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This lesson will show how to find the vertex of a quadratic equation. We will also see why this process is so useful to know and apply it to a real world application that could easily come up in one’s life.

Steps to Solve

We are going to learn how to find the vertex of a quadratic equation. As you may know, the graph of a quadratic equation, y = ax2 + bx + c, is the shape of a parabola, which looks like a U or an upside down U. The vertex of a quadratic equation is the maximum or minimum point on the equation’s parabola.

verquad1


See why this point is so important? It’s always incredibly useful to know the maximum or minimum value of an equation, especially when that equation represents real world phenomena!What’s really nice about the vertex of a quadratic equation is that it’s pretty easy to find! To do so, we use the following steps:

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  1. Get the equation in the form y = ax2 + bx + c.

  2. Calculate –b / 2a. This is the x-coordinate of the vertex.
  3. To find the y-coordinate of the vertex, simply plug the value of –b / 2a into the equation for x and solve for y. This is the y-coordinate of the vertex.

After following these steps, you have your x and y coordinates of the vertex, so you have your vertex! Not too bad, hey? We see that the vertex of a quadratic equation, y = ax2 + bx + c, is the following point;(-b / 2a, a(-b / 2a)2 + b(-b / 2a) + c)

Solution

To find the vertex of a quadratic equation, y = ax2 + bx + c, we find the point (-b / 2a, a(-b / 2a)2 + b(-b / 2a) + c), by following these steps.

  1. Get the equation in the form y = ax2 + bx + c.
  2. Calculate –b / 2a.

    This is the x-coordinate of the vertex.

  3. To find the y-coordinate of the vertex, simply plug the value of –b / 2a into the equation for x and solve for y. This is the y-coordinate of the vertex.

Application

We saw that this process doesn’t look so bad, so let’s go ahead and give it a go with an application! Suppose you and a few of your friends decide to open a math tutoring business. Taking time and travel expenses into account, your business’ monthly profit can be modeled by the quadratic equationy = -0.7x2 + 140xwhere y is the profit when you sell x tutoring packages. We see that this is a quadratic equation, and its graph is shown.

Now, when it comes to the profit of your business, what is the number one question you want to know about it? It’s what is the maximum amount of profit you can make, and how many tutoring packages do you have to sell to get that maximum profit?Well, what did we just learn about the vertex of a quadratic equation? It’s the maximum or minimum point of the parabola of the equation! In this case, we see it is the maximum, so to find the maximum possible profit and how many packages must be sold to get that maximum profit, we just need to find the vertex of the equation y = -0.7x2 + 140x, so let’s get started!First, we notice that the first step is already done, since the quadratic profit equation is already in the form y = ax2 + bx + c. Next, we want to find –b / 2a. We see that in our equation, a = -0.7 and b = 140, so we plug these into –b / 2a to find the x-coordinate of the vertex.


verquad3


We see that the x-coordinate of the vertex is 100. This tells us that to maximize your profit, you must sell 100 tutoring packages. Now, let’s find out what that maximum profit is! To do this, we simply carry out our third step.

That is, we plug x = 100 into our equation.

verquad4


We see that when x = 100, y = 7000. This tells us that your maximum monthly profit is $7000. Not too shabby! Now, you just need to sell 100 tutoring packages to attain that profit! No problem!We see that knowing how to find the vertex of a quadratic equation is extremely useful, because the vertex is the maximum or minimum value of the equation, and this has many uses in the world around us!

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