1. What is the difference between the standard form of the equation of a horizontal hyperbola and a vertical hyperbola?

2. How do changes in **h** and **k** affect the position of the graph of hyperbola?
3.  How do changes in **a and b**  affect the size and shape of the graph of hyperbola? 

### 1. Difference Between Standard Forms of Horizontal and Vertical Hyperbolas

The standard forms of the equations of a hyperbola depend on the orientation of the hyperbola—whether it opens horizontally or vertically.

- **Horizontal Hyperbola:** 
  \[
  \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
  \]
  In this form, the hyperbola opens horizontally, with the foci located along the x-axis.

- **Vertical Hyperbola:** 
  \[
  \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
  \]
  In this form, the hyperbola opens vertically, with the foci located along the y-axis.

**Key Difference:** The placement of the terms determines the orientation. If \((x-h)^2\) comes first (before the minus sign), the hyperbola is horizontal. If \((y-k)^2\) comes first, the hyperbola is vertical.

### 2. Effect of Changes in **h** and **k** on the Position of the Hyperbola

The parameters \( h \) and \( k \) represent the coordinates of the center of the hyperbola \((h, k)\).

- **\( h \)**: Changing \( h \) moves the hyperbola left or right. Increasing \( h \) shifts the hyperbola to the right, while decreasing \( h \) shifts it to the left.
  
- **\( k \)**: Changing \( k \) moves the hyperbola up or down. Increasing \( k \) shifts the hyperbola upward, while decreasing \( k \) shifts it downward.

Thus, the center of the hyperbola moves according to changes in \( h \) and \( k \), altering the graph's position on the coordinate plane.

### 3. Effect of Changes in **a** and **b** on the Size and Shape of the Hyperbola

- **\( a \)**: The value of \( a \) determines the distance from the center to the vertices along the transverse axis (the axis the hyperbola opens along). Increasing \( a \) stretches the hyperbola, making it wider (for a horizontal hyperbola) or taller (for a vertical hyperbola). Decreasing \( a \) compresses the hyperbola, making it narrower or shorter.

- **\( b \)**: The value of \( b \) influences the distance from the center to the co-vertices along the conjugate axis (the axis perpendicular to the transverse axis). While \( b \) doesn’t directly affect the shape of the hyperbola itself, it determines the steepness of the asymptotes. A larger \( b \) makes the asymptotes less steep, while a smaller \( b \) makes them steeper.

**Summary:** 
- \( h \) and \( k \) change the position.
- \( a \) changes the size and the distance between vertices.
- \( b \) changes the steepness of the asymptotes.

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Would you like further details on any of these topics? Here are some related questions you might consider:

1. What are the asymptotes of a hyperbola, and how do they relate to \( a \) and \( b \)?
2. How do you find the foci of a hyperbola given \( a \), \( b \), and the center?
3. How can you derive the standard form of a hyperbola from its general equation?
4. What is the significance of the transverse axis in a hyperbola?
5. How does the equation of a hyperbola change if the center is at the origin \((0,0)\)?

**Tip:** The relationship \( c^2 = a^2 + b^2 \) is crucial for finding the foci of a hyperbola, where \( c \) is the distance from the center to each focus.

Learn about the standard forms of horizontal and vertical hyperbolas, how changes in h and k affect their positions, and how changes in a and b affect their size and shape. This comprehensive guide covers essential concepts in hyperbola geometry, suitable for high school students.

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