Vertex Formula: Definition, Derivation, and Examples

Vertex Formula

Vertex formula provides an easy way to find the coordinates of the point where a parabola intersects its axis of symmetry.  This point is known as the vertex of the parabola. 

Understanding this concept through various examples using the vertex formula discussed here makes it easier for students to use the formula effectively in problem-solving and graphing quadratic functions.

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What is Vertex Formula?

The vertex is the point on the graph of the parabola that intersects the axis of symmetry of the parabola. The vertex formula is used to find the coordinates of the vertex of a parabola represented by a quadratic function. 

The vertex of a parabola is typically denoted as (h, k) where h and k are the x-coordinate and y-coordinate of the vertex respectively. A quadratic function representing a parabola is given as

y = ax² + bx + c

Here, a, b, and c are real numbers.

• If a > 0, the parabola opens upwards, and the vertex represents the minimum point.

• If a < 0, the parabola opens downwards, and the vertex represents the maximum point.

The vertex form of the parabola is y = a (x - h)² + k, where (h, k) are the coordinates of the vertex.

The vertex formula denotes the x and y coordinates of the vertex as follows:

x = -b/2a and y = -(b² - 4ac)/4a

Therefore, we can write: (h, k) = (-b/2a, -D/4a) where D = b² - 4ac.

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Derivation of Vertex Formula

After knowing what the vertex formula is, students may often wonder how to arrive at the formula. We will explain here the derivation of the vertex formula to help them understand where it comes from.

Let’s take the general quadratic equation of a parabola:

y = ax² + bx + c.

To find the vertex, we need to locate the point where the parabola has the maximum or minimum value. Here is a stepwise process to derive the same:

Step 1: Rearrange the equation by subtracting ‘c’ from both sides.

y - c = ax² + bx

Step 2: Factor out ‘a’ from the first two terms on the RHS.

y - c = a (x² + b/a x)

Step 3: Add and subtract b²/4a² on RHS

y – c = a (x² + b/a x + b²/4a² - b²/4a²)

Step 4: We change the above into the square form.

y - c = a{(x + b/2a)² - b²/4a²}[a² + 2ab + b² = (a + b)²]

Step 5: Simplify the RHS

y – c = a (x + b/2a) 2 - b2/4a 

or, y = a (x + b/2a) 2 - b²/4a + c

or, y = a (x + b/2a) 2 - (b² - 4ac) / (4a)

Step 6: Comparing this with y = a (x - h)² + k, we get:

h = -b/2a and k = -(b² - 4ac) / (4a)

So, the coordinates of the vertex are derived as

(h, k) = (-b/2a, -D/4a) where D = b² - 4ac

Students can memorize only the formula for the x-coordinate of the vertex and then just substitute it in the given equation y = ax² + bx + c to get the y-coordinate of the vertex.

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Solved Examples Using Vertex Formulas

Solved examples for the application of the vertex formula can help your child understand the concept better. Let’s find a few examples using vertex formula:

Example 1: Find the vertex of y = 2x² - 4x + 1.

Solution:

Comparing the given equation with y = ax² + bx + c, we get

a = 2, b = - 4, c = 1.

Then the discriminant is D = b² - 4ac = (-4)² - 4(2)(1) = 16 - 8 = 8.

Using the vertex formula (h, k) = (-b/2a, -D/4a)

we get:

h = (-(-4) / (2×2)) = 4/4 = 1

k = D/4a = 8/(4 x 2) = 8/8 = 1

Therefore, the vertex of the given parabola is (1, -1).

Example 2: Find the vertex of x² + 6x - 5.

Solution:

Comparing the given equation with y = ax² + bx + c, we get

a = 1, b = 6, c = -5

Using the vertex formula (h, k) = (-b/2a, -D/4a)

we get:

h = (-6) / (2×1) = -6/2 = -3

It is the value of the x coordinate of the vertex. Substituting the value of x in the given equation, we get:

y = (-3)² + 6(-3) - 5 = 9 - 18 - 5 = -14

Therefore, the vertex of the given parabola is (-3, -14).

Example 3: Find the equation of a parabola with vertex (-2, -3) and passing through the point (0, 5). 

Solution:

Vertex: (h, k) = (-2, -3)

The point (0, 5) lies on the parabola.

In the equation y = a(x - h)² + k, we substitute the values as follows:

h = -2, k = -3, x = 0, y = 5

Therefore,

5 = a(0 + 2)² - 3 

Or, 5 = 4a - 3

Or, 4a = 8

Or, a = 2

So, the equation of the parabola is y = 2(x + 2)² - 3.

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Vertex Formula Real Life Application

The vertex formula is not just a theoretical concept but also finds application in many real-life situations. Let’s look at the practical significance of the vertex formula:

• Projectile Motion: The highest point of a thrown object is determined using the vertex of its quadratic path.

• Economics: Finding maximum profit or minimum cost involves quadratic equations where the vertex represents the optimal point.

• Engineering: Structural elements like arches and bridges are often designed in parabolic shapes, where the vertex formula helps to determine the symmetry.

• Physics: Concepts of motion, optics, and energy optimization often involve parabolic equations and the calculation of the vertex.

The vertex formula is an important concept in algebra and coordinate geometry. The definition, derivation, and examples of using vertex formulas help students handle algebraic problems confidently in school and competitive exams, and also in real-world applications.

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