Use the formula for the axis of symmetry to get the x-coordinate for the vertex. Then, use that to get the y-coordinate, which is the maximum.
Time to Maximum: 0.9375 seconds Maximum Height: 34.125 feet Range: 0 ≤ h ≤ 34.125
Practice makes perfect
This exercise asks us to complete three tasks with the function h=-16t^2+30t+6.
How long will it take to reach its maximum?
What is its maximum height?
What is the range?
In order to find maximums and minimums of a standard quadratic function, y= ax^2+ bx+c, we will need to use the equation for the axis of symmetry.
x=- b/2 a
Once we have the axis of symmetry, we can substitute the x-value into our original equation to find the maximum. The maximum will give us the upper bound for the range of our function.
How long will it take to reach its maximum?
Let's start by comparing our function to the standard quadratic.
ccc
y&=& ax^2&+& bx&+&c
↕ & & ↕ & & ↕ & & ↕
h&=& -16t^2&+& 30t&+&6
We can see that time, t, is our independent variable and that the height, h, is our dependent variable. Then we can see that a = -16, b= 30, and c=6.
When the ball reaches 34.125 feet, it will be at it's maximum height.
Range
Now that we know the maximum, we know the ball will not go any higher than 34.125feet and will stop when it hits the ground at 0. Therefore, we can say that the range is 0 ≤ h ≤ 34.125.
