# How to Find the Maximum or Minimum Value of a Quadratic Function Easily

The y-coordinate of the parabola's vertex (usually represented by k) is also the maximum or minimum value of the quadratic function represented by the parabola. Let's learn how to find it!

### Method 1 If the quadratic is in the form y = ax2 + bx + c

1. 1
Decide whether you're going to find the maximum value or minimum value. It's either one or the other, you're not going to find both.
• The maximum or minimum value of a quadratic function occurs at its vertex.
For y = ax2 + bx + c,
(c - b2/4a) gives the y-value (or the value of the function) at its vertex. • If the value of a is positive, you're going to get the minimum value because as such the parabola opens upwards (the vertex is the lowest the graph can get) • If the value of a is negative, you're going to find the maximum value because as such the parabola opens downward (the vertex is the highest point the graph can get) • The value of a can't be zero, otherwise we wouldn't be dealing with a quadratic function, would we? ### Method 2 If the quadratic is in the form y = a(x-h)2 + k

1. 1
For y = a(x-h)2 + k,
k is the value of the function at its vertex.
• k gives us the maximum or minimum value of the quadratic accordingly as a is negative or positive respectively. ### Method 3 Using differentiation when the quadratic is in the form y = ax^2 + bx + c

1. 1
Differentiate y with respect to x. dy/dx = 2ax + b
2. 2
Determine the differentiation point values in terms of dy/dx. Since dy/dx is the gradient function of a curve, the gradient of a curve at any given point can be found. Thus, the maximum/minimum value can be found by setting these values equal to 0 and find the corresponding values. dy/dx = 0, 2ax+b = 0, x = -b/2a
3. 3
Substitute this value of x into y to get the minimum/maximum point.

### Method 4 Examples

1. 1
Find the maximum or minimum value of the function f(x) = x2 + x + 1.
2. 2
Find the maximum or minimum value of the function f(x) = -2(x-1)2 + 3.

## Tips

• The parabola's axis of symmetry is x = h.
• -h is the value that corresponds to your maximum or minimum value.

## Article Info

Categories: Algebra