Minimum Point

Definition

The minimum point of a quadratic function is the lowest point on the graph of the function, where the function changes from decreasing to increasing. It represents the point at which the function reaches its smallest value.

5 Must Know Facts For Your Next Test

1. The minimum point of a quadratic function is located at the vertex of the parabola.
2. The $x$-coordinate of the minimum point is given by the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function.
3. The $y$-coordinate of the minimum point is found by substituting the $x$-coordinate into the original quadratic function.
4. The minimum point represents the lowest point on the graph of the quadratic function, where the function changes from decreasing to increasing.
5. Identifying the minimum point is crucial for understanding the behavior and properties of a quadratic function, such as its range, domain, and critical points.

Review Questions

• Explain how the minimum point of a quadratic function is related to the vertex of the parabola.
  • The minimum point of a quadratic function is located at the vertex of the parabola. The vertex is the point where the function changes from decreasing to increasing, and it represents the lowest or highest point on the graph, depending on whether the function is concave up or concave down. The $x$-coordinate of the minimum point is given by the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function, and the $y$-coordinate is found by substituting the $x$-coordinate into the original function.
• Describe the significance of the minimum point in the context of graphing quadratic functions.
  • The minimum point of a quadratic function is crucial for understanding the behavior and properties of the function. It represents the lowest point on the graph, where the function changes from decreasing to increasing. Identifying the minimum point allows you to determine the function's range, domain, and critical points, which are essential for accurately graphing the function and analyzing its characteristics. Additionally, the minimum point can be used to determine the function's vertex, which is a key feature in understanding the symmetry and transformations of the parabola.
• Explain how the coefficients of a quadratic function relate to the location of the minimum point.
  • The coefficients of a quadratic function, $a$, $b$, and $c$, directly affect the location of the minimum point. The $x$-coordinate of the minimum point is given by the formula $x = -b/2a$, which means that the value of the coefficient $a$ determines the horizontal position of the minimum point, while the coefficient $b$ influences the direction of the parabola (concave up or concave down). The coefficient $c$ affects the vertical position of the minimum point, as it is used to calculate the $y$-coordinate by substituting the $x$-coordinate into the original function. Understanding the relationship between the coefficients and the minimum point is crucial for accurately graphing and analyzing quadratic functions.

Related terms

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

Vertex: The point on the graph of a quadratic function where the function changes from decreasing to increasing or vice versa. The vertex is the minimum or maximum point of the function.

Parabola: The graph of a quadratic function, which is a U-shaped curve.