Graphing quadratic functions or equation is always a U-shaped graph called a Parabola. The graph tells a lot about the nature of the quadratic equation. The graph of a quadratic equation is beneficial while studying the motion of a body under gravitational force. Quadratic Equation is also a Quadratic Function, hence the graph of quadratic equation and quadratic functions are the same.

In this article, we will learn more about quadratic equations, graphs based on quadratic equations, vertex, and axis of symmetry of the graph along with a few examples based on the topic.

What is a Quadratic Function?

A Quadratic Equation is a polynomial equation whose degree is always 2. It is also known as a second-order polynomial equation. In general or standard form it is represented by

f(x) = ax² + bx + c

Where,

a, b, and c are real numbers and a ≠ 0.

Other forms of quadratic equation are:

• Vertex Form: a(x – h)² + k = 0
• Intercept Form: a(x – p)(x – q) = 0

Learn, Quadratic Functions

What is Graphing Quadratic Functions?

When the quadratic equation is represented graphically, the graph thus obtained is known as the graph of quadratic equation. The graphing of quadratic equation is always a parabola. The orientation/direction of the parabola is completely depended upon the value of ‘a’, the coefficient of x² of the given quadratic equation such that:

• if the value of a < 0, the parabola will be oriented downwards.
• if the value of a > 0, the parabola will be oriented upwards.

Vertex of a Quadratic Graph

The vertex of the graph of quadratic function represents the absolute minima or absolute maxima of the given function.

In Quadratic graph, the point of vertex is given by -b/2a, -D/4a

Axis of Symmetry of a Quadratic Graph

The Axis of symmetry passes through the vertex of the parabola and is always parallel to y-axis.

In Quadratic graph, the axis of symmetry is given by: x = -b/2a

y-intercept

y-intercept is the point on the graph that intersects with y-axis. In simple words, y-intercept is the the point on y axis when the value of x coordinate is 0.

In Quadratic graph y-intercept is given by (0, c).

x-intercept

x-intercepts are the points on the graph when the value of y coordinate is 0. These are the points through which the parabola passes on the x-axis.

In Quadratic graph, there are two x intercepts represented by: [Tex] \frac{-b\pm √ ({b}^2 – 4ac)}{ 2a}, 0 [/Tex].

Graph of Quadratic Function Cases

The graph of quadratic equation has two cases, which are as follows:

Upward Case (a > 0)

The direction of the graph completely depends upon the value of coefficient of x² i.e. ‘a’. if a is greater than zero, then the parabola thus formed will open upwards.

Example : Plot a graph of quadratic equation y = 5x²-5.

Solution:

We have the equation: y = 5x² – 5, on comparing it with f(x) = ax² + bx + c

we have, a = 5, b = 0 and c = -5.

The vertex of the above equation is:

x = -b/(2a)

x = -0/(2(5))

x = 0

Now put x = 0 in the equation y = 5x² – 5

y = 5(0)² – 5

y = 0 – 5

y = -5

The vertex of the above equation is (0, -5).

Now, find the different values of x and y by solving the equation:

x	0	1	-1
y	-5	0	0

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening upwards as a = 5 >0.

Downward Case (a < 0)

The direction of the parabola formed for the given quadratic equation will be oriented downwards if the value of coefficient of x² i.e. ‘a’ is less than zero.

Example: Plot a graph of quadratic equation y = -3(x + 2)² + 4.

Solution:

We have the equation: y = -3(x + 2)² + 4, on comparing it with a(x – h)² + k = 0

we have, h = -2, k = 4 and a = -3

The vertex of the above equation is given by (h, k), so vertex is (-2, 4)

Also a = -3, the negative value represent the downward direction of the parabola and so the vertex (-2, 4) is the point of absolute maxima.

Now, find the different values of x and y by solving the equation:

x	-3	-1	-2
y	1	1	4

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening downwards as a = -3 < 0.

Hence we can conclude that, if

• a > 0, the graph will open upwards.
• a < 0, the graph will open downwards.

Graphing Quadratic Functions in Standard Form

The standard or general form of the quadratic equation is given by f(x) = ax² + bx + c. To plot a graph we need to find the vertex and some other coordinates of the given equation. Following are the steps to plot a graph by standard form of quadratic equation:

• Step 1: Plot x-axis and y-axis respectively.
• Step 2: Find the point of absolute maxima (if a<0) or minima (if a>0) i.e. find the vertex of the equation by the formula -b/2a, -D/4a
• Step 3: Plot the obtained vertex on the graph.
• Step 4: Find at-least 4 more coordinates by putting different values of ‘x’ and find its corresponding ‘y’ values.
• Step 5: Plot all the obtained points on the graph and join the points to obtain a parabola.

