How To Find Increasing And Decreasing Intervals On A Graph Parabola

Savager Indie vor

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23-11-2024

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Meta Description: Learn how to identify increasing and decreasing intervals of a parabola. This guide covers the basics of parabolas, vertex identification, and applying the information to determine intervals where the function increases or decreases, complete with examples and visual aids.

Understanding Parabolas

A parabola is a U-shaped curve representing a quadratic function, typically expressed as f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' doesn't equal zero. The parabola's shape depends on the value of 'a':

• a > 0: The parabola opens upwards (a "cup"). It decreases to a minimum point (the vertex) and then increases.
• a < 0: The parabola opens downwards (an upside-down "cup"). It increases to a maximum point (the vertex) and then decreases.

The vertex is the parabola's highest or lowest point, representing either a maximum or minimum value of the function. Finding the vertex is crucial for determining increasing and decreasing intervals.

Finding the Vertex

There are several ways to find the x-coordinate of the vertex:

• Using the formula: For the quadratic function f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / 2a.

• Completing the square: This algebraic method transforms the quadratic equation into vertex form, f(x) = a(x - h)² + k, where (h, k) is the vertex.

• Graphically: Locate the lowest (for upward-opening parabolas) or highest (for downward-opening parabolas) point on the graph.

Example: Finding the Vertex

Let's find the vertex of the parabola represented by f(x) = 2x² - 8x + 6.

1. Identify a, b, and c: a = 2, b = -8, c = 6.
2. Use the formula: x = -b / 2a = -(-8) / (2 * 2) = 2.
3. Find the y-coordinate: Substitute x = 2 into the equation: f(2) = 2(2)² - 8(2) + 6 = -2.
4. The vertex is (2, -2).

Identifying Increasing and Decreasing Intervals

Once you've located the vertex, you can determine the intervals where the function increases and decreases:

Parabola Opens Upward (a > 0):

• Decreasing Interval: The function decreases from negative infinity to the x-coordinate of the vertex. In the example above, it's decreasing from (-∞, 2).
• Increasing Interval: The function increases from the x-coordinate of the vertex to positive infinity. In the example above, it's increasing from (2, ∞).

Parabola Opens Downward (a < 0):

• Increasing Interval: The function increases from negative infinity to the x-coordinate of the vertex.
• Decreasing Interval: The function decreases from the x-coordinate of the vertex to positive infinity.

Example: Identifying Intervals

For f(x) = 2x² - 8x + 6 (a parabola opening upwards with vertex (2, -2)):

• Decreasing Interval: (-∞, 2)
• Increasing Interval: (2, ∞)

For f(x) = -x² + 4x - 3 (a parabola opening downwards):

1. Find the vertex: a = -1, b = 4, c = -3. x = -b / 2a = -4 / (2 * -1) = 2. y = -(2)² + 4(2) -3 = 1. Vertex is (2,1)

2. Increasing Interval: (-∞, 2)

3. Decreasing Interval: (2, ∞)

Visual Representation

Always visualize! Graphing the parabola helps understand the increasing and decreasing intervals visually. Online graphing tools or graphing calculators can be extremely helpful.

Advanced Cases and Considerations

While this focuses on standard parabolas, remember that more complex functions might have multiple increasing/decreasing intervals. Calculus techniques (like the first derivative test) become necessary for analyzing such functions.

By combining algebraic methods with visual representation, determining the increasing and decreasing intervals of a parabola becomes straightforward. Remember to always consider the parabola's orientation (upward or downward) to correctly identify these intervals.