Graphing Quadratic Functions:
Quadratic equations in standard form are represented as \(ax^2~+~bx~+~c~=~0\)
To plot any function \(y~+~f(x)\)
The graph of a quadratic function is a parabola which can be represented in two forms:
- Standard Form: \(ax^2~+~bx~+~c\)
- Vertex Form or \((a~-~h~-~k)\)
form: \(f(x)~=~a(x~-~h)^2~+~k\) , where \((h,k)\) is vertex of parabola.
The graph opens upwards if \(a > 0\)
Vertex of a quadratic equation is its minimum or lowest point if the parabola is opening upwards or its highest or maximum point if it opens downwards.
On rearranging the standard form, vertex form is obtained by completing the square method.
On comparing both forms,
When a quadratic function is represented in vertex form, following points are to be noted:
- If h > 0, graph shifts right by h units.
- If h < 0, graph shifts left by h units.
- If k > 0, graph shifts upwards by k units.
- If k < 0, graph shifts downwards by k units.
- (h, k) denotes the vertex of function.
For graphing a quadratic function, above steps are followed and further transformations are used.
Using Transformations to Graph Quadratic Functions:
i) Horizontal shifting by m units:
Consider the standard form of quadratic equation \(ax^2~+~bx~+~c~=~0\)
Suppose instead of \(\alpha\)
Similarly, if roots are \((\alpha~-~m)\)
ii) Vertical shifting by k units:
Considering the vertex form i.e.,\(y\)
- If k > 0 , graph shifts upwards by k units.
- If k < 0 , graph shifts downwards by k units.
Let us first consider the graph of \(y\)
Let us go through an example for a better understanding.
Example: Find range of \(x\)
Solution: Let us try to graph the given equations to find out range of \(x\)
From equation (1),\((x~-~1)(x~-~5)~\leq~0\)
From equation (2), \((x~-~0)(x~-~2)~\gt~0\)
Thus, range of \(x\)
Thus graphing quadratic functions becomes easy using transformations.’