Graphing Quadratic Functions in Vertex Form

The vertex of the quadratic equation is given by a(x – h)² + k = 0, here h = -b/2a and k = -(b² – 4ac)/4a. To plot a graph we need to find the vertex and some other coordinates of the given equation. Following are the steps to plot a graph by vertex form of quadratic equation:

• Step 1: Plot x-axis and y-axis respectively.
• Step 2: Find the point of absolute maxima (if a < 0) or minima (if a > 0) i.e. find the vertex of the equation by the formula (h, k)
• Step 3: Plot the obtained vertex on the graph.
• Step 4: Find at-least 4 more coordinates by putting different values of ‘x’ and find its corresponding ‘y’ values.
• Step 5: Plot all the obtained points on the graph and join the points to obtain a parabola.

Note: The direction of the parabola will be determined by the value of ‘a’, if a > 0, the direction will be upwards else downwards.

Graph of Quadratic Functions Examples

Example 1: Draw a graph of quadratic equation y = 3x² + x.

Solution:

We have the equation: y = 3x² + x, on comparing it with f(x) = ax² + bx + c

we have, a = 3, b = 1 and c = 0.

The vertex of the above equation is:

x = -b/(2a)

x = -1/(2(3))

x = -1/ 6

x = -0.166

Now put x = -0.166 in the equation y = 3x² + x

y= 3(-0.166)² + (-0.166).

y = 3(0.0275) – 0.166

y = 0.0825 – 0.166

y = -0.0835

The vertex of the above equation is (-0.166, -0.0835)

Now, find the different values of x and y by solving the equation:

x	0	1	-1
y	0	4	2

Plot the graph with these coordinates, the graph thus obtained will be a parabola opening upwards as a = 3 >0.

Example 2: Determine the axis of symmetry and the y-intercept of the quadratic function f(x) = 5x² + 4x +1.

Solution:

We have the equation: y = 5x² + 4x +1

Here, a = 5, b = 4 and c = 1

The axis of symmetry is given by x = -b/2a, putting values we get,

x = -4/2(5)

x = -4/10

x = -0.4

y-intercept is given by (0, c)

Here c = 1, so y-intercept = (0, 1).

Hence, the axis of symmetry and the y-intercept of the quadratic function f(x) = 5x² + 4x +1 is -0.4 and (0, 1) respectively.

Important Points on Graphing Quadratic Functions

• The graph of a quadratic function is a parabola.
• In the equation f(x)=a(x−h)² +k, the coefficient ? determines if the parabola opens upwards or downwards.
• Quadratic functions can be graphed using both the general form and the vertex form.

Graphing Quadratic Functions – Conclusion

Quadratic Equation
Standard Form	f(x) = ax2 + bx + c
Vertex Form	a(x – h)2 + k = 0
Graph Shape	U-shaped or Parabola
Vertex (h, k)	(-b/2a, -D/4a)
Axis of Symmetry	x = -b/2a
y-intercept	(0, c)
x-intercept	[Tex]\frac{-b\pm √ ({b}^2 – 4ac)}{ 2a}, 0 [/Tex]

Check:

• Factoring Polynomials with Examples
• Roots of Quadratic Equation
• Quadratic Equation

Graphing Quadratic Functions – Practice Questions

1. Draw a graph of quadratic equation y = 16x² – 4.

2. Plot a graph for the quadratic equation y = -x² -2x + 3.

3. Find the x-intercept and y-intercept of the equation 3x² + 5x -2.

4. Find the axis of symmetry for the equation 2x² + 4x + 5.

5. Plot a graph for the equation f(x) = x² , also determine the orientation of the parabola.

Graphing Quadratic Functions – FAQs

What is a Graph of Quadratic Equations?

The graph of a quadratic equation is always a U-shape graph called a Parabola. It is helpful to study the nature of the given quadratic equation.

What is the Standard form of the Quadratic Function?

The standard form of quadratic equation is f(x) = y = ax² + bx + c, where a, b, c are real numbers and a ≠ 0.

What is the Vertex form of the Quadratic Function?

The vertex form of quadratic equation is f(x) = y = a (x-h)²+k.

What is the orientation of the parabola formed by a Quadratic Equation?

A parabola is formed by a quadratic equation f(x) = y = ax² + bx + c, the orientation of the parabola depends on the value of a such that:

• if the value of a < 0, the parabola will be oriented downwards.
• if the value of a > 0, the parabola will be oriented upwards.

What is a Real-Life Example of Quadratic Functions?

Few real-life examples of quadratic equation can be:

• The trajectory of a ball thrown in the air
• The shape of a suspension bridge cables
• The design of certain types of arches
• The motion of any object under the influence of gravity

What is the Axis of Symmetry in the Graph of Quadratic Function?

The axis of symmetry of the quadratic equation is determined by the formula: x= -b/2a