The Vertex Formula and Graphing y = ax² + bx + c
Simple Explanation
Every quadratic y = ax² + bx + c (any a, b, c, with a ≠ 0) can be graphed the same way: find the vertex using x = −b/2a, decide the direction from the sign of a, then plot a few extra points (like the intercepts) to complete the sketch.
Why Do We Need It?
This combines everything from sections 5.1-5.3 into one general method that graphs any quadratic function whatsoever — the complete, general-purpose technique you will use for the rest of the chapter.
See It
An upward-opening, narrow parabola with vertex at (1, 3), that never crosses the x-axis
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
Graph a general quadratic function
Sketch y = 2x² − 4x + 5, describing its key features.
Why Does This Work?
The vertex formula x = −b/2a comes from completing the square on ax² + bx + c: rewriting it as a(x + b/2a)² + (c − b²/4a) shows the expression is smallest (if a>0) or largest (if a<0) exactly when the squared term is zero, i.e. when x = −b/2a.
Real-Life Example
Modeling profit versus price
A company's profit, as a function of the price it charges, often follows a downward-opening quadratic — too low a price or too high a price both reduce profit.
Finding the vertex of that profit function tells the company exactly which price maximizes profit — a direct business use of the general quadratic graphing method.
Practice
What is the vertex of y = 3x² + 12x + 7?
HardCommon mistake
Skipping the sign check on a before sketching — without it, you cannot tell whether the vertex you calculated is the graph's highest point or its lowest point.
Quick Review
- Vertex: x = −b/2a, then substitute back in for y.
- Direction: a > 0 opens up (minimum); a < 0 opens down (maximum).
- Plot the vertex, y-intercept, and use symmetry to sketch the rest.