FIND VERTEX AND AXIS OF SYMMETRY OF PARABOLA

What is vertex ?

The graph of the quadratic function is called a parabola. The point where the graph turns is called the vertex.

If the graph opens upward, the y-coordinate of the vertex is the minimum and the graph is concave upwards.

If the graph opens downward, the y-coordinate of the vertex is the maximum and the graph is concave downwards.

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.

The quadratic function will be represented in three different forms.

• Factored form
• Vertex form and
• Standard form.

Here you can find examples of 

converting from standard into vertex form

Use the vertex, axis of symmetry and y-intercept to graph:

Problem 1 :

y = (x - 1)² + 3

Solution :

Comparing f(x) = a(x - h)² + k and y = (x - 1)² + 3.

h = 1 and k = 3

Vertex of parabola :

(h, k) = (1, 3)

Axis of symmetry :

x = h = 1

x = 1

y- intercept :

x = 0

y = (0 - 1)² + 3

= (-1)² + 3

y = 4

y- Intercept = (0, 4)

Problem 2 :

y = 2(x + 2)² + 1

Solution :

Comparing f(x) = a(x - h)² + k and y = 2(x + 2)² + 1.

h = -2 and k = 1

Vertex of parabola :

(h, k) = (-2, 1)

Axis of symmetry:

x = h = -2

x = -2

y- intercept: x = 0

y = 2(0 + 2)² + 1

y = 8 + 1

y = 9

y- Intercept = (0, 9)

Problem 3 :

y = -2(x - 1)² - 3

Solution :

Comparing f(x) = a(x - h)² + k and y = -2(x - 1)² - 3.

h = 1 and k = -3

Vertex of parabola :

(h, k) = (1, -3)

Axis of symmetry :

x = h = 1

x = 1

y- intercept :

x = 0

y = -2(0 - 1)² - 3

y = -5

y- Intercept = (0, -5)

Problem 4 :

y = 1/2(x - 3)² + 2

Solution :

Comparing f(x) = a(x - h)² + k and y = 1/2(x - 3)² + 2.

h = 3 and k = 2

Vertex of parabola :

(h, k) = (3, 2)

Axis of symmetry :

x = h = 3

x = 3

y- intercept :

x = 0

y = 1/2(0 - 3)² + 2

= 1/2(-3)² + 2

= 1/2 (9) + 2

y = 13/2

y- Intercept = (0, 13/2)

Problem 5 :

y = -1/3(x - 1)² + 4

Solution :

Comparing f(x) = a(x - h)² + k and y = -1/3(x - 1)² + 4.

h = 1 and k = 4

Vertex of parabola :

(h, k) = (1, 4)

Axis of symmetry :

x = h = 1

x = 1

y- intercept :

Put x = 0

y = -1/3(0 - 1)² + 4

= -1/3 + 4

= 11/3

y = 3 2/3

y- Intercept = (0, 3 2/3)

Problem 6 :

y = -1/10 (x + 2)² - 3

Solution :

y = -1/10 (x + 2)² - 3

Comparing f(x) = a(x - h)² + k and y = -1/10(x + 2)² - 3.

h = -2 and k = -3

Vertex of parabola :

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

Axis of symmetry :

x = h = -2

x = -2

y- intercept :

Put x = 0

y = -1/10 (0 + 2)² - 3

y = -1/10(4) - 3

y = -2/5 - 3

y = -17/5

y = -3 2/5

y- Intercept = (0, -3 2/5)