scoresvideos

📈college algebra review

key term - Opening Upward

Citation:

Definition

The term 'opening upward' refers to the orientation or shape of a parabolic curve, where the curve opens or points in an upward direction on a coordinate plane. This characteristic of a parabola is a crucial aspect of understanding the behavior and properties of this type of quadratic function.

5 Must Know Facts For Your Next Test

  1. Parabolas that open upward have a positive leading coefficient (a > 0) in the standard form equation $y = ax^2 + bx + c$.
  2. The vertex of an upward-opening parabola is the lowest point on the curve, representing the minimum value of the function.
  3. The axis of symmetry for an upward-opening parabola is a vertical line that passes through the vertex and divides the curve into two equal halves.
  4. Upward-opening parabolas have a U-shaped appearance, with the curve bending back towards the x-axis as it extends in both the positive and negative directions.
  5. The graph of an upward-opening parabola is often used to model various real-world phenomena, such as the path of a projectile, the shape of a suspension bridge, or the cost-revenue relationship in economics.

Review Questions

  • Explain the significance of the leading coefficient (a) in the standard form equation $y = ax^2 + bx + c$ for an upward-opening parabola.
    • For an upward-opening parabola, the leading coefficient (a) is positive (a > 0). This positive value determines the orientation of the parabola, with the curve opening upward on the coordinate plane. The magnitude of the leading coefficient also affects the steepness and curvature of the parabola, with a larger positive value resulting in a narrower, more sharply curved parabola.
  • Describe the relationship between the vertex and the axis of symmetry for an upward-opening parabola.
    • The vertex of an upward-opening parabola represents the lowest point on the curve, which corresponds to the minimum value of the quadratic function. The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two equal halves. This means that for any point on one side of the axis of symmetry, there is a corresponding point on the other side that is equidistant from the axis, resulting in a symmetrical curve.
  • Analyze the real-world applications and implications of upward-opening parabolas in various fields, such as physics, engineering, or economics.
    • Upward-opening parabolas are commonly used to model and analyze various real-world phenomena. In physics, the path of a projectile, such as a ball or a rocket, can be described by an upward-opening parabola, which helps in understanding and predicting the motion of the object. In engineering, the shape of suspension bridges and the cross-section of certain structural elements often resemble upward-opening parabolas, which are designed to efficiently support and distribute loads. In economics, the cost-revenue relationship for a business can sometimes be represented by an upward-opening parabola, which can inform decision-making processes and strategic planning.