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Vertex and Interval Calculator

Reviewed by Calculator Editorial Team

A quadratic function is a second-degree polynomial that graphs as a parabola. The vertex of a parabola is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. The intervals of a parabola refer to the regions where the function is increasing or decreasing.

What is the Vertex of a Quadratic Function?

The vertex of a quadratic function is the point where the parabola reaches its maximum or minimum value. For a quadratic function in the form f(x) = ax² + bx + c, the vertex (h, k) can be found using the vertex formula:

h = -b/(2a)

k = f(h) = a(h²) + b(h) + c

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and the direction the parabola opens.

If a > 0, the parabola opens upwards, and the vertex is the minimum point. If a < 0, the parabola opens downwards, and the vertex is the maximum point.

How to Find the Vertex of a Quadratic Function

To find the vertex of a quadratic function, you can use the vertex formula or complete the square. Here's a step-by-step guide:

  1. Identify the coefficients a, b, and c in the quadratic function f(x) = ax² + bx + c.
  2. Calculate the x-coordinate of the vertex using h = -b/(2a).
  3. Substitute h back into the function to find the y-coordinate k = f(h).
  4. The vertex is at the point (h, k).

For example, consider the quadratic function f(x) = 2x² - 8x + 3.

a = 2, b = -8, c = 3

h = -(-8)/(2*2) = 8/4 = 2

k = f(2) = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5

Vertex: (2, -5)

You can also find the vertex by completing the square:

  1. Rewrite the quadratic function in the form f(x) = a(x² + (b/a)x) + c.
  2. Complete the square by adding and subtracting (b/(2a))² inside the parentheses.
  3. Factor the perfect square trinomial and write the function in vertex form.

For the same example:

f(x) = 2(x² - 4x) + 3

f(x) = 2(x² - 4x + 4 - 4) + 3 = 2((x - 2)² - 4) + 3 = 2(x - 2)² - 8 + 3 = 2(x - 2)² - 5

Vertex form: f(x) = 2(x - 2)² - 5

Vertex: (2, -5)

Intervals of a Parabola

The intervals of a parabola refer to the regions where the function is increasing or decreasing. The vertex divides the parabola into two intervals:

  • If the parabola opens upwards (a > 0), the function is decreasing on the interval (-∞, h) and increasing on the interval (h, ∞).
  • If the parabola opens downwards (a < 0), the function is increasing on the interval (-∞, h) and decreasing on the interval (h, ∞).

For the quadratic function f(x) = 2x² - 8x + 3 with vertex at (2, -5):

  • The function is decreasing on the interval (-∞, 2).
  • The function is increasing on the interval (2, ∞).

The intervals of a parabola can be determined by analyzing the sign of the derivative of the quadratic function. The derivative f'(x) = 2ax + b. The critical point is at x = h = -b/(2a).

Using the Vertex and Interval Calculator

Our vertex and interval calculator makes it easy to find the vertex and intervals of a quadratic function. Simply enter the coefficients a, b, and c of the quadratic function, and the calculator will display the vertex and intervals.

The calculator also provides a visual representation of the quadratic function and its vertex using Chart.js. This helps you understand the shape and position of the parabola.

To use the calculator:

  1. Enter the coefficients a, b, and c of the quadratic function.
  2. Click the "Calculate" button to find the vertex and intervals.
  3. View the results, including the vertex coordinates and intervals.
  4. Use the chart to visualize the quadratic function and its vertex.

The calculator also includes a "Reset" button to clear the inputs and results.

FAQ

What is the vertex of a quadratic function?

The vertex of a quadratic function is the point where the parabola reaches its maximum or minimum value. It is the highest or lowest point on the graph of the quadratic function.

How do I find the vertex of a quadratic function?

You can find the vertex of a quadratic function using the vertex formula h = -b/(2a) and k = f(h). Alternatively, you can complete the square to rewrite the quadratic function in vertex form.

What are the intervals of a parabola?

The intervals of a parabola refer to the regions where the function is increasing or decreasing. The vertex divides the parabola into two intervals: one where the function is decreasing and one where the function is increasing.

How do I use the vertex and interval calculator?

To use the vertex and interval calculator, enter the coefficients a, b, and c of the quadratic function, click the "Calculate" button, and view the results, including the vertex coordinates and intervals. The calculator also provides a visual representation of the quadratic function and its vertex.

What is the vertex form of a quadratic function?

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and the direction the parabola opens.