The Quadratic Function
The quadratic function, also known as the second-degree polynomial, can be expressed in the canonical form as ax² + bx + c.
Canonical Form of the Quadratic Function
The canonical form of the quadratic function is: a(x - p)² + q, where [p₁q] is the turning point and A > 0.
The Position of the Function's Graph
The general formula for the position of the function's graph is: y = a(x - x₁)(x - x₂), where x₁ and x₂ are the roots, and A = b² - 4ac.
Properties
The quadratic function has various properties such as the domain (D: x € R), the range (ye (-0,5>), the axis of symmetry (x = p), the turning point (H(p, q)), and the values of the function (f(p)).
Examples of Tasks
- Given the function y = 3(x - 4)² + 5, determine the range.
- Calculate the discriminant for the function y = -x² + 2x - 6.
- Find the maximum interval of monotonicity for the function f(x) = -3(x + 5)² - 1.
Solutions
a) The range of the function y = 3(x - 4)² + 5 is: y < 5 and y > -∞.
b) The discriminant for the function y = -x² + 2x - 6 is 20.
c) The maximum interval of monotonicity for the function f(x) = -3(x + 5)² - 1 is: (-∞, -5> and <-5, +∞).
Additional Examples
- If the minimum value of the function is -10 for x = 4 and the graph intersects the x-axis at (4, -10) and (2,3), find the equation of the quadratic function.
- Determine the range of the function if the graph decreases in the interval (-∞, 6) and intersects the x-axis at (0,1).
- If the function takes positive values in the interval (-2, 8) and the maximum value is 10, find the equation of the quadratic function.
By understanding the canonical form, the position of the function's graph, and its properties, one can effectively solve problems related to the quadratic function. Regular practice and solving various types of tasks are essential to master the concept.
For more exercises and solutions related to the quadratic function, refer to additional resources such as PDF files with quadratic function exercises and solutions. Understanding the quadratic function in its canonical form and various properties such as the axis of symmetry and values of the function is crucial for further progress in mathematical studies and problem-solving.
