Video: Finding the Vertex of a Quadratic Function in a Real-World Context

Video Transcript

The height in feet, 𝑦, of a golf ball can be found using the equation 𝑦 equals negative 16.1𝑡 squared plus 137𝑡 plus three, where 𝑡 is the time in seconds after it was struck. Will the ball reach a height of 301 feet?

In this question, we’ve been given a quadratic equation which models the motion of this golf ball 𝑡 seconds after it was struck. Now, quadratic equations with negative coefficients of 𝑡 squared or 𝑥 squared are really nice to model the motion of something like a ball or a projectile as they look a little something like this. And our job in this question is to establish whether the ball itself will reach a height of 301 feet.

So, let’s imagine the curve does indeed describe the motion of the ball. There are three different scenarios. Either the ball never quite reaches the height of 301 feet. In other words, the curve 𝑦 equals negative 16.1𝑡 squared plus 137𝑡 plus three never intersects the line 𝑦 equals 301. The second scenario is the curve and the line 𝑦 equals 301 intersect exactly once. In this scenario, the maximum height of the ball is 301 feet. And then, our third scenario is if the ball reaches the height of 301 feet, goes a little bit higher, and then drops back down to 301 feet again. In this case, the line 𝑦 equals 301 intersects the curve 𝑦 equals negative 16.1𝑡 squared plus 137𝑡 plus three twice.

And to establish which of these three scenarios is happening, we’re going to create an equation. We’re going to replace 𝑦 in the equation for our curve with 301 such that negative 16.1𝑡 squared plus 137𝑡 plus three is equal to 301. The number of solutions to this equation will tell us which scenario we’re interested in. In other words, there will be no solutions, one solution, or two solutions. Now, of course, before we solve the quadratic equation, we need to make it equal to zero. So, we’re going to subtract 301 from both sides. And we see that negative 16.1𝑡 squared plus 137𝑡 minus 298 is equal to zero. And we could go ahead and use something like the quadratic formula or completing the square to solve this.

Alternatively, we can use the discriminant. Now, for a quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, its discriminate is 𝑏 squared minus four 𝑎𝑐. And we have to use a triangle to denote this. If the discriminant is less than zero, the equation has no solutions, or certainly no real solutions. If the discriminant is equal to zero, there’s exactly one solution. And if the discriminant is greater than zero, we have two real solutions. So, we need to find the discriminant of our equation.

We can see that the coefficient of 𝑡 squared is negative 16.1, so we’ll let 𝑎 be equal to negative 16.1. And don’t worry that our equation is in terms of 𝑡 whereas the general form is in terms of 𝑥. That’s absolutely fine. 𝑏 is 137 and 𝑐, our constant, is negative 298. The discriminant is, therefore, 137 squared minus four times negative 16.1 times negative 298, which is equal to negative 422.2. This is less than zero, so there are no solutions, or real solutions, to the equation negative 6.1𝑡 squared plus 137𝑡 plus three equals 301. Since our equation has no real solutions, we can conclude that the first scenario is true. The curve 𝑦 equals negative 16.1𝑡 squared plus 137𝑡 plus three never intersects the line 𝑦 equals 301. This means the ball never reaches a height of 301 feet, and the answer’s no.