How to Find Vertex of a Quadratic Function?

The vertex of a quadratic function is the point on the graph (parabola) where the function reaches its maximum or minimum value. It is the turning point of the parabola, where the graph changes direction.

For a quadratic function f(x) = ax² + bx + c, the vertex is (h, k), where:
h = -b / (2a)
k = -(b² − 4ac) / (4a)

• If a > 0, the vertex gives the minimum value of the function.
• If a < 0, the vertex gives the maximum value of the function.

Methods to Find the Vertex of a Quadratic Function

Different methods to find the vertex of a Quadratic Function are:

• Using the Vertex Formula
• Completing the Square Method
• By Graphical Analysis

Using the Vertex Formula

The vertex of a quadratic function of standard form f(x) = ax² + bx + c can be found using the vertex formula:

• h = -b/2a
• k = f(h)

The (h, k) coordinates represent the vertex of the quadratic function.

The x-coordinate of the vertex is h, and the y-coordinate of the vertex is k. h is found by substituting -b/2a in the function, and the y-coordinate is the value of the function at h.

Steps:

Step 1: Identify the coefficients a and b in the quadratic function f(x) = ax² + bx + c

Step 2: Calculate the x-coordinate of the vertex using h = -b / (2a)

Step 3: Substitute the value of h into the function f(x) to find k, i.e., k = f(h)

Step 4: The coordinates of the vertex are (h, k).

Completing the Square Method

Using the completing the square method,we can convert the quadratic equation of form ax2 + bx + c to vertex form a(x - h)2 + k, where (h, k) is the vertex.

Steps:

Step 1: Start with the standard form of a quadratic equation:
f(x) = ax² + bx + c

Step 2: If a ≠ 1, factor out a from the quadratic and linear terms:
f(x) = a(x² + (b/a)x) + c

Step 3: Add and subtract the square of half the coefficient of x:
f(x) = a(x² + (b/a)x + (b/2a)² − (b/2a)²) + c

Step 4: Form a perfect square:
f(x) = a(x + b/2a)² − a(b/2a)² + c

Step 5: Simplify the equation:
f(x) = a(x + b/2a)² + c − b²/4a

Step 6: Compare with vertex form:
f(x) = a(x − h)² + k

Step 7: Identify the values:
h = −b/2a
k = c − b²/4a

Step 8: Therefore, the vertex is: (h, k)

By Graphical Analysis

In graphical analysis we plot the quadratic function and identify the vertex visually,i.e. by looking. This method is very useful for understanding the behaviour of the quadrafunction, and it is also used to confirm the results obtained using algebraic methods.

Steps for Graphical Analysis

Step 1: Plot the quadratic function. We will use graphing calculator or software to plot the function: f(x) = ax² + bx + c

Step 2: Identify the Axis of symmetry. The axis of symmetry is the vertical line that passes through the vertex. It can be found using the formula x = -b/2a

Step 3: Vertex is the point where the parabola intersects the axis of symmetry. The coordinates of vertex is (h, k).

👁 vertex

Also Read:

• Graph of Quadratic Function
• Standard Form to Vertex Form
• Quadratic Equations: Formula, Method and Examples

Practice Questions

Q1. Find the vertex of the quadratic function f(x) = 3x² - 6x + 2

Solution:

Identify the coefficient in given quadratic function: a = 3, b = -6 and c = 2

Calculate the x-coordinate of the vertex using formula h = -b/2a:

h = -(-6/2 × 3) = 6/6 = 1

Substitute the value of h = 1 in given function to find the y-coordinate:

k = f(1) = 3(1)² - 6(1) + 2 = 3 - 6 + 2 = -1

Coordinates of vertex are (1 , -1).

Q2. Find the vertex of the quadratic function f(x) = -2x² + 8x - 5

Solution:

Identify the coefficient in given quadratic function: a = -2, b = 8 and c = -5

Calculate the x-coordinate of the vertex using formula h = -b/2a:

h = -b/2a = -8 / (2 × -2) = -8 / -4 = 2

Substitute the value of h = 2 in given function to find the y-coordinate:

k = f(2) = -2(2)² + 8(2) - 5 = -8 + 16 - 5 = 3

Coordinates of vertex are (2 , 3).

Q3. Find the vertex of the quadratic function f(x) = x² - 4x + 6

Solution:

Identify the coefficient in given quadratic function: a = 1, b = -4 and c = 6

Calculate the x-coordinate of the vertex using formula h = -b/2a:

h = -(-4/2 × 1) = (4/2) = 2

Substitute the value of h = 1 in given function to find the y-coordinate:

k = f(2) = (2)² - 4(2) + 6 = 4 - 8 + 6 = 2

Coordinates of vertex are (2 , 2).

Q4. Find the vertex of the quadratic function f(x) = -x² + 2x - 1

Solution:

Identify the coefficient in given quadratic function: a = -1, b = 2 and c = -1

Calculate the x-coordinate of the vertex using formula h = -b/2a:

h = -(-2/2 × (-1)) = -(2/-2) = 1

Substitute the value of h = 1 in given function to find the y-coordinate:

k = f(1) = -(1)² + 2(1) - 1 = -1 + 2 - 1 = 0

Coordinates of vertex are (1 , 0).

Q5. Find the vertex of the quadratic function f(x) = 2x² + 3x + 1

Solution:

Identify the coefficient in given quadratic function: a = 2, b = 3 and c = 1

Calculate the x-coordinate of the vertex using formula h = -b/2a:

h = -(3/2 × 2) = -3/4

Substitute the value of h = -3/4 in given function to find the y-coordinate:

k = f(-3/4) = 2(-3/4)² + 3(-3/4) + 1

k = 2 × (9/16) - 9/4 + 1

k = 18/16 - 36/16 + 16/16

k = 18 - 36 + 16/16

k = -2/16 = -1/8

Coordinates of vertex are (-3/4 , -1/8).