The graph of any quadratic function will be a parabola.

The general form of a quadratic function is

y = ax^{2} + bx + c

To explore the graphs of quadratic functions, we have to be aware of the following stuff.

(i) Open upward or open downward parabola

(ii) Vertex

(iii) x - intercepts

(x - intercepts are nothing but the points where the curve cuts x - axis)

(iv) y - intercept

(y - intercepts are nothing but the points where the curve cuts y - axis)

Let us see the above stuff in detail.

**(i) Open upward or downward :**

y = ax^{2} + bx + c

If the leading coefficient or the sign of 'a' is positive, the parabola is open upward and 'a' is negative, the parabola is open downward.

**(ii) Vertex :**

To find the vertex of the parabola which is given by the quadratic function y = ax^{2} + bx + c, we have to find the value x using the formula given below.

x = -b / 2a

And the vertex is

[-b/2a, y(-b/2a)]

**(iii) X - intercepts :**

To know the x - intercepts, we have to plug y = 0 in

y = ax^{2} + bx + c

and find the values of x.

**Note : **

If the values of x are imaginary, the curve will not intersect x - axis.

**(iii) Y - intercept :**

To know the y - intercept, we have to plug x = 0 in

y = ax^{2} + bx + c

and find the value of y.

Based on the results which we get from the above four points, we have to sketch parabola.

**Example 1 : **

Graph the quadratic equation that is given below.

x^{2} + 5x + 6 = 0

**Solution : **

Let y = x^{2} + 5x + 6

**Step 1 :**

In the given quadratic function, since the leading coefficient (x^{2}) is positive, the parabola will be open upward.

**Step 2 :**

When the given quadratic function is compared to

y = ax^{2} + bx + c,

we get

a = 1

b = 5

c = 6

x coordinate of the vertex is

= -b / 2a

= -5 / 2(1)

= -5 / 2

= -2.5

y coordinate of the vertex is

= y(-2.5)

= (-2.5)^{2} + 5(-2.5) + 6

= 6.25 - 12.5 + 6

= -0.25

So, the vertex is (-2.5, -0.25).

**Step 3 :**

To know x - intercept, we have to substitute y = 0 in the given quadratic function.

Then, we have

0 = x^{2} + 5x + 6 or x^{2} + 5x + 6 = 0

(x + 2)(x + 3) = 0

x + 2 = 0 or x + 3 = 0

x = -2 or x = -3

Therefore, the parabola cuts x - axis at

x = -2 and x = -3

**Step 4 : **

To know y - intercept, we have to substitute x = 0 in the given quadratic function.

Then, we have

y = (0)^{2} + 5(0) + 6

y = 6

Therefore, the parabola cuts y - axis at y = 6.

Now, let us sketch the parabola.

**Example 2 : **

Graph the quadratic equation that is given below.

-2x^{2} + 5x - 7 = 0

**Solution : **

**Let y = -2**x^{2}** + 5x - 7**

**Step 1 :**

In the given quadratic function, since the leading coefficient (x^{2}) is negative, the parabola will be open downward.

**Step 2 :**

When the given quadratic function is compared to

y = ax^{2 }+ bx + c,

we get

a = -2

b = 5

c = -7

x coordinate of the vertex is

= -b / 2a

= -5 / 2(-2)

= -5 / (-4)

= 1.25

y coordinate of the vertex is

= y(1.25)

= -2(1.25)^{2} + 5(1.25) - 7

= -3.125 + 6.25 - 7

= -3.875

So, the vertex is (1.25, -3.875).

**Step 3 :**

To know x - intercept, we have to plug y = 0 in the given quadratic function.

Then, we have

0 = -2x^{2} + 5x -7 or -2x^{2} + 5x - 7 = 0

The above quadratic equation can not be solved using factoring.

So let us try to solve the equation using quadratic formula as given below

Clearly, the two values of x are imaginary.

Therefore, the parabola does not cut x - axis.

**Step 4 : **

To know y - intercept, we have to substitute x = 0 in the given quadratic function.

Then, we have

y = -2(0)^{2} + 5(0) - 7

y = -7

Therefore, the parabola cuts y - axis at y = -7.

Now, let us sketch the parabola.

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