Maximum or Minimum Values
Maximum or minimum values is a Grade 7 math skill from Yoshiwara Intermediate Algebra focused on finding the vertex of a quadratic function to determine where the function reaches its highest or lowest point. If a > 0 in ax^2 + bx + c, the vertex is a minimum; if a < 0, the vertex is a maximum.
Key Concepts
Property If the equation relating two variables is quadratic, then the maximum or minimum value is easy to find: It is the value at the vertex. If the parabola opens downward, there is a maximum value at the vertex. If the parabola opens upward, there is a minimum value at thevertex.
Examples A company's weekly profit is modeled by $P(x) = 2x^2 + 80x 300$, where $x$ is the price of a product. The maximum profit is at the vertex. The price that maximizes profit is $x = \frac{ 80}{2( 2)} = 20$ dollars.
The height of a diver above water is given by $h(t) = 5t^2 + 10t + 3$, where $t$ is time in seconds. The minimum height doesn't apply here, but the maximum is at $t = \frac{ 10}{2( 5)} = 1$ second.
Common Questions
How do you find the maximum or minimum value of a quadratic?
The maximum or minimum occurs at the vertex. Find the vertex x-coordinate with x = -b/(2a), then substitute to find the y-value.
How do you know if a quadratic has a maximum or minimum?
If the leading coefficient a > 0, the parabola opens upward and has a minimum. If a < 0, it opens downward and has a maximum.
What is the minimum value of y = x^2 - 4x + 3?
x = -(-4)/(2·1) = 2. y = 4 - 8 + 3 = -1. The minimum value is -1 at x = 2.
Where are maximum/minimum values used in real life?
In business (maximizing profit), physics (peak height of a projectile), and engineering (minimizing cost or material usage).