key term - Vertex Form

Definition

The vertex form of a quadratic equation is a way to express the equation in a form that highlights the vertex of the parabolic graph. It is a useful representation that provides information about the maximum or minimum point of the parabola, as well as its orientation and symmetry.

5 Must Know Facts For Your Next Test

1. The vertex form of a quadratic equation is written as $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.
2. The value of $a$ determines the orientation of the parabola, with $a > 0$ indicating an upward-opening parabola and $a < 0$ indicating a downward-opening parabola.
3. The value of $h$ represents the $x$-coordinate of the vertex, while the value of $k$ represents the $y$-coordinate of the vertex.
4. The vertex form allows for easy identification of the maximum or minimum value of the function, as well as the point of symmetry.
5. Transforming a quadratic equation from standard form ($y = ax^2 + bx + c$) to vertex form involves completing the square.

Review Questions

• Explain how the vertex form of a quadratic equation is related to the graph of a parabola.
  • The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, directly relates to the graph of a parabola. The vertex $(h, k)$ represents the point where the parabola changes direction, either from decreasing to increasing or vice versa. This point corresponds to the maximum or minimum value of the function, and the parameter $a$ determines the orientation of the parabola (upward or downward). The vertex form provides a clear representation of the parabola's key features, such as its symmetry and the location of its turning point.
• Describe the process of transforming a quadratic equation from standard form to vertex form.
  • To transform a quadratic equation from standard form ($y = ax^2 + bx + c$) to vertex form ($y = a(x - h)^2 + k$), you need to complete the square. This involves rearranging the equation to isolate the $x^2$ term, then adding and subtracting a constant to create a perfect square trinomial. The constant added is $(\frac{b}{2a})^2$, and the $h$ and $k$ values are determined by this process. Completing the square allows you to identify the coordinates of the vertex $(h, k)$ and the coefficient $a$, which are essential for understanding the properties of the parabolic graph.
• Analyze how the vertex form of a quadratic equation can be used to determine the maximum or minimum value of the function.
  • The vertex form of a quadratic equation, $y = a(x - h)^2 + k$, provides a direct way to determine the maximum or minimum value of the function. The value of $k$ represents the $y$-coordinate of the vertex, which is the point where the parabola changes direction. If $a > 0$, the vertex represents the minimum value of the function, and if $a < 0$, the vertex represents the maximum value. By analyzing the values of $a$, $h$, and $k$, you can quickly identify the key characteristics of the parabolic graph, including the location of the maximum or minimum point, the orientation of the parabola, and the point of symmetry.

Related terms

Parabola:

A parabola is a U-shaped curve that is the graph of a quadratic function. It is defined by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Quadratic Equation:

A quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.

Vertex:

The vertex of a parabola is the point where the parabola changes from decreasing to increasing (or vice versa). It represents the maximum or minimum value of the function.