Graph quadratic functions and show key features including the vertex, axis of symmetry, intercepts, maximum or minimum, and direction of opening (TN A1.F.IF.D.7a).

What this topic is asking

Standard A1.F.IF.D.7a asks you to graph a quadratic and show its key features: the vertex, the axis of symmetry, the intercepts, whether it has a maximum or minimum, and its direction of opening. A parabola is symmetric about a vertical line through its vertex, and reading these features (by hand in simple cases) is a core Functions-category skill that overlaps the Quadratic strand.

The features and how to find them

Symmetry makes graphing fast

Because a parabola is symmetric about its axis, every point has a mirror image the same distance on the other side. Once you have the vertex and one extra point (often the $y$-intercept), reflecting gives a third point for free, enough to sketch the curve. This is also why the two $x$-intercepts are equidistant from the axis, so the axis sits exactly halfway between them.

How TNReady examines this topic

• Numeric response. Find the axis, a vertex coordinate, or an intercept.
• Multiple choice. Identify the direction of opening, max or min, or match a graph to an equation.
• Graphing. Plot the vertex and intercepts of a parabola.

A clarifying idea is that the zeros from the solving topic are precisely the $x$-intercepts here, and the vertex from completing the square is the same vertex you locate with $x = \frac{-b}{2a}$. The solving and graphing strands describe one object from two angles.

Why the sign of a decides so much

The leading coefficient $a$ controls the parabola's shape and orientation before any other feature. A positive $a$ opens the curve upward into a valley, so the vertex is the lowest point (a minimum) and the function decreases then increases. A negative $a$ flips it into a hill, so the vertex is the highest point (a maximum) and the function increases then decreases. The magnitude of $a$ also sets how narrow or wide the parabola is, a larger $|a|$ makes it steeper. On a multiple-choice item that asks for direction and max-or-min, you do not need the vertex at all: the single sign of $a$ answers it, which is the fastest possible route to the point. This is why checking the sign of $a$ is always the first move when you meet a quadratic graph.

Try this

Q1. For $f(x) = x^2 + 2x - 8$, find the axis of symmetry and the vertex. [2 points]

• Cue. Axis $x = \frac{-2}{2} = -1$; $f(-1) = 1 - 2 - 8 = -9$, vertex $(-1, -9)$.

Q2. Does $f(x) = -3x^2 + 6x$ open up or down, and is the vertex a max or min? [1 point]

• Cue. $a = -3 < 0$: opens down, vertex is a maximum.