Interval of Increase of A Parabola Calculator

What is the Interval of Increase of a Parabola?

The interval of increase of a parabola is the set of x-values where the function is increasing. For a parabola defined by y = ax² + bx + c, this interval depends on the coefficients a, b, and c.

For a parabola that opens upwards (a > 0), the interval of increase is all x-values to the left of the vertex. For a parabola that opens downwards (a < 0), the interval of increase is all x-values to the right of the vertex.

How to Calculate the Interval of Increase

To find the interval of increase for a parabola:

1. Identify the coefficients a, b, and c in the equation y = ax² + bx + c.
2. Calculate the x-coordinate of the vertex using the formula x = -b/(2a).
3. Determine the interval based on the sign of a:
   • If a > 0, the parabola increases for x < -b/(2a).
   • If a < 0, the parabola increases for x > -b/(2a).

The Formula

The interval of increase for the parabola y = ax² + bx + c is determined by the following:

This formula comes from the fact that the vertex of a parabola is the point where the function stops increasing and starts decreasing (for a > 0) or vice versa (for a < 0).

Worked Example

Let's find the interval of increase for the parabola y = 2x² - 4x + 1.

1. Identify the coefficients: a = 2, b = -4, c = 1.
2. Calculate the x-coordinate of the vertex: x = -(-4)/(2*2) = 4/4 = 1.
3. Since a = 2 > 0, the parabola increases for x < 1.

Therefore, the interval of increase is (-∞, 1).

Interpreting the Results

The interval of increase tells you where the parabola is rising. For a parabola that opens upwards (a > 0), this is all x-values to the left of the vertex. For a parabola that opens downwards (a < 0), this is all x-values to the right of the vertex.

Understanding this interval helps in analyzing the behavior of the parabola and can be useful in various applications such as optimization problems and graph analysis.