CONVERTING BETWEEN DIFFERENT FORMS OF QUADRATIC FUNCTIONS

Converting From Standard Form

Convert the quadratic function from standard form to

i) Vertex form

ii) Factored form

Problem 1 :

y = x² - 8x + 15

Solution:

Vertex Form:

y = a(x - h)² + k

y = x² - 8x + 15

y = x² - 8x  + 4² + 15 - 4²

y = (x - 4)² - 1

Factored Form:

y = a(x - p) (x - q)

y = x² - 8x + 15

y = x² - 5x - 3x + 15

y = x(x - 5) - 3(x - 5)

y = (x - 3) (x - 5)

Problem 2 :

y = x² - 4x

Solution:

Vertex Form:

y = a(x - h)² + k

y = x² - 4x + 2² - 2²

y = (x - 2)² - 4

Factored Form:

y = a(x - p) (x - q)

y = x² - 4x

y = x(x - 4)

Problem 3 :

y = x² + 4x + 3

Solution:

Vertex Form:

y = a(x - h)² + k

y = x² + 4x + 3

y = x² + 4x + 2² + 3 - 2²

y = (x + 2)² - 1

Factored Form:

y = a(x - p) (x - q)

y = x² + 4x + 3

y = x² + x + 3x + 3

y = x(x + 1) + 3(x + 1)

y = (x + 1) (x + 3)

Problem 4 :

y = 3x² - 6x + 3

Solution:

Vertex Form:

y = a(x - h)² + k

y = 3x² - 6x + 3

y = 3(x² - 2x + 1)

y = 3(x² - 2x + 1² + 1 - 1²)

y = 3(x - 1)² + 0

Factored Form:

y = a(x - p) (x - q)

y = 3x² - 6x + 3

y = 3(x² - 2x + 1)

y = 3(x - 1)(x - 1)

Problem 5 :

y = x² + 6x + 5

Solution:

Vertex Form:

y = a(x - h)² + k

y = x² + 6x + 5

y = x² + 6x + 3² + 5 - 3²

y = (x + 3)² - 4

Factored Form:

y = a(x - p) (x - q)

y = x² + 6x + 5

y = x² + x + 5x + 5

y = x(x + 1) + 5(x + 1)

y = (x + 1) (x + 5)

Converting From Factored Form

Convert the quadratic function from factored to

i) Standard form

ii) Vertex form

Problem 6 :

y = (x + 4)(x + 3)

Solution:

Standard Form:

y = ax² + bx + c

y = (x + 4)(x + 3)

y = x² + 3x + 4x + 12

y = x² + 7x + 12

Vertex Form:

y = a(x - h)² + k

y = x² + 7x + 12

Problem 7 :

y = -2(x + 2)(x - 1)

Solution:

Standard Form:

y = ax² + bx + c

y = -2(x + 2)(x - 1)

y = -2(x² - x + 2x - 2)

y = -2(x² + x - 2)

y = -2x² - x + 4

Vertex Form:

y = a(x - h)² + k

y = -2x² - x + 4

Problem 8 :

y = 3(2x - 1)(x - 1)

Solution:

Standard Form:

y = ax² + bx + c

y = 3(2x - 1)(x - 1)

y = 3(2x² - 2x - x + 1)

y = 3(2x² - 3x + 1)

y = 6x² - 9x + 3

Vertex Form:

y = a(x - h)² + k

y = 6x² - 9x + 3

Problem 9 :

y = (5x + 1)(x - 3)

Standard Form:

y = ax² + bx + c

y = (5x + 1)(x - 3)

y = 5x² - 15x + x - 3

y = 5x² - 14x - 3

Vertex Form:

y = a(x - h)² + k

y = 5x² - 14x - 3

Problem 10 :

y = (x - 10)(x - 6)

Solution:

Standard Form:

y = ax² + bx + c

y = (x - 10)(x - 6)

y = x² - 6x - 10x + 60

y = x² - 16x + 60

Vertex Form:

y = a(x - h)² + k

y = x² - 16x + 60

y = x² - 16x + 8² - 8² + 60

y = (x - 8)² - 4

Converting From Vertex Form

Convert the quadratic function from vertex form to

i) Standard form

ii) Factored form

Problem 11 :

y = (x - 4)² - 9

Solution:

Standard Form:

y = ax² + bx + c

y = (x - 4)² - 9

y = (x² + 16 - 8x) - 9

y = x² - 8x + 7

Factored Form:

y = a(x - p) (x - q)

y = x² - 8x + 7

y = x² - x - 7x + 7

y = x(x - 1) -7(x - 1)

y = (x - 1) (x - 7)

Problem 12 :

y = (x + 2)² - 9

Standard Form:

y = ax² + bx + c

y =  (x + 2)² - 9

y = x² + 4 + 4x - 9

y = x² + 4x - 5

Factored Form:

y = a(x - p) (x - q)

y = x² + 4x - 5

y = x² - x + 5x - 5

y = x(x - 1) + 5(x - 1)

y = (x - 1) (x + 5)

Problem 13 :

Solution:

Standard Form:

y = ax² + bx + c

Factored Form:

y = a(x - p) (x - q)

y = x² + 7x + 12

y = (x + 4)(x + 3)

Problem 14 :

Solution:

Standard Form:

y = ax² + bx + c

Factored Form:

y = a(x - p) (x - q)

y = 6x² - 9x + 3

y = 6x² - 6x - 3x + 3

y = 6x(x - 1) - 3(x - 1)

y = (6x - 3) (x - 1)

y = 3(2x - 1) (x - 1)

Problem 15 :

Solution:

Standard Form:

y = ax² + bx + c

Factored Form:

y = a(x - p) (x - q)

y = 5x² - 14x - 3

y = 5x² + x - 15x - 3

y = x(5x + 1) - 3(5x + 1)

y = (5x + 1) (x - 3)