Vertex, Axis of Symmetry, and Parabola Graphing
Find the vertex using x = −b/2a and evaluate f at that x. Identify the axis of symmetry, plot additional symmetric points, and use all features to graph a parabola accurately without a calculator.
A lesson of Algebra 1, in Quadratic Functions, about 30 minutes.
Vertex, Axis of Symmetry, and Graphing
By the end of this lesson, you will be able to find a vertex with $x = -\tfrac{b}{2a}$, name the axis of symmetry, use symmetry to get points free, and sketch an accurate parabola without a calculator.
One Formula for the Vertex
$$x = -\frac{b}{2a}$$
That gives the $x$-coordinate of the vertex. Put it back into the function to get the $y$-coordinate, and you have the turning point.
$$f(x) = x^2 - 6x + 5$$
$$x = -\frac{-6}{2 \cdot 1} = 3 \qquad f(3) = 9 - 18 + 5 = -4$$
The vertex is $(3, -4)$.
Two steps, and the second is easy to forget. The formula gives $x$ only; $-4$ came from substituting. Answering "the vertex is $3$" is answering half.
Where the formula comes from. The two zeros of a quadratic sit symmetrically either side of the vertex, so the vertex is at their average. The quadratic formula gives the zeros as $\frac{-b \pm \sqrt{\ }}{2a}$, and averaging those two makes the $\pm$ parts cancel, leaving exactly $-\frac{b}{2a}$. It works even when there are no real zeros, because the symmetry does not depend on them.
The Axis of Symmetry Is the Same Number
$$x = -\frac{b}{2a}$$
The axis of symmetry is the vertical line through the vertex, and its equation is that same $x$-value.
$$\text{vertex } (3, -4) \quad \Rightarrow \quad \text{axis of symmetry } x = 3$$
Write it as an equation, not a number. The axis is a line, so it is $x = 3$; $3$ on its own is a coordinate. A vertex is a point, an axis is a line, and the two answers look confusingly similar.
What the axis buys you is mirror points. If $(1, 0)$ is on the parabola and the axis is $x = 3$, then $(5, 0)$ is too, because $1$ and $5$ are both two units from $3$. Every point you compute gives you a second one free, and that is what makes an accurate sketch quick.
Example 1: Finding the Vertex
Here is the whole method on one function, worked all the way through. $$f(x) = 2x^2 - 8x + 5$$
- What is a? 2
- What is b? -8
- Work out -b over 2a. What is the x-coordinate of the vertex? 2
- Substitute x = 2 into the function. What is the y-coordinate? -3
- Write the equation of the axis of symmetry x = 2