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

Four Methods:If the quadratic is in the form y = ax2 + bx + cIf the quadratic is in the form y = a(x-h)2 + kUsing differentiation when the quadratic is in the form y = ax^2 + bx + cExamples

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. Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 1
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    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.
      Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 1Bullet1
    • 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)
      Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 1Bullet2
    • 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)
      Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 1Bullet3
    • The value of a can't be zero, otherwise we wouldn't be dealing with a quadratic function, would we?
      Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 1Bullet4

Method 2
If the quadratic is in the form y = a(x-h)2 + k

  1. Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 2
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    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.
      Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 2Bullet1

Method 3
Using differentiation when the quadratic is in the form y = ax^2 + bx + c

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    Differentiate y with respect to x. dy/dx = 2ax + b
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    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
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    Substitute this value of x into y to get the minimum/maximum point.

Method 4
Examples

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    Find the maximum or minimum value of the function f(x) = x2 + x + 1.
  2. Image titled Find the Maximum or Minimum Value of a Quadratic Function Easily Step 7
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    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