STANDARD FORM TO VERTEX FORM

To convert standard form to vertex form, we may follow the different ways.

Complete the square to convert the standard form quadratic function into vertex form. Then find the vertex.

Example 1 :

f(x) = x² + 4x - 14

Solution :

Coefficient of x² is 1. So, don't have to factorize.

Write the coefficient of x as multiple of 2.

f(x) = x² + 2 ⋅ x ⋅ 2 + 2² - 2²  - 14

f(x) = (x+2)² - 4 - 14

f(x) = (x+2)² - 18

Vertex of the parabola is (-2, -18).

Example 2 :

f(x) = 2x² + 9x

Solution :

Coefficient of x² is 2.

Write the coefficient of x as multiple of 2.

Vertex of the parabola is (9/4, -81/8).

Example 3 :

f(x) = 5x² - 4x + 1

Solution :

Coefficient of x² is 5. So, we have to factorize 5.

Vertex of the parabola is (2/5, 1/5).

Example 4 :

f(x) = x² - 16x + 70

Solution :

Coefficient of x² is 1. So, don't have to factorize.

Write the coefficient of x as multiple of 2.

f(x) = x² - 2 ⋅ x ⋅ 8 + 8² - 8² + 70

f(x) = (x - 8)² - 64 + 70

f(x) = (x - 8)² + 6

Vertex of the parabola is (8, 6).

Example 5 :

f(x) = -3x² + 48x - 187

Solution :

Coefficient of x² is -3. So, factorize -3.

f(x) = -3[x² - 16x] - 187

Write the coefficient of x as multiple of 2.

f(x) = -3[x² - 2 ⋅ x ⋅ 8 + 8²- 8²] - 187

f(x) = -3[(x - 8)²- 64] - 187

f(x) = -3(x - 8)²+ 192 - 187

f(x) = -3(x - 8)²+ 5

Vertex of the parabola is (8, 5).

Example 6 :

The parabola shows the path of your first golf shot, where x is the horizontal distance (in yards) and y is the corresponding height (in yards).

The path of your second shot is modeled by the function

f(x) = −0.02x(x − 80)

Which shot travels farther before hitting the ground? Which travels higher?

Solution :

By observing the graph, it is clear that the ball will reach the horizontal distance of 100 yards 

f(x) = −0.02x(x − 80)

Midpoint of the x-coordinates of x-intercepts is the x-coordinate of the vertex.

−0.02x(x − 80) = 0

x = 0 and x = 80

= (0 + 80)/2

= 80/2

= 40

Applying x = 40 in f(x) = −0.02x(x − 80)

f(40) = −0.02(40)(40 - 80)

= 0.8(40)

= 32

The maximum height of the second shot is 32 yards.

Because 100 yards > 80 yards, the first shot travels farther.

Because 32 yards > 25 yards, the second shot travels higher.

To check that the second shot travels higher, graph the function representing the path of the second shot and the line y = 25, which represents the maximum height of the first shot. The graph rises above y = 25, so the second shot travels higher.

Example 7 :

Tell whether the function f(x) = −4x² − 24x − 19 has a minimum value or a maximum value. Then find the value.

Solution :

f(x) = −4x² − 24x − 19

a = -4, b = -24 and c = -19

x = -b/2a

= 24/-8

x = -3

Applying x = -3. we get

f(-3) = −4(3)² − 24(3) − 19

= -4(9) - 72 - 19

= -36 - 72 - 19

= -127