Graphing y = −x² + bx + c
Simple Explanation
When a = −1, the graph of y = −x² + bx + c is a downward-opening parabola (upside-down U) with a maximum point at its vertex, instead of a minimum. Everything else — the vertex formula, the axis of symmetry — works exactly the same way.
Why Do We Need It?
Recognizing the sign of a instantly tells you whether you are looking for a highest point (maximum, like a ball's peak height) or a lowest point (minimum) — a critical distinction in real applications.
See It
A downward-opening parabola with vertex at (1, 4), crossing the x-axis at x = -1 and x = 3
Formula
Vertex Formula of a Quadratic Function
Vertex = (−b/2a, f(−b/2a)), for y = ax² + bx + c
Gives the exact coordinates of the turning point (vertex) of a parabola directly from its coefficients, without needing to complete the square.
- a, b, c
- — the coefficients of y = ax² + bx + c (a ≠ 0)
- −b/2a
- — the x-coordinate of the vertex, and the equation of the axis of symmetry
- f(−b/2a)
- — the y-coordinate of the vertex — the minimum or maximum value of the function
When to use it: Whenever you need the highest or lowest point of a parabola, or its axis of symmetry, directly from the equation y = ax² + bx + c.
Worked Example
Find the vertex of a downward parabola
Find the vertex of y = −x² + 6x − 5, and state whether it is a maximum or minimum.
Why Does This Work?
Multiplying x² by a negative number flips every nonnegative value it would have produced into a nonpositive one, so instead of curving upward from a low point, the graph curves downward from a high point — the vertex becomes the highest value the function reaches.
Real-Life Example
The path of a thrown ball
A ball thrown into the air follows a path that can be modeled by a downward-opening parabola, height versus time.
The vertex of that parabola gives the ball's maximum height and the exact time it is reached — a direct real-world use of the downward parabola's vertex as a maximum.
Practice
The graph of y = −x² + 4x has a vertex at (2, 4). Is this a maximum or minimum?
MediumCommon mistake
Assuming every parabola opens upward — always check the sign of a first: positive a opens upward (minimum vertex), negative a opens downward (maximum vertex).
Quick Review
- When a = −1 (or any negative a), the parabola opens downward, with a maximum vertex.
- The vertex formula, x = −b/2a, still applies — only the direction of opening changes.
- Sign of a: positive → minimum (opens up); negative → maximum (opens down).