Graph a quadratic function and identify and interpret its key features: vertex, axis of symmetry, x- and y-intercepts, direction of opening, and maximum or minimum value (MA.912.AR.3.7, MA.912.F.1.3).

What this topic is asking

MA.912.AR.3 asks you to graph a parabola and read its key features: the vertex, the axis of symmetry, the intercepts, the direction of opening, and the maximum or minimum value. Because the B.E.S.T. Algebra 1 EOC calculator is scientific (it does not graph), you compute these features from the equation by hand, which is exactly what the standard rewards.

Direction of opening

The sign of $a$ tells you everything about the shape's orientation:

• $a > 0$: opens upward, the vertex is the lowest point (a minimum).
• $a < 0$: opens downward, the vertex is the highest point (a maximum).

The size of $|a|$ controls width: larger $|a|$ is narrower, smaller is wider.

The axis of symmetry and the vertex

This formula is not on the reference sheet, so memorize it. The axis is the vertical line of symmetry through the vertex; every point on the parabola has a mirror image across it.

Intercepts

The $y$-intercept is $f(0) = c$, the constant term, found instantly. The $x$-intercepts (also called zeros or roots) are where $f(x) = 0$, found by factoring, square roots, or the quadratic formula. A parabola can have two, one, or no $x$-intercepts depending on whether it crosses, touches, or misses the $x$-axis.

How the B.E.S.T. EOC examines this topic

• Equation editor. Compute the vertex, axis of symmetry, or an intercept.
• GRID and hot-spot. Plot the vertex or click intercepts on a coordinate plane.
• Multiple choice. Identify the direction of opening, the max/min, or the axis.

A clarifying idea: a parabola is fully pinned down by its vertex and direction, plus one more point. Once you have $x = \frac{-b}{2a}$ and the sign of $a$, the rest of the graph follows by symmetry, which is why the vertex calculation is the centerpiece.

Why the axis of symmetry is at x = -b/2a

The formula is not arbitrary; it locates the parabola's mirror line. The $x$-intercepts of $f(x) = ax^2 + bx + c$, when they exist, come from the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, which is a pair of values spaced equally on either side of $\frac{-b}{2a}$ (the $\pm$ term adds and subtracts the same amount). The vertex, by symmetry, sits exactly halfway between them, at that center value $\frac{-b}{2a}$. Even when there are no real $x$-intercepts, the algebra still centers the parabola there, because the $\pm$ part is the only thing that shifts left or right of the center. So $\frac{-b}{2a}$ is the average of the two roots, which is why it is both the axis of symmetry and the vertex's $x$-coordinate.

Reading a max or min in context

When a quadratic models height, profit, or area, the vertex answers "what is the most (or least)?" and "when?". The $y$-coordinate of the vertex is the maximum or minimum value, and the $x$-coordinate is where it occurs. For a downward parabola modeling height, the vertex is the peak height and the time it is reached. Always report the value as the $y$ and the location as the $x$, and attach units, the EOC interpretation credit depends on it.

Try this

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

• Cue. $x = \frac{-8}{2(2)} = -2$.

Q2. Does $f(x) = -x^2 + 4$ open up or down, and is the vertex a max or min? [1 point]

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