This week in Precalc 11 we started the intro to quadratic functions. The graph of a quadratic function is a parabola, which is a U-shaped curve. The direction and openness of the parabola depend on the sign of the coefficients. If a > 0, the parabola opens upwards, and if 0>a the parabola opens downwards.

Vocabulary :

parent function:

• the parent function spacing follows the 1 – 3 – 5 – 7 … pattern
• y=x^2 
  

Vertex: the turning point of the parabola (also known as the most important point on a parabola)

Maximum of quadratic function: the highest point on a graph when a parabola opens down

Minimum of quadratic functions: the lowest point on a graph when a parabola opens up

The axis of symmetry: the equation of a line that divides the figure into two equal parts where one is the mirror image of the other.

The standard form: 

• Vertical Translations: Adding a number moves the graph up. Subtracting a number moves the graph down.
• Horizontal Translations: Adding a number inside the function shifts the graph left. Subtracting a number inside the function shifts the graph right.
• Vertical Stretching/Compression: Multiplying the function by a number: If the number is greater than 1, it stretches the graph vertically. If the number is between 0 and 1, it compresses the graph vertically

Here is an example:  f(x)= (x-2)^2+3

The blue parabola is the parent function, and the green parabola is the example. In the green parabola we can see that the minimum value is y=3 but in the blue parabola it is 0. this is because in the equation we have a vertical translation of +3 this made the parabola move up 3 units. Another difference is that the green parabolas axis of symmetry is at x=2 rather than 0. Again this is due to the horizontal translation. Even though it says -2 in the brackets we are moving to the right when a negative sign is seen. If a positive sign is seen in the brackets we move to the left.