Quadratic Expression and Function

A cricket ball is hit straight up with a speed of 20 m/s. Its height at time t is h(t) = 20t - 5t^2 metres. When does the ball reach its highest point, and how high does it get? You could try different values of t by hand, but there is a faster way: the expression 20t - 5t^2 is a quadratic, and every quadratic traces out a curve called a parabola. The shape of that curve answers both questions at a glance — the highest point is the peak of the parabola.

This idea — treating a quadratic expression as a machine that takes in a number and returns a number — turns the expression into a function: f(x) = ax^2 + bx + c. And once you have a function, you have a graph. Where is the function at its lowest (or highest)? Where does it equal zero? Where is it positive, where negative? The parabola answers all of these questions visually.

The shape of a parabola

Every parabola y = ax^2 + bx + c has the same basic shape — a smooth U-curve (or an upside-down U). The sign of a decides the orientation.

• When a > 0, the parabola opens upward — it has a lowest point, and both arms go up to infinity. Think of it as a bowl.
• When a < 0, the parabola opens downward — it has a highest point, and both arms go down to negative infinity. Think of it as a hill.

The magnitude of a controls the width. A large |a| makes the parabola narrow (steep arms); a small |a| makes it wide (gentle arms). The value a = 1 gives the "standard" parabola y = x^2; a = 3 gives a narrower version, a = 0.5 gives a wider one.

When a is negative, the picture flips. The parabola opens downward, the bowl becomes a hill, and the extreme point is a maximum instead of a minimum.

The vertex and axis of symmetry

Every parabola is perfectly symmetric — the left half is a mirror image of the right half. The vertical line that divides the parabola into two mirror halves is the axis of symmetry. The point where the axis meets the parabola — the tip of the U (or the peak of the hill) — is the vertex.

The axis of symmetry always passes through x = -\frac{b}{2a}. This is the x-coordinate of the vertex. To find the y-coordinate, plug this value back into the function:

So the vertex is at \left(-\frac{b}{2a},\; c - \frac{b^2}{4a}\right).

Where does the formula x = -b/(2a) come from? If you remember the quadratic formula, the two roots are \frac{-b + \sqrt{D}}{2a} and \frac{-b - \sqrt{D}}{2a}, where D = b^2 - 4ac. Their average is \frac{-b}{2a} — the \pm\sqrt{D} parts cancel. The vertex sits exactly at the midpoint of the two roots. Even when the roots are complex (and the parabola doesn't cross the axis at all), the formula still gives the correct location of the vertex.

A connection you already know

If the roots of ax^2 + bx + c = 0 are \alpha and \beta, then by Vieta's formulas, \alpha + \beta = -b/a. The midpoint of the roots is (\alpha + \beta)/2 = -b/(2a). This is exactly the x-coordinate of the vertex. The vertex, the midpoint of the roots, and the axis of symmetry are all the same x-value — three names for one idea.

Vertex form

There is a way to rewrite f(x) = ax^2 + bx + c that makes the vertex visible at a glance. It is called the vertex form:

where (h, k) is the vertex.

The conversion from standard form to vertex form is completing the square — the same trick that derived the quadratic formula. Start with f(x) = ax^2 + bx + c. Factor out a from the first two terms:

Complete the square inside the bracket by adding and subtracting \left(\frac{b}{2a}\right)^2:

And there it is: a(x - h)^2 + k with h = -b/(2a) and k = c - b^2/(4a).

The vertex form tells you three things immediately:

1. The vertex is at (h, k) — you read it off without computing anything.
2. The extreme value is k — the smallest value of f (when a > 0) or the largest (when a < 0).
3. Why k is the extreme: (x - h)^2 \ge 0 for every real x. So a(x - h)^2 \ge 0 when a > 0, which means f(x) = a(x - h)^2 + k \ge k. The minimum value is k, achieved when x = h (so the squared term vanishes). The argument flips for a < 0.

Maximum and minimum values

This is the payoff. The question "what is the largest (or smallest) value this function can take?" has a one-line answer from vertex form.

• If a > 0: f(x) \ge k for all x, with equality at x = h. The minimum value is k.
• If a < 0: f(x) \le k for all x, with equality at x = h. The maximum value is k.

No calculus needed. The answer comes straight from the algebra of completing the square.

