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Chapter 13: Problem 9

CONCEPTS A. What is the standard form of the equation of a parabola opening upward or downward? B. What is the standard form of the equation of a parabola opening to the right or left?

Short Answer

Expert verified
A. Equation: \( y = ax^2 + bx + c \) B. Equation: \( x = ay^2 + by + c \)

Step by step solution

01

Understanding the Problem

We need to find the standard forms of parabolas depending on their orientation: either opening upward/downward or right/left.
02

Parabola Opening Upward or Downward

For a parabola that opens upward or downward, the standard form of the equation is \[ y = ax^2 + bx + c \].Here, 'a' determines the direction and the width of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
03

Parabola Opening Right or Left

For a parabola that opens to the right or left, the standard form of the equation is \[ x = ay^2 + by + c \].In this equation, 'a' controls the direction and the width. If 'a' is positive, the parabola opens to the right; if 'a' is negative, it opens to the left.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Parabola
The equation of a parabola is a mathematical expression that represents the shape and position of the parabola on a coordinate plane. **A parabola** can be visualized as a U-shaped curve, and its equation helps to precisely define this curve. In general, there are two key forms for the equations of parabolas based on their orientation:
  • Vertical Orientation: This is when the parabola opens upwards or downwards. The standard form of the equation is \[ y = ax^2 + bx + c \]. Here, 'a', 'b', and 'c' are constants, with 'a' affecting the direction and width of the parabola.
  • Horizontal Orientation: This is when the parabola opens to the right or to the left. The standard form is \[ x = ay^2 + by + c \]. Similar to the vertical form, 'a', 'b', and 'c' are constants, influencing the parabola's direction and shape.
Understanding these equations allows you to predict and graph the parabola accurately on a coordinate plane.
Parabola Orientation
The orientation of a parabola describes the direction in which it "opens". Recognizing the orientation is crucial in graphing and understanding the relationship of a parabola's equation.
  • Upward Opening: When a parabola opens upwards, it resembles a "smile." This occurs when the coefficient 'a' in the equation \( y = ax^2 + bx + c \) is positive.
  • Downward Opening: Conversely, an "upside-down smile" or a downward-opening parabola results from a negative 'a'.
  • Right Opening: For parabolas that open to the right, the equation \( x = ay^2 + by + c \) has a positive 'a'. This orientation resembles a "greater-than" shape.
  • Left Opening: If the parabola opens to the left, 'a' will be negative in the horizontal equation, making a "less-than" shape.
These simple rules about 'a' help in quickly determining how the parabolas will behave when graphed.
Standard Form of a Parabola
The standard form of a parabola provides a structured way to express and understand the nature of parabolas easily. It provides insights into the direction, vertex, and axis of symmetry of the parabola.For an **upward or downward opening parabola**, the standard form is \( y = ax^2 + bx + c \). This form is particularly useful for recognizing:
  • The **vertex** of the parabola, which represents the highest or lowest point.
  • **Axis of symmetry**, which is a vertical line through the vertex, expressed as \( x = -\frac{b}{2a} \).
  • The **direction of opening**, dictated by the sign of 'a'.
For a **right or left opening parabola**, the standard form is \( x = ay^2 + by + c \). This configuration is helpful for identifying:
  • The **horizontal axis of symmetry** which is \( y = -\frac{b}{2a} \).
  • The **vertex** as a point of minimum or maximum x-coordinate.
  • The **directional opening**, either right or left based on 'a'.
Using these standard forms efficiently facilitates deeper analysis and a sharper understanding when solving problems involving parabolas.

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