Standard Parabola and Basic Transformations

This section introduces the fundamental concepts of the standard parabola and its transformations. The normal parabola serves as the foundation for understanding more complex quadratic functions.

Definition: The standard parabola is defined by the equation y=x², with its vertex at the origin (0,0).

Highlight: The a-value in y=ax² determines whether the parabola opens upward (a>0) or downward (a<0).

Example: For a=0.5, the parabola is stretched vertically, making it appear wider than the standard parabola.

Vocabulary: The vertex (Scheitelpunkt) is the highest or lowest point of the parabola, depending on its orientation.

Transformations in X and Y Directions

This section explores how parabolas can be shifted horizontally and vertically, demonstrating the versatility of quadratic functions.

Definition: Vertical shifts are achieved using y=x²+e, where e determines the direction and magnitude of the shift.

Example: When e=1, the parabola shifts up one unit; when e=-1, it shifts down one unit.

Highlight: Horizontal shifts use the form y=$x-d$², where d>0 shifts right and d<0 shifts left.

Quote: "Es geht auch, dass man die Normalparabel spiegeln/strecken, in y-Richtung und x-Richtung verschieben kann."

Point Testing and Extreme Values

This section covers methods for verifying points on parabolas and finding maximum/minimum values.

Definition: Point testing involves substituting x-values into the quadratic equation to verify if points lie on the parabola.

Example: For y=4.5x², testing point P(5/35) involves calculating y=4.5(5²).

Highlight: Every parabola has either a maximum or minimum value, which is crucial for solving real-world problems.

General Form and Vertex Form

This section examines the relationship between different forms of quadratic functions and methods for converting between them.

Definition: The general form ax²+bx+c can be converted to vertex form a$x-d$²+e through completing the square.

Vocabulary: The "Satz vom Nullprodukt" (Zero Product Property) states that a product is zero if and only if at least one factor is zero.

Example: Converting y=x²-3x+3 to vertex form involves shifting and finding zeros of the resulting function.

Highlight: The vertex always lies halfway between the zeros of a quadratic function.