Shortcut: Vertex formula

Vertex Formula

Before we get into using the vertex formula shortcut, as well as how to convert standard form to vertex form, let’s do a quick review of some fundamental concepts we must grasp before continuing on. First, make sure you’re familiar with some foundational concepts on solving linear quadratic system, system of quadratic equations, and graphing quadratic inequalities in two variables.

At this point, you should be very familiar with quadratic equations and the types of graphs these equations produce. But, to make sure you’re up to speed, a parabola is a type of U-Shaped curve that is formed from equations that include the term $x^2$. Oftentimes, the general formula of a quadratic function in standard form is written as: y= (x – h)^2+k or as y= ax^2+bx+c

Another way of writing quadratic equations is to use vertex form. This form, when read, allows us to easily figure out the vertex of a parabola, which makes graphing a much easier task. The vertex form of a parabola is as follows, where (d,f) is the vertex point and (x,y) is another point:

$(y \pm d) = a(x \pm f )^2$

Vertex form can also be written in its more “proper” form, as:

$y=a(x \pm f)^2 \mp d$

Like mentioned before, this form of a quadratic function allows us to find the vertex more easily. But, the tricky part to his form is how to convert standard form to vertex form and vice-versa, which we’ll get into in the next section. Before that, however, make sure you’re familiar with all of the above information by taking a look at the lessons on vertex form and general form of a quadratic equation.

What is Vertex Formula?

The vertex formula is a very useful little shortcut that allows us to easily determine the vertex of a quadratic equation in standard form. With the vertex, we are then able to also easily write out the vertex form of the quadratic equation!

The vertex form shortcut formula is as follows:

$Vertex:\; (\frac{-b}{2a},\;\frac{-(b^2-4ac)}{4a})$

The variables of this formula refer to the values found in the standard form of quadratic equations that is written as: y= ax^2+bx+c. Using the values for a, b, and c, we can easily determine the vertex!

How to Find the Vertex of a Parabola

The best way to become familiar with how to use the vertex form shortcut formula is to take a look at some practice problems.

Example 1:

Find the vertex of the quadratic function: $y=2x^2-12x+10$

Step 1: Recall the Vertex Formula

As we defined earlier, the vertex formula is: $(\frac{-b}{2a},\;\frac{-(b^2-4ac)}{4a})$

Step 2: Plug Numbers into Formula

Now, to find the vertex of the quadratic function, all we need to do is find the values of a, b, and c from the given quadratic function and solve! Looking at the initial function we were given, a=2, b=-12, and c=10.

To find the x-coordinate of the vertex, put numbers into the vertex formula part that is negative b over 2a.

$x=\frac{-b}{2a}=\frac{-(-12)}{2(2)}=\frac{12}{4}=3$

To find the y-coordinate of the vertex, put numbers into the vertex formula part that is negative b^2-4ac all over 4a.

$y=\frac{-(b^2-4ac)}{4a}=\frac{-[(-12)^2-4(2)(10)]}{4(2)}=\frac{-(144-80)}{8}=\frac{-64}{8}=-8$

Step 3: Write the Vertex

Now that we’ve found both the x and y-coordinates, therefore, the vertex = (3, -8)

NOTE:

If you can’t remember the more complex part of this formula $y=\frac{-(b^2-4ac)}{4a}$, just try to memorize that $x=\frac{-b}{2a}$. Once we have the x value of the vertex using negative b/2a, we can simply plug in this x-value into the original quadratic equation and solve for y, which will give us the y-coordinate of the vertex as follows:

$x=\frac{-b}{2a}=\frac{-(-12)}{2(2)}=\frac{12}{4}=3$

$y=2x^2-12x+10$

$y=2(3)^2-12(3)+10$

$y=-8$

How to Convert Standard Form to Vertex Form

Remember, as we discussed earlier, the shortcut method of using vertex formula isn’t just useful for finding the vertex of a quadratic function, is it also extremely helpful to make converting between standard form and vertex form much easier!

The best way to demonstrate the power of the vertex formula shortcut is to go over some examples.

Example 1:Example 1:

Convert the following quadratic function from general form to vertex form by using the vertex formula: $y=-3x^2-60x-50$

Step 1: Find Constants and Recall Vertex Formula

Remember, as we defined earlier, the vertex formula is: $(\frac{-b}{2a},\;\frac{-(b^2-4ac)}{4a})$

Next, all we need to do is find the values of a, b, and c from the given quadratic function and solve! Looking at the initial function we were given, a=-3, b=-60, c=-50.

Step 2: Plug Numbers into Formula

Now, to find the vertex of the quadratic function, all we need to do is use the values of a, b, and c from the given quadratic function and solve!

$x=\frac{-b}{2a}=\frac{-(-60)}{2(3)}=\frac{60}{-6}=-10$

In this example, we’re going to sub the x-value of the vertex into the original quadratic function instead of using the vertex formula shortcut that is negative b^2-4ac all over 4a. Sometimes, this is a little easier anyways!

$y=-3(-10)^2-60(-10)-50$

$y=-300+600-50$

$y=250$

This image below gives us a visual of what we’ve just accomplished:

Step 3: Write the Vertex

Now that we’ve found both the x and y-coordinates, therefore, the vertex = (-10, 250)

Step 4: Write the Vertex Form of Quadratic Equation

For the last step, we need to remember that we were initially asked to construct the vertex form of this quadratic equation. To do so, recall vertex form:

$(y \pm d)=a(x \mp f )^2$

Using the coordinates of our vertex and the value of a:

$y=250$

$(y-250)=0$

$x=-10$

$(x+10)$

$a=-3$

$(y-250)=-3(x+10)^2$

And then, in proper vertex form of a parabola, our final answer is:

$y=-3(x+10)^2+250$

Example 2:

This example is a much trickier one! In this case, we’ll go over how vertex formula is derived by completing the square. It is very unlikely that you will ever be asked to do this, but, for completion’s sake, we’ll briefly go over how to accomplish this. Before we complete this example, however, make sure you know how to completing the square.

Derive the vertex formula by completing the square:

Step 1: Complete the Square

$y=ax^2+bx+c$

$y-c=ax^2+bx$

$y-c=a(x^2 + \frac{b}{a}x)$

$y-c+a(\frac{b}{2a})^2=a[x^2+\frac{b}{a}x+(\frac{b}{2a})^2]$

$y-c+\frac{b^2}{4a}=a(x+\frac{b}{2a})^2$

$y-\frac{4a}{4a} \cdot c+\frac{b^2}{4a}=a(x+\frac{b}{2a})^2$

$y-\frac{4ac}{4a}+\frac{b^2}{4a}=a(x+\frac{b}{2a})^2$

$(y+\frac{b^2-4ac}{4a})=a(x+\frac{b}{2a})^2$

Step 2: Solve for x and y

Solve x and y using: $y+\frac{b^2-4ac}{4a}=0$ and $x+\frac{b}{2a}=0$

Remember, the x and y values make up the vertex! the vertex. Therefore, the vertex is equal to:

$[\frac{-b}{2a},\;\frac{-(b^2-4ac)}{4a}]$

And that’s all there is to it! Also, please take a look at how we can use a similar method of completing the square to convert standard form to vertex form. Now, do some more practice to make sure you’ve mastered using vertex formula and finding vertex form!