An object is projected such that it follows a parabolic path given by 𝑦 equals negative half 𝑥 squared plus 80𝑥, where 𝑥 is the horizontal distance traveled in feet and 𝑦 is the height. Determine how far along the horizontal the object traveled to reach the maximum height.
Let’s break down what we know. The horizontal line is the distance traveled in feet. The vertical line represents the height that the object reaches. So we’ll just sketch a parabola that faces downward to represent negative one-half 𝑥 squared plus 80𝑥.
The point that the object is the highest is called the vertex of the parabola. We use the letters ℎ and 𝑘 to represent the coordinates of the vertex: ℎ for the horizontal value and 𝑘 for the vertical value.
Our question is specifically asking how long along the horizontal the object traveled when it reached its maximum height. We are primarily interested in this ℎ coordinate. The ℎ coordinate of the vertex can be found by using the formula negative 𝑏 over two 𝑎.
When we’re given our function, where 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, for us 𝑎 equals negative half and 𝑏 equals 80. Plugging that in, we get negative 80 over two times negative half. Negative 80, two times negative half equals negative one. Negative 80 over negative one equals positive 80.
Our ℎ value, our distance along the horizontal, is 80. We’ve already been told that our horizontal is calculated in feet.
So we say at a horizontal distance of 80 feet, the object reached its maximum height.