Algebra
Maximum and Minimum – The Highest or Lowest Value of a Parabola
The vertex's y-coordinate is the parabola's maximum or minimum value – the single highest or lowest value that y = ax² + bx + c can ever reach. Which one it is depends on the orientation you already know: if a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum.
This connects the vertex formula directly to real-world optimisation: whenever a problem asks you to find the “best,” “highest,” “lowest,” “most,” or “least” value of something that follows a quadratic pattern, the answer is exactly the vertex's y-coordinate.
Finding a Maximum or Minimum
If a > 0, the vertex gives the minimum value. If a < 0, the vertex gives the maximum value. Find it by substituting the vertex's x-coordinate into the equation.
The dashed green line marks the vertex's height in each case. On the left (a > 0), the curve never dips below that line — every other point on the curve is higher, so the vertex is the smallest y-value the function ever takes: a minimum. On the right (a < 0), the curve never rises above that line — every other point is lower, so the vertex is the largest y-value: a maximum. That's the entire idea in one picture: the vertex is always the "extreme" point the parabola never goes past.
The coefficient of x² is −2, which is negative, so the parabola opens downward: it has a maximum.
Vertex x = −(−4) ÷ (2 × 1) = 2. y = 2² − 4(2) + 9 = 4 − 8 + 9 = 5.
Real-Life Application
- Business: finding the price that maximises profit or minimises cost.
- Sports: finding the maximum height a ball reaches during its flight.
- Engineering: finding the minimum material needed for a given design constraint.
Key Takeaways
- The vertex's y-value is the parabola's maximum or minimum.
- a > 0 gives a minimum; a < 0 gives a maximum.
- This is the key idea behind real-world optimisation word problems.