The graph of the function is a parabola opening downwards, so its vertex is the maximum value of the function. Find the vertex's coordinates using the method of completing the square.
G
Practice makes perfect
It is given that the height of a baseball t seconds after being thrown is given by the following formula.
h(t)=- 16t^2+105t+5
This function is quadratic, so its graph is a parabola. Moreover, the coefficient a, which gives the direction of the parabola, is negative, so the parabola opens downward. This means that the vertex of the parabola is the maximum of the function.
If we find its coordinates we will be able to tell the time at which the ball reaches its maximum height. Let's rewrite the equation into vertex form.
y=a(x-h)^2+k
We will use the method of completing the square.
To complete the square inside the parentheses, we need to add a constant of ( b2)^2. In our case, b is - 10516, so the constant that completes the square is the following.
(- 10516/2)^2=(105/32)^2
Multiplying it by - 16, we can find what constant we should add to both sides of the equation.
- 16(105/32)^2
As we can see, the constant is negative, so instead of adding it we will subtract its absolute value of 16( 10532)^2.
Let's now compare our equation with the standard quadratic equation in vertex from.
y &= a(x- h)^2 + k
h(t)&=- 16(t- 3.3)^2+ 177.3
In this formula (h,k) are the coordinates of the vertex, which in our case are (3.3,177.3). Let's recall that the x-coordinate represents the time in seconds, and the y-coordinate represents the height of the baseball. We conclude that the baseball reaches a maximum height of 177.3 feet 3.3 seconds after it was thrown. The answer is G.
