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Graphing Quadratic Functions




 
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In this lesson, we shall learn

  • how to graph of quadratic functions by plotting points
  • how to graph quadratic function of the form y = ax2
  • the properties of the graph y = ax2
  • how to graph a quadratic function given in general form.
  • how to graph a quadratic function given in factored form.
  • how to graph a quadratic function given in vertex form.


Quadratic Graphs Of The Form y = ax2 ( a ≠ 0 )

Example:

Draw the graph of y = 2x2 for ≤ x ≤ 3, using a scale of 1 cm to 1 unit on the x-axis and 1 cm to 5 units on the y-axis.

Solution:

Step 1 : Construct the table of values.

x

–3

–2

–1

0

1

2

3

y

18

8

2

0

2

8

18

Step 2 : Plot the points on the graph.

Step 3 : Draw a smooth curve passing through the points.

The properties of quadratic graphs y = ax2

The curves of the functions you have drawn so far are called parabolas.

From the example above, you may have noticed the following properties.
Refer to the following diagram when you study these properties.

1. The graphs of y = ax2 (a ≠ 0) pass through the origin (0, 0).

2. The y-axis is the line of symmetry

3. (a) When a is positive, each graph has a lowest point (origin) and opens upwards. This point is known as the minimum point.

(b) The smaller the value of a, the wider the graph opens.

4. (a) When a is negative, each graph has a highest point (the origin) and opens downwards. This point is known as the maximum point.

(b) The smaller the value of , the wider the graph opens.



Activity to Explore the Graph of a Quadratic Equation




Graphing Quadratic Functions in General Form

The general form of a quadratic equation is y = ax2 + bx + c where a, b and c are real numbers and a is not equal to zero.

The following videos show how to Graph Quadratic Functions given in General Form.
Graphing Quadratic Functions - Example 1.
This video outlines a little recipe of things to examine when graphing a quadratic function by hand. An example of graphing a quadratic function is also shown


 

Graphing Quadratic Functions in Factored Form

The general form of a quadratic equation isy = a(x + b)(x + c) where a, b and c are real numbers and a is not equal.

Graphing Parabolas in Factored Form y = a(xr)(xs)

Graphing a Quadratic Function in Factored Form


Graphs of parabolas in Factored Form.
If you have a quadratic in factored form, it is easy to see when it crosses the x-axis. You can also see the shape of the curve, which allows you to get the vertex quickly.

Graphing Quadratic Functions in Vertex Form

The vertex form of a quadratic equation is
y
= a(x h)2 + k where a, h and k are real numbers and a is not equal to zero.

We can convert quadratic functions from general form to vertex form or factored form.

The following videos show how to Graph Quadratic Functions given in Vertex Form.

Graphing a Quadratic Function in Vertex Form



Graphing Quadratic Functions in Standard Form.
The vertex form is also sometimes called the standard form.
This video explains how to graph quadratic functions in the form y = a(x - h)2 + k.


 

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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