key term - Horizontal Parabola

Definition

A horizontal parabola is a type of parabolic curve that opens either to the left or to the right, rather than upwards or downwards. It is defined by a quadratic equation in the form $x = a(y - h)^2 + k$, where $a$ is the horizontal stretch or compression factor, and $(h, k)$ is the vertex of the parabola.

5 Must Know Facts For Your Next Test

1. The graph of a horizontal parabola is symmetric about its axis of symmetry, which is a vertical line.
2. The vertex of a horizontal parabola represents the point of minimum or maximum value, depending on the sign of the coefficient $a$.
3. Horizontal parabolas can be used to model various real-world phenomena, such as the path of a projectile or the shape of a bridge.
4. The direction of the parabola's opening (left or right) is determined by the sign of the coefficient $a$: positive $a$ opens to the right, negative $a$ opens to the left.
5. Transformations, such as horizontal and vertical shifts, can be applied to a horizontal parabola to change its position and orientation on the coordinate plane.

Review Questions

• Explain how the equation $x = a(y - h)^2 + k$ defines a horizontal parabola and describe the role of the parameters $a$, $h$, and $k$.
  • The equation $x = a(y - h)^2 + k$ defines a horizontal parabola because it represents a quadratic function in the $x$-variable, rather than the $y$-variable. The parameter $a$ determines the horizontal stretch or compression of the parabola, with $a > 0$ resulting in a parabola that opens to the right and $a < 0$ resulting in a parabola that opens to the left. The parameters $h$ and $k$ represent the coordinates of the vertex, $(h, k)$, which is the point of minimum or maximum value for the parabola, depending on the sign of $a$.
• Describe the relationship between the axis of symmetry and the vertex of a horizontal parabola, and explain how this relationship can be used to sketch the graph of a horizontal parabola.
  • The axis of symmetry of a horizontal parabola is a vertical line that passes through the vertex of the parabola. This means that the vertex is the point where the parabola changes direction and is the point of minimum or maximum value. When sketching the graph of a horizontal parabola, the first step is to identify the vertex by using the equation $x = a(y - h)^2 + k$. Once the vertex is known, the axis of symmetry can be drawn as a vertical line passing through the vertex. The parabola will then open either to the left or right, depending on the sign of the coefficient $a$, and the shape of the parabola can be sketched accordingly.
• Explain how transformations, such as horizontal and vertical shifts, can be applied to a horizontal parabola and discuss the effect these transformations have on the graph.
  • Transformations can be applied to a horizontal parabola to change its position and orientation on the coordinate plane. A horizontal shift, represented by a change in the $h$ parameter, will move the parabola left or right without affecting its orientation or shape. A vertical shift, represented by a change in the $k$ parameter, will move the parabola up or down without affecting its orientation or shape. Additionally, changes to the $a$ parameter will affect the horizontal stretch or compression of the parabola, causing it to open more or less widely. By understanding how these transformations impact the graph of a horizontal parabola, you can manipulate the equation to model a variety of real-world situations and phenomena.

Related terms

Vertex:

The point where the parabola changes direction, representing the minimum or maximum value of the function.

Axis of Symmetry:

The vertical line that passes through the vertex of a horizontal parabola, dividing the parabola into two symmetric halves.

Concavity:

The direction in which a parabola opens, either upwards (positive concavity) or downwards (negative concavity).