[MUSIC PLAYING] Let's look at our objectives for today. We'll start by looking at the definition of a vertex. We'll then introduce the vertex formula. And finally, we'll do some examples finding the vertex using the vertex formula.
Let's start by looking at the definition of a vertex. A "parabola" is the name of a graph of a quadratic equation, which can generally be written as y equals ax squared plus bx plus c. If the coefficient a is positive, the graph is a U shape pointing up. If a is negative, the graph is a U shape pointing down.
A parabola will have a maximum point if the graph is pointing down or a minimum point if the graph is pointing up. The maximum or minimum point of a parabola is called the "vertex." Therefore, the vertex provides useful information about the highest or lowest value in a quadratic relationship. These highest or lowest values are related to optimization problems, such as maximum profit or minimum cost.
Now, let's talk about the vertex formula. We can use the equation for a parabola to find the vertex without looking at the graph. Again, a parabola is the graph of a quadratic equation in the form y equals ax squared plus bx plus c.
When a is positive, the parabola will be upward-facing. And when a is negative, the parabola will be downward-facing. The vertex is the maximum or minimum point on the parabola.
The x-coordinate of the point of the vertex can be found using the formula x equals negative b over 2a, where a and b are the coefficients in our equation. Once we have used the formula to find the x-coordinate of the vertex, we can substitute this value in for x in the equation to determine the y-coordinate of the vertex.
Now, let's do some examples using the vertex formula to find the vertex of a parabola. An Olympic diver is competing for a medal. Her dive can be modeled by the equation and graph shown where x is the time in seconds after she begins the dive and y is the height in feet above the water.
In this example, the vertex of the parabola is a maximum point. We start with our formula for the x coordinate of the vertex. We substitute r values for b and a.
This gives us x equals negative a over 2 times negative 2. Multiplying the values, then our denominator gives us negative 8 over negative 4. And dividing these values gives us x equals 2, so the x-coordinate of our vertex is 2.
We then substitute 2 into our original equation for x to find the y value of the vertex. This gives us negative 2 times 2 squared plus 8 times 2 plus 20. Simplifying with our exponent gives us negative 2 times 4. And then multiplying, negative 2 times 4 is negative 8 and 8 times 2 is 16.
Adding from left to right, negative 8 plus 16 plus 20 gives us 28. So the y-coordinate of our vertex is 28, and we see that our vertex is at the point 2, 28. This means that 2 seconds after she begins the dive she reaches her maximum height of 28 feet above the water.
Here is our last example. Natasha kicks a soccer ball during a game. The flight of the ball can be modeled by the equation and graph shown, where x is the time in seconds after she kicks the ball and y is the height of the ball in feet. Here, again, the vertex is a maximum point. We start by using our formula for the x-coordinate of the vertex.
Substituting our values in for b and a gives us negative 32 over 2 times negative 16. Simplifying the denominator gives us negative 32 over negative 32, and dividing these values gives us a positive 1. So the x-coordinate of the vertex is 1.
We then substitute 1 in for x into our original equation to find the y-coordinate of the vertex. This gives us negative 16 times 1 squared plus 32 times 1. Simplifying our exponent gives us negative 16 times 1.
Multiplying gives us negative 16 plus 32, which is 16. So the y-coordinate of our vertex is 16. Therefore, our vertex is at the point 1, 16. And in the context of a problem, the maximum height of the ball is reached after one second and the maximum height is 16 feet.
Let's go over our important points from today. Make sure you get them in your notes so you can refer to them later. A parabola, the graph of a quadratic equation, will have a maximum point if the graph is pointing down or a minimum point if the graph is pointing up.
The maximum or minimum point of a parabola is called the "vertex." The x-coordinate of the point of the vertex can be found using the formula x equals negative b over 2a. Once we have the x-coordinate of the vertex, we can substitute this value in for x in the equation to determine the y-coordinate of the vertex.
So I hope that these important points and examples helped you understand a little bit more about the vertex formula. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
