Graph quadratic functions and identify the vertex, axis of symmetry, intercepts, and direction of opening from standard, factored, and vertex forms (Ohio F-IF.7a, F-IF.8a).

What this topic is asking

Ohio standard F-IF.7a asks you to graph quadratic functions and read their key features: the vertex, the axis of symmetry, the intercepts, and the direction of opening. A quadratic graphs as a parabola. The axis of symmetry $x = \frac{-b}{2a}$ and vertex form are not on the reference sheet, so they must be memorized. This is a high-value Functions and Quadratics skill across both parts.

Direction of opening and the vertex as max or min

The leading coefficient $a$ controls the shape before you plot anything.

So $y = 3x^2$ is a narrow upward parabola with a minimum, and $y = -\frac{1}{2}x^2$ is a wide downward parabola with a maximum.

The axis of symmetry and the vertex

The axis of symmetry is the vertical line through the vertex; the parabola is a mirror image across it.

Reading features from the three forms

Each algebraic form hands you a different feature for free.

• Standard form $ax^2 + bx + c$: the $y$-intercept is $c$, and the axis is $x = \frac{-b}{2a}$.
• Factored form $a(x - p)(x - q)$: the $x$-intercepts (zeros) are $x = p$ and $x = q$; the axis is halfway between them.
• Vertex form $a(x - h)^2 + k$: the vertex is $(h, k)$ directly.

How Ohio examines this topic

• Numeric and equation response. State the axis of symmetry, the vertex, or an intercept.
• Graphing. Plot the vertex and points to draw the parabola on the grid.
• Multiple choice and multiple-select. Match a quadratic to its graph, or pick the opening direction, max/min, or vertex.

Why the axis runs through the vertex

A parabola is symmetric: for every point on one side there is a mirror point at the same height on the other side. The single line that this mirror folds along is the axis of symmetry, and the one point that lies on the axis, with no partner, is the vertex, the turning point. That is why the vertex's $x$-coordinate equals the axis value $\frac{-b}{2a}$: the vertex is where the two symmetric halves meet. This symmetry is also a practical tool: once you know the axis and one point, you immediately know its mirror point at the same height, which lets you plot a parabola from just the vertex and a couple of points.

Why the sign of a decides max versus min

Whether the vertex is a maximum or a minimum follows directly from which way the parabola opens, and that is set by the sign of $a$. When $a > 0$, the $ax^2$ term grows large and positive as $x$ moves away from the vertex in either direction, so the arms rise and the vertex is the lowest point, a minimum. When $a < 0$, the $ax^2$ term grows large and negative away from the vertex, so the arms fall and the vertex is the highest point, a maximum. This is why so many application problems, maximum height, maximum area, maximum revenue, reduce to finding the vertex of a downward parabola: the sign of $a$ tells you a maximum exists, and $\frac{-b}{2a}$ locates it.

Try this

Q1. Find the axis of symmetry of $f(x) = x^2 + 8x + 3$. [1 point]

• Cue. $x = \frac{-8}{2(1)} = -4$.

Q2. Does $y = -x^2 + 2x + 5$ open up or down, and is the vertex a max or min? [1 point]

• Cue. $a = -1 < 0$, so opens down, vertex is a maximum.