FIND THE VERTEX AND AXIS OF THE SYMMETERY OF PARABOLA FROM GRAPH

What is vertex of the parabola ?

If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.

If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. 

The vertex form of a quadratic polynomial is 

y = a(x - h)² + k

Here (h, k) is vertex.

What is axis of symmetry ?

The axis of symmetry is the vertical line, which divides the parabola into two equal parts.

The axis of symmetry will always pass through the vertex of the parabola.

Axis of symmetry can be found using the formula x = -b/a

Identify the vertex and the axis of symmetry of each parabola.

Problem 1 :

Solution :

While tracing the curve, the minimum point of the parabola is at (0, 1).

Equation of axis of symmetry is at x = 0

Problem 2 :

Solution :

The minimum is at (3, 0). So, the vertex is (3, 0).

Axis of symmetry at x = 3.

Problem 3 :

Solution :

The minimum is at (0, -2). So, the vertex is (0, -2).

Axis of symmetry at x = -2.

Graph each function. Label the vertex and the axis of symmetry.

Problem 4 :

y = x² –2x – 3

Solution :

Finding Vertex :

y = x² –2x – 3

Decomposing coefficient of x as multiple of 2, we get

y = x² –2 ⋅ x ⋅ 1 + 1² - 1² – 3

y = (x - 1)² - 1 – 3

y = (x - 1)² - 4

Vertex (h, k) ==> (1, -4)

a = 1, b = -2 and c = -3

Since a is positive 1, the parabola opens up.

Finding axis of symmetry :

Axis of symmetry x = -b/2a

x = -(-2)/2(1)

x = 1

Problem 5 :

y = -3x² +12x – 8

Solution :

Finding Vertex :

y = -3x² +12x – 8

y = -3[x² -4x] – 8

y = -3[x² - 2⋅ x⋅ 2 + 2² - 2²] – 8

y = -3[(x-2)² - 2²] – 8

y = -3(x-2)² + 12 – 8

y = -3(x - 2)² + 4

Vertex (h, k) ==> (2, 4)

a = -3, b = 12 and c = -8

Since a is negative, the parabola opens down.

Finding axis of symmetry :

Axis of symmetry x = -b/2a

x = -12/2(-3)

x = 12/6

x = 2