general quadratic f(x) = ax^2 + bx + c
so in our case t of vertex = -10/(2(-5)) = 1
h(1) = -5(1) + 10(1) + 1 = 6

Quadratic function:

y = a x ^ 2 + b x + c

has minimum or maximum in point

x = - b / 2 a

If a > 0 quadratic function has minimum

If a < 0 quadratic function has maximum

In your case :

a = - 5 , b = 10 , c = 1

Function has maximum in point :

- b / 2 a = - 10 / [ 2 * ( - 5 ) ] = -10 / - 10 = 1

h ( max ) = h ( 1 ) = - 5 * 1 ^ 2 + 10 * 1 + 1 =

- 5 + 10 + 1 = 6 m

A baseball player swings and hits a pop fly straight up in the air to the catcher. The height of the baseball in meters t seconds after it is hit is given by a quadratic function h(t) = −4.9t^2 + 9.8t + 1. What is the maximum height of the baseball?

To find the maximum height of the baseball, we need to determine the vertex of the quadratic function. The vertex represents the peak of the parabolic curve, which corresponds to the highest point.

The equation of the quadratic function is h(t) = -5t^2 + 10t + 1.

To find the vertex of a quadratic function in the form h(t) = at^2 + bt + c, we can use the formula t = -b / (2a).

In this case, a = -5 and b = 10. Plugging these values into the formula, we have t = -10 / (2 * -5) = -10 / -10 = 1.

Now that we have found the time when the ball reaches its maximum height, we can substitute t = 1 back into the original equation to find the maximum height.

h(1) = -5(1)^2 + 10(1) + 1
= -5 + 10 + 1
= 6

Therefore, the maximum height of the baseball is 6 meters.