Hyperbolas: Key Characteristics and Graphing

This page focuses on hyperbolas, a type of conic section with unique properties and graphical representations. It provides essential information for students learning about more advanced function types.

The page begins by introducing a specific hyperbola example: y = 1/$x+3$ - 3. This function is used to illustrate key characteristics of hyperbolas.

Highlight: Hyperbolas have asymptotes, which are lines that the graph approaches but never touches.

Several important properties of this hyperbola are discussed:

1. The function can never equal -3 or 0, which relates to its vertical and horizontal asymptotes.
2. The graph starts downward and then goes up, which can be verified by plugging in x = 0.
3. It has two lines of symmetry.
4. There is no x-intercept.

Vocabulary: Asymptotes are lines that a curve approaches as it heads towards infinity.

The domain and range of this hyperbola are also explained:

Example: For this hyperbola, the domain is all real numbers except -3, and the range is all real numbers except 0.

The page emphasizes the behavior of the function:

Highlight: This hyperbola is always decreasing, which is a characteristic of negative hyperbolas.

Finally, the asymptotes are explicitly stated:

1. Vertical asymptote: x = -3
2. Horizontal asymptote: y = 0

This concise overview provides students with a clear understanding of hyperbola characteristics, which is crucial for graphing and analyzing these functions.

Parabolas and Quadratic Functions: Understanding Transformations

This page delves into parabolas and quadratic functions, focusing on how various parameters affect their shape and position. It's an essential topic for students learning about function transformations and graphing techniques.

The page starts by introducing the parent function of a parabola: y = x². This serves as the basis for understanding all transformations.

Definition: The parent function of a parabola is y = x², which has its vertex at (0,0) and opens upward.

The concept of "flipping" a parabola is introduced, which occurs when the coefficient 'a' is negative.

Highlight: When 'a' is negative, the parabola opens downward, effectively flipping the graph vertically.

The page then explains the three main parameters that affect a parabola's shape and position:

1. 'a': The multiplier that affects the shape and direction of the parabola.
2. 'k': The vertical shift of the parabola.
3. 'h': The horizontal shift of the parabola.

Example: In the equation y = a$x-h$² + k, 'h' shifts the parabola horizontally, and 'k' shifts it vertically.

The effect of the 'a' parameter is explored in more detail:

Highlight: When |a| > 1, the parabola stretches vertically. When 0 < |a| < 1, the parabola compresses vertically.

An example of a compressed parabola is given: y = 0.5x² or y = $1/2$x².

The page concludes by emphasizing that parabolas are symmetrical, with answers mirroring each other on either side of the vertex.

This comprehensive overview provides students with a solid understanding of how to manipulate and graph parabolas, which is crucial for analyzing quadratic functions.

Domain and Range: Understanding Inequalities and Functions

This page introduces the fundamental concepts of domain and range, focusing on their representation using inequalities. It provides essential information for students learning about function analysis and graphing.

The page begins by explaining the basic notation for domain and range using inequalities. Domain inequality examples are provided, showing how to express the input values of a function. Similarly, range inequality examples demonstrate how to represent the output values.

Definition: Domain refers to the set of all possible input values $x$ for a function, while range encompasses all possible output values $y$.

Several examples are presented to illustrate different scenarios:

Example: For the domain -2 ≤ x < 4, x is greater than or equal to -2 and less than 4.

The page also covers special cases, such as discrete domains and ranges:

Example: A function with domain {-4, 1, 3} and range {1, 2, 3, 4}.

Importantly, the concept of "all real numbers" is introduced for both domain and range, represented by the infinity symbol (∞).

Highlight: When expressing domain and range as inequalities, use "less than or equal to" (≤) and "greater than or equal to" (≥) symbols to include endpoint values.

The page concludes with more complex examples, including radical and quadratic functions, demonstrating how to write domain and range as inequalities for these cases.

Example: For f$x$ = ³√x, the domain is x ≥ 0, and the range is y ≥ 0.

This comprehensive overview provides students with a solid foundation for understanding and expressing domains and ranges using inequality notation.