How do you find the greatest value of a quadratic function?

The greatest value of a quadratic function, if it exists, is found at the vertex of the parabola. According to the provided reference (YouTube video "Finding Maximum Value of a Quadratic Function"), if a quadratic function has a maximum point, that point is simply the vertex, where the parabola changes direction.

Here's a breakdown:

Understanding Quadratic Functions and Their Graphs

A quadratic function is typically expressed in the form:

f(x) = ax² + bx + c

Where a, b, and c are constants and a is not equal to zero. The graph of a quadratic function is a parabola.

Finding the Maximum Value

The maximum value exists when the parabola opens downwards, meaning the coefficient a is negative. In this case, the vertex represents the highest point on the graph. To find the maximum value:

• Identify the Vertex: The x-coordinate of the vertex can be found using the formula:

  x = -b / 2a

• Calculate the Maximum Value: Substitute the x-coordinate of the vertex back into the original quadratic function to find the corresponding y-coordinate, which represents the maximum value of the function.

  f(-b / 2a) = a(-b / 2a)² + b(-b / 2a) + c

Example

Let's say you have the quadratic function:

f(x) = -x² + 4x - 1

1. Identify a, b, and c: a = -1, b = 4, c = -1

2. Find the x-coordinate of the vertex:

   x = -b / 2a = -4 / (2 * -1) = 2

3. Calculate the maximum value (y-coordinate of the vertex):

   f(2) = -(2)² + 4(2) - 1 = -4 + 8 - 1 = 3

Therefore, the maximum value of the quadratic function f(x) = -x² + 4x - 1 is 3.