**Determining maximum and minimum values of a quadratic function :**

To find maximum or minimum value of a function, mostly students use first and second order derivatives.

But, to find maximum or minimum values of a quadratic function, we do not have to use derivatives.

We can determine the maxim or minimum value of the quadratic function using the vertex of the parabola (graph the quadratic function).

The general form of a quadratic function is

**f(x) = ax² + bx + c**

Here, if the leading coefficient or the sign of "a" is positive, then the graph of the quadratic function will be a parabola which opens up.

If the leading coefficient or the sign of "a" is negative, then the graph of the quadratic function will be a parabola which opens down.

The quadratic function f(x) = ax² + bx + c will have only the minimum value when the the leading coefficient or the sign of "a" is positive.

When "a" is positive, the graph of the quadratic function will be a parabola which opens up.

The minimum value is "y" coordinate at the vertex of the parabola.

**Note : **

There is no maximum value for the parabola which opens up.

The quadratic function f(x) = ax² + bx + c will have only the maximum value when the the leading coefficient or the sign of "a" is negative.

When "a" is negative the graph of the quadratic function will be a parabola which opens down.

The maximum value is "y" coordinate at the vertex of the parabola.

**Note :**

There is no minimum value for the parabola which opens down.

To find the vertex of the parabola which is given by the quadratic function f(x) = ax² + bx + c, we have to plug

**x = -b / 2a**

And the vertex is

**[ ****-b/2a,****f(-b/2a) ]**

Hence, the maximum or minimum value of the quadratic function is,

**"y" coordinate = f(-b/2a)**

**Example 1 : **

Find the minimum or maximum value of the quadratic equation given below.

f(x) = 2x² + 7x + 5

**Solution : **

In the given quadratic function, since the leading coefficient (2x²) is positive, the function will have only the minimum value.

When we compare the given quadratic function with f(x) = ax² + bx + c, we get

a = 2

b = 7

c = 5

"x" coordinate of the vertex = -b / 2a

"x" coordinate of the vertex = -7 / 2(2)

"x" coordinate of the vertex = -7 / 4

"x" coordinate of the vertex = -1.75

Minimum value = f(-1.75)

Minimum value = 2(-1.75)² + 7(-1.75) + 5

Minimum value = 2(3.0625) - 12.25 + 5

Minimum value = 6.125 - 12.25 + 5

**Minimum value = -1.125**

**Example 2 : **

A golfer attempts to hit a golf ball over a gorge from a platform above the ground. The function that models the height of the ball is

h(t) = -5t² + 40t + 100

where is the height in meters at time "t" seconds after contact. Find the maximum height reached by the golf ball.

**Solution : **

It is clear that the path of the golf ball is a parabola which opens up.

It has been illustrated in the picture given below.

When we compare the given quadratic function with f(x) = ax² + bx + c, we get

a = -5

b = 40

c = 100

"x" coordinate of the vertex = -b / 2a

"x" coordinate of the vertex = -40 / 2x(-5)

"x" coordinate of the vertex = -40 / (-10)

"x" coordinate of the vertex = 4

Maximum height = h(4)

Maximum height = -5(4)² + 40(4) + 100

Maximum height = -5(16) + 160 + 100

Maximum height = -80 + 160 + 100

Maximum height = 180

**Hence,the maximum height reached by the golf ball is 180 meters. **

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