Education

Quadratic Maximum and Minimum Calculator

Find the maximum or minimum value of a quadratic function and where it occurs. The calculator returns the vertex, axis of symmetry, range, and whether the parabola opens up or down.

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Result
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Vertex—
Axis of symmetry—
Range—
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How to know whether a quadratic has a maximum or minimum

A quadratic function has one turning point called the vertex. If a > 0, the parabola opens upward and the vertex is the minimum. If a < 0, the parabola opens downward and the vertex is the maximum. OpenStax states this relationship in its explanation of quadratic functions, vertex form, and maximum or minimum values.

This calculator is built for the way students actually see these problems: an equation in standard form, a question asking for the max/min value, and often a follow-up asking for the x-value where it happens. It returns both, because saying “maximum is 11” is incomplete if the question asks “when does it occur?”

Formula used

For f(x) = ax² + bx + c: x-coordinate of vertex: h = -b / (2a) maximum or minimum value: k = f(h) vertex: (h, k) If a > 0: minimum value = k If a < 0: maximum value = k

The value of k is the maximum or minimum output of the function, not the x-coordinate. This distinction is important in word problems. If x represents time and f(x) represents height, then h is the time when the object reaches its highest or lowest point, while k is the actual height.

Worked example

Find the maximum value of f(x) = -2x² + 8x + 3 a = -2, b = 8, c = 3 h = -b / (2a) h = -8 / (2 × -2) = 2 f(2) = -2(2²) + 8(2) + 3 f(2) = -8 + 16 + 3 = 11 Since a is negative, the parabola opens downward. Answer: maximum value is 11 at x = 2.

How this appears in word problems

Maximum and minimum questions often appear in projectile motion, profit, area, and optimization problems. The equation may describe height, revenue, cost, area, or distance. The calculator can find the turning point, but the meaning of the answer depends on the units in the problem. A vertex of (4, 64) could mean 64 feet after 4 seconds, $64 profit after selling 4 units, or an area of 64 square meters when a side length is 4 meters.

Common questions

  • First find the x-coordinate of the vertex using h = -b/(2a). Then substitute h back into the function to get k = f(h). If a is positive, k is the minimum value. If a is negative, k is the maximum value. The answer is usually written as both a value and a location: for example, the minimum value is -4 when x = 3.
  • Look at the sign of a in ax² + bx + c. If a is positive, the parabola opens upward, so the vertex is the lowest point and gives a minimum. If a is negative, the parabola opens downward, so the vertex is the highest point and gives a maximum. You do not need to graph the full parabola to decide this.
  • The maximum or minimum value of the function is the y-value of the vertex. The x-value tells where that maximum or minimum occurs. Many mistakes happen because students report only the x-coordinate. If the vertex is (2, 11), the maximum or minimum value is 11, and it occurs at x = 2.
  • A standard quadratic function has only one turning point. It has either a maximum or a minimum, not both. A parabola that opens upward has no maximum because it continues upward forever. A parabola that opens downward has no minimum because it continues downward forever.
  • Yes. If the equation is y = a(x - h)² + k, the vertex is already (h, k). If a is positive, k is the minimum value; if a is negative, k is the maximum value. Standard form needs the -b/(2a) formula, while vertex form makes the vertex visible immediately.
  • The vertex often represents the best, highest, lowest, or most efficient value in a real situation. In a projectile problem, it may be the maximum height. In a business problem, it may be maximum profit or minimum cost. The calculator gives the mathematical point, but you should always attach the correct unit and meaning from the original problem.