Vertex Formula

Definition

The vertex formula is a mathematical equation used to determine the coordinates of the vertex of a parabolic function. The vertex is the point on the parabola where the function changes from increasing to decreasing or vice versa, and it represents the minimum or maximum value of the function.

5 Must Know Facts For Your Next Test

1. The vertex formula is $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function $f(x) = ax^2 + bx + c$.
2. The $y$-coordinate of the vertex can be found by substituting the $x$-coordinate into the original quadratic function.
3. The vertex formula is used to find the coordinates of the minimum or maximum point of a parabolic function.
4. The vertex formula is essential for understanding the behavior of a parabolic function, such as its direction of opening, its range, and its critical points.
5. The vertex formula is a key concept in the study of parabolas and their applications in various fields, such as physics, engineering, and economics.

Review Questions

• Explain how the vertex formula is used to determine the coordinates of the vertex of a parabolic function.
  • The vertex formula is used to find the $x$-coordinate of the vertex of a parabolic function, which is given by the equation $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function $f(x) = ax^2 + bx + c$. Once the $x$-coordinate of the vertex is known, the $y$-coordinate can be found by substituting the $x$-coordinate into the original function. The vertex represents the minimum or maximum value of the parabolic function, depending on the sign of the coefficient $a$.
• Describe the relationship between the vertex formula and the axis of symmetry of a parabolic function.
  • The vertex formula is closely related to the axis of symmetry of a parabolic function. The axis of symmetry is the vertical line that passes through the vertex of the parabola, dividing the parabola into two equal halves. The $x$-coordinate of the vertex, as determined by the vertex formula, represents the $x$-coordinate of the axis of symmetry. This means that the vertex formula provides the location of the axis of symmetry, which is an important characteristic of a parabolic function and is used to analyze its behavior, such as its range and critical points.
• Analyze how the vertex formula can be used to predict the behavior and characteristics of a parabolic function.
  • The vertex formula is a powerful tool for understanding the behavior and characteristics of a parabolic function. By using the vertex formula to determine the coordinates of the vertex, you can gain valuable insights into the function's behavior. For example, the sign of the coefficient $a$ in the quadratic function $f(x) = ax^2 + bx + c$ determines whether the parabola opens upward (for $a > 0$) or downward (for $a < 0$), which affects the function's range and critical points. Additionally, the vertex formula provides the location of the axis of symmetry, which is crucial for analyzing the function's symmetry and identifying its minimum or maximum value. Understanding the vertex formula and its relationship to the parabolic function's characteristics is essential for solving problems and making predictions in various fields that involve the study of parabolas.

Related terms

Parabola:

A parabola is a U-shaped curve that is the graph of a quadratic function, which is a polynomial function of degree two.

Quadratic Function:

A quadratic function is a polynomial function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers, and $a \neq 0$.

Axis of Symmetry:

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola, dividing the parabola into two equal halves.