Graph quadratic functions and identify key features: the vertex, the axis of symmetry, the y-intercept, the x-intercepts (zeros), and the direction of opening (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard opens the quadratic work by asking you to graph a parabola and read its key features: the vertex, the axis of symmetry, the intercepts, and the direction of opening. Quadratics, with the algebra of solving them, make up a large share of the Functions and Algebra domains on the Georgia Milestones EOC and are often where the Proficient and Distinguished levels are decided. This topic is heavily graphing-tool driven: hot-spot items ask you to plot the vertex or click the zeros, and selected-response items ask for the direction of opening or the max/min.

The shape and direction

A quadratic function $f(x) = ax^2 + bx + c$ (with $a \neq 0$) graphs as a parabola, a symmetric U-shaped curve. The leading coefficient $a$ controls the direction and width:

• $a > 0$: opens upward, vertex is the lowest point (a minimum).
• $a < 0$: opens downward, vertex is the highest point (a maximum).
• Larger $|a|$ makes a narrower parabola; smaller $|a|$ makes it wider.

The axis of symmetry and the vertex

The axis of symmetry is the vertical line that splits the parabola into mirror halves, and the vertex lies on it.

The intercepts

• The y-intercept is $f(0) = c$, the constant term. So $f(x) = x^2 - 6x + 5$ crosses the y-axis at $(0, 5)$.
• The x-intercepts (the zeros or roots) are where $f(x) = 0$. Solve the quadratic equation to find them; they are symmetric about the axis of symmetry.

A parabola can have two, one, or no x-intercepts depending on whether and how it crosses the x-axis (the discriminant decides this, covered in the quadratic-formula page).

How the Milestones examines this topic

• Hot spot / graphing. Plot the vertex, or click the x-intercepts of a graphed parabola.
• Numeric entry. Find the vertex, the axis of symmetry, or the y-intercept.
• Multiple choice. Identify the direction of opening and whether the vertex is a maximum or minimum.

Why the vertex lies on the axis of symmetry

Understanding why $x = \frac{-b}{2a}$ locates both the axis and the vertex ties the picture together. A parabola is symmetric: for any height, there are two $x$-values equidistant from the center that share that output, like the two x-intercepts. The one place with a single point is the turning point, the vertex, and it must sit exactly on the line of symmetry halfway between any mirror pair. The formula $x = \frac{-b}{2a}$ is precisely that center line, which is why substituting it back gives the vertex's height. This also explains a fast graphing trick: once you have the vertex and the y-intercept, the mirror image of the y-intercept across the axis gives a third point for free, so three points and the symmetry sketch the whole parabola.

Reading a parabola from its graph

The reverse skill is also tested: given a graphed parabola, read its features. The vertex is the turning point; the axis of symmetry is the vertical line through it; the y-intercept is where it crosses the y-axis; the x-intercepts are where it crosses the x-axis. The direction of opening tells you the sign of $a$, and whether the vertex is the highest or lowest point tells you max versus min. Being able to extract these from a picture is exactly what a hot-spot item rewards, and it confirms that the algebraic features and the visual ones describe the same curve.

Try this

Q1. For $f(x) = x^2 + 4x - 1$, find the axis of symmetry and vertex. [2 points]

• Cue. $x = \frac{-4}{2} = -2$; $f(-2) = 4 - 8 - 1 = -5$, vertex $(-2, -5)$.

Q2. Does $f(x) = -3x^2 + x + 2$ have a maximum or minimum? [1 point]

• Cue. $a = -3 < 0$, opens down, so a maximum.