Physical meaning

If you throw a ball upward, its height at time t is a quadratic function of the form h(t) = -\frac{1}{2}gt^2 + v_0 t + h_0, where g is gravitational acceleration, v_0 is the initial velocity, and h_0 is the initial height. Since the coefficient of t^2 is negative (a = -g/2 < 0), the parabola opens downward. The vertex is the highest point of the trajectory — the maximum height. The t-coordinate of the vertex tells you when the ball reaches its peak; the h-coordinate tells you how high it goes. All from completing the square.

Interactive: exploring the vertex

Drag the red point along the parabola below. The readout shows the coordinates and the function value at each point. Notice how the function value decreases as you approach the vertex and increases as you move away from it — the vertex is the turning point.

Common confusions

• "The vertex is always on the x-axis." Only when the discriminant is zero. In general, the vertex can be above, on, or below the x-axis. Its y-coordinate is k = c - b^2/(4a), which can be any real number.

• "A parabola that opens upward has a maximum." Backwards. An upward-opening parabola (a > 0) has a minimum at the vertex. Its values go up to infinity from there — there is no largest value. A downward-opening parabola (a < 0) has a maximum.

• "To find the vertex, set f(x) = 0." No — setting f(x) = 0 finds the roots (where the parabola crosses the axis), not the vertex. The vertex is at x = -b/(2a), regardless of whether the function is zero there.

• "Vertex form and standard form are different functions." They are the same function, written differently. 2(x - 3)^2 - 4 and 2x^2 - 12x + 14 give exactly the same output for every input x. The vertex form is just more informative about the shape of the graph.

• "Completing the square only works when a = 1." It works for any a \neq 0. Factor a out first, complete the square inside the bracket, then distribute a back. The worked example above shows this for a = 2.

Going deeper

The parabola as a transformed y = x^2

Every quadratic function f(x) = a(x - h)^2 + k is a transformed version of the simplest parabola y = x^2. The transformation is built from three steps:

1. Horizontal shift by h units: x^2 \to (x - h)^2. This moves the vertex from (0, 0) to (h, 0).
2. Vertical stretch by factor |a| (and reflection if a < 0): (x - h)^2 \to a(x - h)^2. This changes the width and possibly flips the parabola.
3. Vertical shift by k units: a(x - h)^2 \to a(x - h)^2 + k. This moves the vertex from (h, 0) to (h, k).

Every parabola you will ever meet is just y = x^2 slid, stretched, and flipped. This is why all parabolas look the same up to scaling — they are all the same curve, living in different places.

The focus-directrix definition

In coordinate geometry, a parabola has a precise definition: it is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). For the standard parabola y = x^2/(4p), the focus is at (0, p) and the directrix is the line y = -p. Every point on the curve is exactly as far from the focus as it is from the directrix.

This definition is the reason parabolas appear in satellite dishes and headlight reflectors: parallel rays hitting a parabolic mirror all bounce through the focus, concentrating energy at a single point. The shape is not just algebraically clean — it is physically useful.

Discriminant and the vertex

The y-coordinate of the vertex is k = c - b^2/(4a) = -D/(4a), where D = b^2 - 4ac is the discriminant. This connects two ideas that might have seemed separate:

• If D > 0, then k and a have opposite signs (since k = -D/(4a)). When a > 0, k < 0: the vertex is below the axis, the parabola crosses the axis at two points. When a < 0, k > 0: the vertex is above the axis, still two crossings.
• If D = 0, then k = 0: the vertex sits exactly on the axis. One root, repeated.
• If D < 0, then k and a have the same sign. When a > 0, k > 0: the vertex is above the axis, the entire parabola floats above the axis. No real roots.

The discriminant and the vertex position are two views of the same fact.

Where this leads next

• Range of Quadratic Expression — the full set of values f(x) can take, including over restricted intervals. The vertex gives you the extreme value; the range article gives you the complete picture.
• Discriminant and Nature of Roots — a deeper look at b^2 - 4ac and how it controls the position of the parabola relative to the axis.
• Quadratic Inequalities — once you know the graph, you can read off where f(x) > 0 and where f(x) < 0 directly from the picture.
• Coordinate Geometry Basics — the parabola as a conic section, with focus, directrix, and the geometric definition.
• Quadratic Equations — Introduction — the equation f(x) = 0, which this article extends to the full function story.