How To Find Increasing And Decreasing Intervals On A Quadratic Graph

Waren Burn

2 min read

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24-11-2024

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Quadratic functions, represented graphically as parabolas, exhibit distinct increasing and decreasing intervals. Understanding how to identify these intervals is crucial for comprehending the behavior of the function and solving related problems in calculus and algebra. This guide provides a clear, step-by-step method for finding these intervals.

Understanding the Parabola

A parabola, the graphical representation of a quadratic function (generally in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and a ≠ 0), is either concave up (opening upwards) or concave down (opening downwards). This concavity directly impacts the increasing and decreasing intervals.

• Concave Up (a > 0): The parabola opens upwards, meaning the function decreases to a minimum point (the vertex) and then increases.

• Concave Down (a < 0): The parabola opens downwards, meaning the function increases to a maximum point (the vertex) and then decreases.

Locating the Vertex

The vertex of the parabola is the turning point—where the function transitions from increasing to decreasing or vice versa. The x-coordinate of the vertex is found using the formula: x = -b / 2a.

Once you've found the x-coordinate, substitute it back into the quadratic equation to find the y-coordinate of the vertex.

Identifying Increasing and Decreasing Intervals

With the vertex located, identifying the increasing and decreasing intervals becomes straightforward:

• Concave Up (a > 0): The function is decreasing for all x-values less than the x-coordinate of the vertex and increasing for all x-values greater than the x-coordinate of the vertex.

• Concave Down (a < 0): The function is increasing for all x-values less than the x-coordinate of the vertex and decreasing for all x-values greater than the x-coordinate of the vertex.

Example

Let's consider the quadratic function f(x) = x² - 4x + 3.

1. Identify 'a' and 'b': Here, a = 1 and b = -4. Since a > 0, the parabola opens upwards (concave up).

2. Find the x-coordinate of the vertex: x = -b / 2a = -(-4) / 2(1) = 2

3. Find the y-coordinate of the vertex: Substitute x = 2 into the equation: f(2) = (2)² - 4(2) + 3 = -1. The vertex is (2, -1).

4. Determine the intervals: Because the parabola opens upwards, the function is decreasing for x < 2 and increasing for x > 2.

Therefore, the increasing interval is (2, ∞) and the decreasing interval is (-∞, 2).

Conclusion

Finding increasing and decreasing intervals for quadratic functions involves a systematic approach: determine the concavity based on the value of 'a', locate the vertex, and then define the intervals based on the vertex's x-coordinate and the parabola's orientation. This process is fundamental to understanding the behavior of quadratic functions and is a building block for more advanced calculus concepts.