Read a parabola from the three forms of a quadratic, find the vertex, axis of symmetry, intercepts and direction of opening, and identify maximum or minimum values (Functions).

What this topic is asking

A quadratic function $f(x) = ax^{2} + bx + c$ graphs as a parabola. The ACT tests reading its key features, the vertex, axis of symmetry, intercepts and direction of opening, and identifying the maximum or minimum value. Each of the three algebraic forms exposes a different feature, so choosing the right form is the core skill.

The three forms

Each form is best for a different question.

Finding the vertex and axis of symmetry

The vertex is the turning point; the axis of symmetry passes through it.

Direction of opening and max or min

The sign of $a$ controls the shape. If $a > 0$, the parabola opens upward and the vertex is the lowest point (a minimum). If $a < 0$, it opens downward and the vertex is the highest point (a maximum). The value of the maximum or minimum is the $y$-coordinate of the vertex, $k$ in vertex form. This is exactly what "find the maximum height" or "minimum cost" word problems ask: compute the vertex and read its $y$-value.

Intercepts and the discriminant

The $y$-intercept is $f(0) = c$ (in standard form). The $x$-intercepts are the zeros, found by factoring or the quadratic formula, and the discriminant $b^{2} - 4ac$ counts them: two distinct intercepts if positive, one (the vertex on the axis) if zero, none if negative. So a parabola with a negative discriminant sits entirely above or below the $x$-axis.

Converting between forms

Because each form shows a different feature, the ACT may give one form and ask for a feature visible in another, so you convert. To go from factored to standard, expand: $2(x - 1)(x + 3) = 2(x^{2} + 2x - 3) = 2x^{2} + 4x - 6$. To go from standard to vertex, complete the square or use $h = -\frac{b}{2a}$ and $k = f(h)$. To go from vertex to standard, expand the squared binomial. You rarely need full conversions on the ACT; more often you read the feature you need directly from whichever form is given, choosing not to convert when you can avoid it.

Matching a graph to an equation

A frequent ACT task shows a parabola and asks which equation fits, or vice versa. Use the cheap-to-read features: the direction (up or down tells the sign of $a$), the $y$-intercept (where it crosses the vertical axis is $c$), and the $x$-intercepts (the zeros give the factored form). Checking just two of these usually eliminates every wrong choice. For instance, a downward parabola rules out any equation with positive $a$, and a $y$-intercept of $-4$ rules out any standard form whose constant is not $-4$.

Symmetry as a shortcut

Because a parabola is symmetric about its axis, two points with the same $y$-value are mirror images equidistant from the axis. If $f(1) = f(5)$, the axis is halfway between, at $x = 3$, which gives the vertex's $x$-coordinate without the formula. This symmetry also means once you have the vertex and one other point, you get a second point for free by reflecting across the axis, which is handy for sketching or for matching a graph to its equation.

Try this

Q1. Find the vertex of $f(x) = (x + 2)^{2} - 5$. [1 point]

• Cue. Vertex form gives $(h, k) = (-2, -5)$.

Q2. What is the $y$-intercept of $f(x) = 2x^{2} - 3x + 7$? [1 point]

• Cue. The $y$-intercept is $c = 7$ (the value at $x = 0$).