Vertex

Definition

The vertex of a function or graph is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing. It is the turning point of the graph and represents the maximum or minimum value of the function.

5 Must Know Facts For Your Next Test

1. The vertex of a parabola or quadratic function represents the maximum or minimum value of the function.
2. The x-coordinate of the vertex is given by the formula $x = -b/2a$, where $a$ and $b$ are the coefficients in the quadratic function $f(x) = ax^2 + bx + c$.
3. The y-coordinate of the vertex is found by substituting the x-coordinate of the vertex into the original quadratic function.
4. The vertex can be used to determine the direction of the parabola (opening upward or downward) and the range of the function.
5. Understanding the properties of the vertex is crucial for graphing quadratic functions and solving applications involving quadratic equations.

Review Questions

• Explain the role of the vertex in the graph of a quadratic function.
  • The vertex of a quadratic function is the point where the graph changes direction, either from decreasing to increasing or from increasing to decreasing. It represents the maximum or minimum value of the function and is the turning point of the graph. The x-coordinate of the vertex is given by the formula $x = -b/2a$, where $a$ and $b$ are the coefficients in the quadratic function $f(x) = ax^2 + bx + c$. The y-coordinate of the vertex is found by substituting the x-coordinate into the original function. Understanding the properties of the vertex is crucial for graphing quadratic functions and solving applications involving quadratic equations.
• Describe how the vertex is used to solve quadratic equations using the square root property and the quadratic formula.
  • When solving quadratic equations using the square root property or the quadratic formula, the vertex plays a key role. The square root property is used to solve equations of the form $x^2 = k$, where the vertex represents the point (0, k). The quadratic formula, $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$, relies on the coefficients $a$, $b$, and $c$ to determine the x-coordinates of the vertex, which are the solutions to the quadratic equation. The vertex is essential for understanding the nature of the solutions, whether they are real, imaginary, or repeated.
• Analyze how the vertex is used to graph quadratic functions and solve applications involving quadratic equations.
  • The vertex is a critical component in graphing quadratic functions and solving applications involving quadratic equations. By identifying the vertex, you can determine the maximum or minimum value of the function, the direction of the parabola (opening upward or downward), and the range of the function. This information is crucial for sketching the graph of a quadratic function and understanding its behavior. Additionally, in applications such as optimization problems, the vertex represents the point where the function attains its maximum or minimum value, which is essential for finding the optimal solution. Understanding the properties of the vertex and how to use it in the context of quadratic functions and equations is a fundamental skill in intermediate algebra.

Related terms

Parabola:

A parabola is a U-shaped curve that can be described by a quadratic function. The vertex of a parabola is the point where the curve changes direction, either from decreasing to increasing or from increasing to decreasing.

Quadratic Function:

A quadratic function is a polynomial function of degree two, written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers. The vertex of a quadratic function is the point where the graph of the function changes direction.

Axis of Symmetry:

The axis of symmetry of a parabola or quadratic function is the vertical line that passes through the vertex of the graph. It divides the graph into two symmetric halves.