FROM THE GIVEN VERTEX AND POINT OF THE PARABOLA FIND EQUATION

If V is the vertex, find the equation of the quadratic function with graph:

Problem 1 :

Solution:

Vertex form:

y = a(x - h)² + k

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

y = a(x - 2)² + 4

It passes through (0, 0). Substitute (x, y) = (0, 0)

0 = a(0 - 2)² + 4

0 = 4a + 4

4a = -4

a = -1

Vertex form equation of the parabola:

y = -1(x - 2)² + 4

Problem 2 :

Solution:

Vertex form:

y = a(x - h)² + k

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

y = a(x - 2)² - 1

It passes through (0, 7). Substitute (x, y) = (0, 7)

7 = a(0 - 2)2 - 1

7 = 4a - 1 

4a = 8

a = 2

Vertex form equation of the parabola:

y = 2(x - 2)² - 1

Problem 3 :

Solution:

Vertex form:

y = a(x - h)² + k

Vertex (h, k) = (3, 8)

y = a(x - 3)^{2 +} 8

It passes through (1, 0). Substitute (x, y) = (1, 0)

0 = a(1 - 3)² + 8

0 = a(-2)² + 8 

0 = 4a + 8

4a = -8

a = -2

Vertex form equation of the parabola:

y = -2(x - 3)² + 8

Problem 4 :

Solution:

Vertex form:

y = a(x - h)² + k

Vertex (h, k) = (4, -6)

y = a(x - 4)² - 6

It passes through (7, 0). Substitute (x, y) = (7, 0)

0 = a(7 - 4)² - 6

0 = a(3)² - 6 

0 = 9a - 6

9a = 6

a = 2/3

Vertex form equation of the parabola:

y = 2/3(x - 4)² - 6

Problem 5 :

Solution:

Vertex form:

y = a(x - h)² + k

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

y = a(x - 2)² + 3

It passes through (3, 1). Substitute (x, y) = (3, 1)

1 = a(3 - 2)² + 3

1 = a(1)² + 3 

1 = a + 3

a = -2

Vertex form equation of the parabola:

y = -2(x - 2)² + 3

Problem 6 :

Solution:

Vertex form:

y = a(x - h)² + k

Vertex form equation of the parabola: