FROM THE VERTEX FORM OF A QUADRATIC FUNCTION FINDING THE PARAMETERS

Problem 1 :

A parabola has a vertex of (-5, 8) and passes through the point (-7, -4). In the y = a(x - h)² + k form of the parabola, what is the value of a ?

Solution :

The equation of parabola in vertex form is y = a(x − h)² + k

(h, k) being vertex here h = -5, k = 8

Equation of parabola in vertex form is y = a(x + 5)² + 8 . 

The parabola passes through the point (-7, −4) . So the point (-7, −4) will satisfy the equation.

−4 = a(-7 + 5)² + 8

−4 = a(-2)² + 8 

-4 = 4a + 8

4a = 8 + 4

4a = 12

a = 12/4 

a = 3

So, the value of a is 3.

Problem 2 :

A parabola has a vertex of (-3, -21) and passes through the point (-5, 1). In the y = a(x - h)² + k form of the parabola, what is the value of a ?

Solution :

The equation of parabola in vertex form is y = a(x − h)² + k

(h, k) being vertex here h = -3, k = -21

Equation of parabola in vertex form is y = a(x + 3)² - 21 . 

The parabola passes through the point (-5, 1) . So the point (-5, 1) will satisfy the equation.

1 = a(-5 + 3)² - 21

1 = a(-2)² - 21 

1 = 4a - 21

4a = -21 - 1

4a = -22

a = -22/4 

a = -5.5

So, the value of a is -5.5.

Problem 3 :

If

f(x) = x² + 8x - 2 = a(x - h)² + k

then what is the value of k ?

Solution :

f(x) = x² + 8x - 2

f(x) = x² + 8x +16 - 16 - 2

It can be written as

f(x) + 16 = (x + 4)² - 2

f(x) = (x + 4)² - 2 - 16

f(x) = (x + 4)² - 18

So, the value of k is -18.

Problem 4 :

If

f(x) = x² + 10x - 23 = a(x - h)² + k

then what is the value of h ?

Solution :

f(x) = x² + 10x - 23

f(x) = x² + 10x + 25 - 25 - 23

It can be written as

f(x) + 25 = (x + 5)² - 23

f(x) = (x + 5)² - 23 - 25

f(x) = (x + 5)² - 48

So, the value of h is -5.

Problem 5 :

A parabola has a vertex of (-2, 10) and passes through the point (-3, 6). In the y = a(x - h)² + k form of the parabola, what is the value of a ?

Solution :

The equation of parabola in vertex form is y = a(x − h)² + k

(h, k) being vertex here h = -2, k = 10

Equation of parabola in vertex form is y = a(x + 2)² + 10 . 

The parabola passes through the point (-3, 6) . So the point (-3, 6) will satisfy the equation.

∴ 6 = a(-3 + 2)² + 10

6 = a(-1)² + 10

6 = a + 10

a = 6 - 10

a = -4

So, the value of a is -4.

Problem 6 :

If

f(x) = x² + 12x + 41 = a(x - h)² + k

then what is the value of k ?

Solution :

f(x) = x² + 12x + 41

f(x) = x² + 12x + 36 - 36 + 41

It can be written as

f(x) + 36 = (x + 6)² + 41

f(x) = (x + 6)² - 36 + 41

f(x) = (x + 6)² + 5

So, the value of k is 5.

Problem 7 :

If

f(x) = x² - 14x + 39 = a(x - h)² + k

then what is the value of h ?

Solution :

f(x) = x² - 14x + 39

f(x) = x² - 14x + 49 - 49 + 39

It can be written as

f(x) + 49 = (x - 7)² + 39

f(x) = (x - 7)² - 49 + 39

f(x) = (x - 7)² - 10

So, the value of h is 7.

Problem 8 :

The graph of h(x) = -x² + 22x - 40 models the height of one of the arches under a bridge.

a) What is the maximum height of the arch ?

b) How wide is the arch ?

Solution :

h(x) = -x² + 22x - 40

h(x) = -(x² - 22x + 40)

= -(x² - 22x + 121 - 121 + 40)

h(x) + 121 = -(x - 11)² + 40

h(x) = -(x - 11)² + 40 - 121

h(x) = -(x - 11)² - 81

a) Maximum height = 11