Definition of a Quadratic Function

In the function , if is expressed as a quadratic equation in like:

this function is called a quadratic function in .

• •For to be a quadratic function in , must not be , though or can be .
• •To identify a quadratic function: rewrite it as an expression in and check if the right-hand side is a quadratic expression.

Reference:

• •Quadratic expression:
• •Quadratic equation:
• •Quadratic function:

Graph of the Quadratic Function

1. Graph of
1. A curve passing through the origin and opening upwards.
2. Symmetrical about the -axis.
3. For , as increases, decreases.
   For , as increases, also increases.
4. Apart from the origin, the curve is entirely above the -axis.
5. It is symmetrical to the graph of with respect to the -axis.


2. Graph of
1. A curve passing through the origin and opening downwards.
2. Symmetrical about the -axis.
3. For , as increases, increases.
   For , as increases, decreases.
4. Apart from the origin, the curve is entirely below the -axis.
5. It is symmetrical to the graph of with respect to the -axis.

3. Key Points for and Graphs
1. Equation of the axis: (the -axis)
2. Vertex coordinates:


4. Parabolas
   A curve shaped like the graphs of or is called a parabola.
1. A parabola is a symmetric figure, and the line of symmetry is called the axis.
2. The intersection point of the parabola and its axis is called the vertex.

Graph of the Quadratic Function

1. Shape of the graph depends on the sign of :
1. If , the parabola opens upwards.
2. If , the parabola opens downwards.

2. Vertex coordinates :
3. Equation of the axis : (symmetry about the -axis).
4. Width of the graph depends on the absolute value of :
   The larger the absolute value of , the narrower the graph.
5. The graph of is symmetric with respect to the -axis.

Graph of the Quadratic Function

1. The graph of is the graph of shifted vertically by .
   
   
   
2. Vertex coordinates :
3. Equation of the axis : (symmetry about the -axis).

Note:
If , the graph shifts upwards.
If , the graph shifts downwards.

Graph of the Quadratic Function

1. The graph of is the graph of shifted horizontally by .
   
   
   
2. Vertex coordinates :
3. Equation of the axis :

Notes:
If , the graph shifts to the right.
If , the graph shifts to the left.

Graph of the Quadratic Function

1. The graph of is the graph of shifted horizontally by and vertically by .
   
   
   
2. Vertex coordinates :
3. Equation of the axis :