Graph and analyze quadratic functions, identifying the vertex, axis of symmetry, intercepts, and direction of opening, and connecting standard, vertex, and factored forms (A.F.5).

What this topic is asking

A.F.5 asks you to graph and analyze quadratic functions: find the vertex, axis of symmetry, intercepts, and direction of opening, and connect the three forms of the equation. On the Virginia Algebra I SOL these are Functions items: identify the vertex or axis, decide which way a parabola opens, or read features from a form. They appear as multiple choice, fill-in-the-blank, and coordinate-plane items.

The parabola and its direction

A quadratic function $f(x) = ax^2 + bx + c$ graphs as a parabola, a symmetric U-shaped curve. The leading coefficient $a$ controls two things:

• Direction: $a > 0$ opens upward (a minimum at the vertex); $a < 0$ opens downward (a maximum at the vertex).
• Width: a larger $|a|$ makes a narrower parabola; a smaller $|a|$ makes a wider one.

The axis of symmetry and the vertex

The axis of symmetry is the vertical line that splits the parabola into mirror halves. It passes through the vertex, and its equation is

This formula is not on the Algebra I formula sheet, so memorize it. The vertex is the turning point; its $x$-coordinate is the axis value, and its $y$-coordinate is found by evaluating the function there.

The three forms

The same parabola can be written three ways, each displaying a different feature:

• Standard form $f(x) = ax^2 + bx + c$: the constant $c$ is the $y$-intercept, and $a$ shows the direction.
• Vertex form $f(x) = a(x - h)^2 + k$: the vertex is $(h, k)$ directly (watch the sign: $(x - 3)$ means $h = 3$).
• Factored form $f(x) = a(x - r_1)(x - r_2)$: the zeros are $r_1$ and $r_2$.

Completing the square converts standard to vertex form; factoring converts standard to factored form.

Reading intercepts

The $y$-intercept is $f(0) = c$ in standard form. The $x$-intercepts (zeros) come from setting $f(x) = 0$ and solving (factoring, square roots, or the quadratic formula). A parabola can have two $x$-intercepts, one (vertex on the axis, a double root), or none (it never reaches the $x$-axis), matching the discriminant.

Why every parabola is symmetric about the vertex

The symmetry of a parabola, and the axis formula, both come from the structure of a quadratic. Two inputs that are equally far from the axis produce the same output, because the squared term treats equal distances on either side identically: $a(x - h)^2$ gives the same value for $h + d$ and $h - d$. That mirror symmetry is why the parabola folds onto itself along the line $x = h$, the axis of symmetry, and why the vertex sits exactly at the fold, where the two halves meet. The formula $x = \frac{-b}{2a}$ is just the axis location written in terms of the standard-form coefficients (it is the $h$ you would get by completing the square). This symmetry is practically useful: once you know the vertex and one other point, you automatically know its mirror image, so you can sketch a parabola from very little, and it explains why a parabola's two zeros are always equidistant from the axis.

How the SOL examines this topic

• Multiple choice. Identify the direction of opening, the vertex, or the axis of symmetry.
• Fill-in-the-blank. Compute the axis of symmetry or the vertex coordinates.
• Coordinate-plane items. Plot the vertex, or match a parabola to its equation form.

Try this

Q1. Find the axis of symmetry of $f(x) = x^2 + 4x - 1$. [1 point]

• Cue. $x = \frac{-4}{2(1)} = -2$.

Q2. Does $f(x) = 3x^2 - 5$ open up or down? [1 point]

• Cue. $a = 3 > 0$, so up (a minimum).