Vertex as Minimum or Maximum Value
The vertex of a parabola is either the minimum or maximum value of a quadratic function — a core concept in California Reveal Math, Algebra 1 (Grade 9). For f(x) = ax²+bx+c: if a > 0 the parabola opens upward and the vertex is the minimum; if a < 0 it opens downward and the vertex is the maximum. To find the vertex from standard form, compute the axis of symmetry x = -b/(2a), then substitute back to find the y-coordinate. For example, f(x) = 2x²-4x+1 has axis x=1 and minimum value f(1) = -1. For f(x) = -x²+6x-5, axis x=3 and maximum value f(3) = 4. This skill is directly applied to real-world optimization problems like maximum height of a projectile.
Key Concepts
For a quadratic function in standard form $f(x) = ax^2 + bx + c$, the vertex represents either the minimum or maximum value of the function.
If $a 0$, the parabola opens upward and the vertex is the minimum point. The minimum value is $f\!\left( \dfrac{b}{2a}\right)$. If $a < 0$, the parabola opens downward and the vertex is the maximum point. The maximum value is $f\!\left( \dfrac{b}{2a}\right)$.
Common Questions
When is the vertex of a parabola a minimum vs maximum?
When a > 0 the parabola opens upward and the vertex is the minimum value. When a < 0 it opens downward and the vertex is the maximum value.
How do you find the minimum or maximum value of f(x) = ax²+bx+c?
Compute the axis of symmetry x = -b/(2a), then substitute that x-value into f(x). The result is the minimum (if a>0) or maximum (if a<0).
What is the minimum value of f(x) = 2x²-4x+1?
Axis of symmetry: x = -(-4)/(2·2) = 1. Minimum value: f(1) = 2(1)-4(1)+1 = -1.
What is the maximum value of f(x) = -x²+6x-5?
Axis of symmetry: x = -6/(2·(-1)) = 3. Maximum value: f(3) = -9+18-5 = 4.
How does the sign of a determine the shape of a parabola?
Positive a means the parabola opens upward (U-shape, minimum at vertex). Negative a means it opens downward (∩-shape, maximum at vertex).
What is the axis of symmetry formula for a quadratic in standard form?
x = -b/(2a). This vertical line passes through the vertex and divides the parabola into two symmetric halves.
How is the vertex's y-value used in real-world problems?
It gives the minimum or maximum output — for example, the maximum height of a ball thrown in the air or the minimum cost in a business problem.