Graph of Quadratic Function

Graphing quadratic functions or equations results in a U-shaped curve known as a parabola. This graph shape provides insights into the behavior of the quadratic equation. Analyzing the parabola is essential when studying the motion of objects under the influence of gravity, where the trajectory forms a parabolic path. Both quadratic equations and quadratic functions share this graphical representation, making the parabola a fundamental concept in various applications across mathematics and physics.

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

Key Characteristics of Quadratic Functions

When the quadratic equation/function is represented graphically, the graph thus obtained is known as the graph of quadratic equation/function. The graphing of a quadratic equation is always a parabola.

1) Direction:

The orientation/direction of the parabola is completely dependent 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.

2) Vertex:

The vertex is the peak or the lowest point of the parabola, depending on whether the parabola opens upwards or downwards. The vertex represents the maximum or minimum value of the function.

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

3) Axis of Symmetry:

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves and is always parallel to the y-axis.

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

4) Domain and Range:

• Domain: The domain of a quadratic function is always all real numbers, represented as (−∞, ∞). This means that the function can take any real value as input (x-coordinate).

• Range: The range of a quadratic function depends on whether the parabola opens upwards or downwards:
  • If the parabola opens upwards, the range is all y-values greater than or equal to k (the y-coordinate of the vertex). This is written as [k, ∞).
  • If the parabola opens downwards, the range is all y-values less than or equal to k, written as (−∞, k].

5) y-intercept:

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

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

6) x-intercept:

x-intercepts are the points on the graph when the value of y the 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: \frac{-b\pm √ ({b}^2 – 4ac)}{ 2a}, 0.

How to Graph a Quadratic Function

Follow these steps to plot a Quadratic Function:

• Find the Vertex:
  • Use the formula x = −b/2a to find the x-coordinate of the vertex. Then, substitute this value into the function to find the y-coordinate.
  • The vertex is at (h, k) where h = −b/2a​ and k = f(h).
• Determine the Axis of Symmetry:
  • The axis of symmetry is the vertical line x = h, where h is the x-coordinate of the vertex.
• Find the X-Intercepts:
  • Solve ax² + bx + c = 0 to find where the graph crosses the x-axis (if real solutions exist).
• Find the Y-Intercept:
  • Set x = 0 in the function and solve for y. The y-intercept is at (0, f(0)).
• Plot the Points:
  • Plot the vertex, intercepts, and any additional points you find for a more accurate graph.
• Draw the Parabola:
  • Connect the points smoothly to form the parabola, ensuring it is symmetric around the axis of symmetry.

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 a of quadratic equation:

• Step 1: Plot the 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.

Solved Examples of Graphing Quadratic Functions

Example 1: Draw the 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.

Points to Remember

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

Important Formulas Related to Quadratic Functions

Related Reads:

• Roots of Quadratic Equation
• Quadratic Equation
• Practice Questions on Quadratic Equations

Unsolved Practice Questions on Graphing Quadratic Functions

Question 1: Draw a graph of quadratic equation y = 16x² - 4.

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

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

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

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